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10.5446/14935 (DOI) | Statistical Mechanics Lecture 10 | https://av.tib.eu/media/14935 | https://tib.flowcenter.de/mfc/medialink/3/de078b7231ca7fd6881be978f7a7db5e7991e5726cd7fc266b22463a0434a89a11/StatisticalMechanicsLecture10__720p__flash9_1_1.mp4 | CC Attribution 3.0 Germany:
You are free to use, adapt and copy, distribute and transmit the work or content in adapted or unchanged form for any legal purpose as long as the work is attributed to the author in the manner specified by the author or licensor. | Physics | Lecture | 2013 | Susskind, Leonard | null | Professor Susskind continues the discussion of phase transitions beginning with a review of the Ising model and then introduces the physics of the liquid-gas phase transition. | Stanford University. You all know about the water at the steam. And it has been understood qualitatively for more than 100 years. When was Vandervals? Vandervals was the birth of India in the 1970s. Somebody will have quit Vandervals. My cue is usually the one to go through for that. Okay, it doesn't matter. It's been understood qualitatively from both mining statistical mechanics for a well over a century, I think. And we'd like to work with a little bit. I'm going to show you how the liquid gas phase transition works, not in the standard way that Vandervals approached it. So the way Vandervals approached it is very elegant, very nice, and is equivalent to what I'm going to show you. What I'm going to show you is based on a system that we've already studied, magnets. So the Ising magnet. What I'm going to show you is in what way the Ising magnet is a model for the liquid gas phase transition. But before we do it, let me just remind you what we learned last time by studying the mean field approximation. To the, not the liquid gas, but to the Ising model. We also know that it doesn't work very well in one dimension. In fact, it gives the wrong answer in one dimension. Mean field approximation always tells you that it's a phase transition, and this happens not to be true in one dimension, but it is true in two dimensions and higher. And by the time you get to three dimensions, every particle in the lattice has enough nearest neighbors, basically six of them, that the mean field approximation becomes very good. Not accurate in the sense of one percent, but probably accurate in the sense of ten percent, I would say. And so the mean field approximation is a good way to qualitatively understand how phase transitions take place. Now, it has to do with these little magnets, which can either be up or down. They would describe replacing at each point on a lattice a variable, what we call sigma. And sigma could easily be the last one in which case we thought of it as pointing up or minus one in which case we thought it was pointing down. And we talked about several, well, at least two different kinds of models to magnets. The Ising model was the more difficult of the models. It said there's an energy not for every particle, every site. Not for every site, but for every pair of neighboring sites. That the energy was stored in the relationship between neighboring sites. And in particular, energy was lower if the sites are parallel and higher if they're anti-parallel. You can think of the energy as being located in some sense in the links between the sites. If two neighbors are parallel, we'll call them an unbroken bond. If they're parallel, then the energy is low. If they're anti-parallel, we'll call that a broken bond. If the bond is broken, then the energy is higher. So we wrote an expression to represent that. And the representation was that the energy is the sum not over the sites of the lattice, but the bonds, the links, they're usually called links. Links of the product of sigma at one end of the link, let's call it sigma i and sigma j. Let's not try to make a fancy notation. i and j just mean the two opposite ends of a link. We put in here a minus sign, whatever you want to admit, and a constant j. A constant j is just associated with how tightly bound, how tightly, how hard it is to break apart. How much energy does it cost to break a bond? Well, if we start with two spins to sign us a line, then sigma dot sigma is plus one. If they're anti-aligned, it's minus one, so the difference in this expression here is two units. Not one unit, but two units to go from minus one to plus one. And so that means every broken bond costs you an energy plus j. Two j. Two j. Every broken bond costs an energy. That's how much energy it takes to flip it. So first thing we introduced, and the second thing was another possible term in the energy that would correspond to a uniform magnetic field, which is just a term which acts not on the neighbor, not on the links, but acts on the individual sites. It's an extra energy if a site is up or down. It doesn't depend on what its neighbors are doing. And so let's put that in there as something proportional to a magnetic field. That just means a number here. I think I probably called it capital H last time. I'm going to use standard notation, and then there's a notation that's in the statistical mechanics. Note that I'm going to call it lil H. You called it b last time. I called it b, you're right. You're right. I called it b. But I'm not going to now because in the notes it's called h. But h looks a lot like b. It just looks like a family drawn slide. So let's not belabor the difference. Yeah. So let's say if h is positive, then this term is lower if sigma is negative. So if this was all there was, which was incidentally the first model that we studied, only h, then this would favor spin. This would favor sigma being negative or being down. That would lower the energy. Lower the energy by this speed. Down configuration, in other words the minus one. But of course thermal fluctuations would cause it to fluctuate up sometimes, and a very, very high energy would be a very high temperature. It would be up as much as down. And so there's a pattern according to energy. So this term here tends to want to make things lie in the same direction, and this one wants to make them lie down. OK. Since you're summing over links, shouldn't there be a h-sigma j as well? No, that coefficient is called j. Yeah. This one? No, no, no. You're summing over links. Yes. So shouldn't there be an h-sigma j as well? OK, so I'm going to have a science. So h-sigma j. Sorry. I'm missing your point. If you're summing over the links, here's something on the links. Right, and there's something over it. Here's something on the sides. Quick question, please. It seems that the larger j when sigma is one, I mean the product, it's the lower energy. When they're parallel, it's the lower energy. Not only when they're parallel, but also when j is large. Yes, j will. Yes, but j is not a variable, the problem. j is just a number. I understand. But it's a number that varies from one substance to another. Yes, yes, it's a number that varies from one substance to another. So does that mean that it is harder to break a bond when the energy is lower? Is there physical meaning to that? Do you mean when j is lower? When j is bigger, it's a lower energy. And at the same time, when j is bigger, it's harder to break. Both are consequences of larger j. I don't think the zero energy matters. It's the energy difference between the broken and the unbroken bond. We are perfectly free to add any constant that we want. And in fact, the material may tell us what constant to add. For example, this energy stored in mass, you know, little v equals mc squared. A lump of uranium would have a bigger number here than a lump of frozen hydrogen. But the point is, this is just a numerical constant and an additive constant in the energy. That doesn't matter. And also, the ground state energy here, how much energy you have, let's calculate how much energy you have. Let's forget this for the moment. It has how much energy you have if all the spins are aligned. That's the lowest possible energy. If they're all aligned, then it's just a number of links times minus j. Minus j times the number of links. And, Sinect, how many links are there in a, let's take a three-dimensional lattice. If the number of sites is in, how many links are there? Everybody know? Sixth and seventh. Sixth and seventh. How many people say six? How many people say three? Three is the way. I'll show you why, but two, because I can't do it for you so well. Is that a sum of infinite? Yeah, yeah, a sum of infinite. And it's a word of infinite. So each site is connected to four links. So you might think the answer is four times the number of sites, but that's not the way it works. So the way to draw it is to draw, for each site, draw two links. Now, the next slide here also draws two links. For the site over here, draw two links. For the site over here, draw two links. Numbers will fill up all the links. So all you have to do is put two links at each site, and that fills up everything. Three dimensions, it's three links at each site, and that covers everything. The reason, of course, is that the link connects two sites. So it's only half. The half of what we might have thought. Okay, but in any case, you'll have in here an energy that's associated with the ground state, which means when everybody is aligned. Well, that's a constant. We could subtract it out. We could hand it over, subtract it in such a way that the ground state has zero energy, and group it together with this constant here. It wouldn't make any difference. Only energy differences are important with statistical mechanics. Or, in almost everything, energy differences. So the ground state energy is not so important. What is important is the energy difference between the up spin and the down spin, or a bond and a broken bond. Let's just remember what we found out from mean field theory. Excuse me, because I need to be a J by the sum over the sites. No, no J there. No J there. Alright, so we're running out of the recalculated partition function. E to the minus beta times the E. That means every time you see a J, it'll come together with a beta. And also every time you see an H, it'll come together with a beta. They'll always be there. Alright, we calculate the partition function. We go through the manipulations that we went through. And if you remember, we calculated in the mean field approximation, just to remind you, if anybody wasn't here, just what the mean field approximation was. So here's a site. It's surrounded by a bunch of neighbors. The number of neighbors across this pool. The neighbors have an average sigma. That average sigma might be zero, or I have a net average value when averaged over the neighboring sites. The more neighboring sites that you have, the better approximation it is to just say each spin here, each neighbor has a spin which is equal to the value equal to the average. So on a high enough dimension, which turns out to only mean two, you can make a reasonable approximation by saying each spin here sees around it a collection of spins, each one of which has the average spin. Alright, then we'll say, alright, good. Then what is the energy of the spin on the average? It is minus j times the average spin of the neighbors times the spin itself of that particular particle. Sigma bar means the average, minus j, sigma bar, sigma. And then what about this time over here? Well, this time doesn't care about the neighbors, and so it's just, oh, and this one here, what about this? The number of neighbors. That's where twice the dimension came from. Twice the dimension, whatever the dimension is, minus twice the dimension of space and dimension of the wireless, j, average of the spins, whatever happens to be times the spin itself, and then another term from here, which is just h times the spin itself. And we pretended that the average here was just a number and did the statistical mechanical problem of one spin with an energy given by this expression. And of course, this is just a number, times the spin. Okay, so that's a calculation that we did when we first looked at these magnets, and we found out that the average spin, the average spin that we get, I'm going to call it for the moment sigma double bar, it's the average of the spin that we get. If all of the neighbors happen to be frozen and had the value of sigma bar, then if you remember, that's equal to the hyperbolic tangent of the coefficient of being in the bracket, minus 2 dj sigma bar, plus h times beta times the inverse temperature, and that was it. Yes, that's it. Now, to complicate the expression, we started a little bit, but the important thing was we now make the mean field assumption that sigma bar, the average of this spin is just equal to the average of all of its neighbors. This, of course, would be true if we had a big enough sample and we would deepen the interior of it. It wouldn't matter which one of the spins we looked at, they all have the same average, so we now do the self-consistency trick of saying, that's our equation that determines sigma bar. 2 dj and h are all just numbers, so this is now an equation for sigma bar. Just remind ourselves what the solution looked like. What we did, we did a trick. We defined a new variable. We redefined this thing by calling it y. We wound up with an equation that looked like y divided by 2 dj, which is, let's say, y, or this bigger, equals tanh of y. That was our equation. Let's forget the rest of it. Let's just start right there and remind ourselves what we learned. Oh, and y plus h. Yes, y plus beta h, I think. Aren't you missing a minus sign on this? Yeah, y minus beta h, I think. Y minus beta h. What was y equal to 2 dj beta or minus 2 dj beta? Yes, plus, if you had a minus sign, the thing to worry about is tan minus 2 dj. That's what I'm saying. That's why the minus sign is here, I think. No. Oh, OK. There's a relative minus sign between that. OK, all right. OK, so this was our equation, our mean field equation, our consistency equation, which tells us what y is. Y is just the average spin divided by 2 dj. One more element, one more thing. Let's just remember that beta is 1 over the temperature, so let's put the temperature over here, the temperature over here. First exercise, what happens if h is equal to 0? Then it's just y times t over 2 dj is equal to tanh of y. We can get this, and now we can just, by inspection, get a pretty good idea of what things look like. h function looks like this. h of y, y on this axis, and on the vertical axis, the hyperbolic tangent. On the left side, you have a linear function of y, and the linear function of y passes through the origin, and it has a slope. It has a slope, which is the temperature divided by 2 dj. So in particular, when the temperature is very high, the slope is steep. And the only place where these curves intersect is right in the origin, and therefore, for large temperature, there is no magnetization. y is equal to 0. y is proportional to sigma bar, and therefore, sigma bar is equal to 0 when the temperature is high, as expected. Now we start lowering the temperature. The slope of this curve decreases, and at some point, it becomes exactly tangent for the hyperbolic tangent. Tangent, two different uses of the term tangent. Tangent being hugged closely. Hyperbolic tangent means, yes, what is the slope of the hyperbolic tangent function at the origin? Anybody know? It's exactly one. Easy to differentiate. You differentiate the hyperbolic tangent and you do it yourself, and you discover that the slope of the hyperbolic tangent at the origin is exactly equal to 0. That means when the slope of the left-hand side is adjusted to 1, that's the magic point where the two curves lie on top of each other. So past that, decrease the temperature even more. You decrease the slope of this line even more, and all of a sudden, you'll have a solution over here and a solution over here. Those solutions are magnetized. Those solutions have non-zero y, and therefore non-zero sigma bar over 2dj, I believe. That's a phase transition. At that point, the system doesn't quite know what to do. Should it go here? It doesn't go there. It doesn't go there. Should it go here? Or should it go here? In other words, there's an ambiguity. It seems like the spins want to have an average value. What is that average value up, or is it down? That's the question. The average value is up, sigma bar is positive, here it's down, and the problem is there's a complete symmetry in the problem. If you take all the spins and turn them over, whatever they are, turn everyone over, it's the same problem back again. So there's no way to decide whether it does this or it does that. To decide, we add a tiny, tiny, tiny little bit of h. And the effect of adding a tiny little bit of h is to shift this curve, even to the left or the right. But once you shift it, that point of the solution is longer there, and only one of these two will remain. In that case, that's what happens. That's magnetization. One of the wins, and the one that wins, of course, is if you put a little magnetic field which is trying to push the spins to be up, then upward-away. If you put this magnetic field such that they want to be down, then downward-away. So we can now draw a graph of what the system looks like with the function of two variables. The two variables are the temperature and the external magnetic field. Those are the two things we can vary in the problem. We can turn up the temperature or down the temperature. We do that by turning up the heat bath, by turning it down, putting a little fire under it. Or we can change the magnetic field by turning on the current through a coil and making the magnetic field larger or smaller. Those are the two things about disposal. What about J? J is also something we can change, but in order to change it, what we usually have to do is change the material. Now, in fact, you can also change it by changing the pressure and other things, but we're going to take J to be a fixed number. Fixed chemical composition, fixed pressure, fixed distance between the atoms. J is a well-defined number that you can look up in there. I can't put the physics, chemistry, or whatever. But beta and J are the things that you... sorry, beta and H are the things that are easily under your disposal. So you might want to ask, is an experimental fact, what does the magnetization look like as a function of temperature and magnetic field? So first of all, let's turn off H altogether. No H. No H and go to large temperature. Oh, oh, oh, let me come back a second. One second. Let's go back to this formula. The question is, at what temperature does the transition happen? The transition from having a solution like this to not having a solution like that. So the transition happens when the slope of the left side is equal to one. When the slope of the left side is equal to one, that's where you're hugging the two curves, huh? Okay, so the slope of the left side is one when the temperature is equal to 2DJ. That's called the critical temperature. It's not exact, this is a mean field approximation, it becomes exact when the D becomes large. And I'm sorry, I forgot to look up what the correct answer is for D equals 2 and 3. But for D equals 2, it's off by a significant amount. For D equals 3, it's fairly good. But it captures qualitatively what it does really happen. Okay, so at temperature 2DJ, that's called the critical temperature. And let's write it as C R I T, the critical temperature below which the system is spontaneously magnetized. Everybody wants to line up and the lining up tendency wins and everybody lines up. Might absolutely. They're not 100% lined up, they fluctuate but there's a net tendency for everybody to line up. Above that temperature, thermal fluctuations win and it doesn't magnetize. Okay, so now let's draw a picture here. Is that the Curie temperature? What's that? The Curie temperature? Curie temperature. Yeah, yeah, yeah, same thing. Curie temperature is a term that's used for magnets, that's strictly. If we're talking about the general phenomena of phase transitions, this would just be called critical temperature. So Curie temperature is a special case of a critical thing. A lambda point in liquid helium is another one and they're very similar to each other. In fact, they fall into a class of phase transitions which are almost identical in many ways. I'm not sure why they're also related or at least how this is related to another interesting transition. Okay, so some critical temperature over here, T critical. Now, what is the magnetization? Let's think about the value of the magnetization on this plane. First of all, if you're at higher temperatures than T critical and there's no magnetic field, then magnetization is just zero. There's no bias loop in the magnetic field but at higher temperatures, the statistics of ups and bounds being equal will win. And so over here, let's call it average of sigma bar is equal to zero along this line. Along the line, the average of sigma bar is equal to zero. Now you start lowering the temperature and you come to this critical point. When you get to the critical point, it just doesn't know what to do. Should it go up or should it go down? But the tiniest little bit of stray magnetic field H will tell you which way to go. If H is positive, the magnetization will be positive and right above this line, it won't be zero. And right below the line, it won't be zero. Spontaneous magnetization has happened and so all along here, sigma is positive, plus, plus, plus, plus, plus, plus, on the average, the average is positive. And down below is negative. Corresponds to the two possible roots where the red line either intersected on the right side or did the second one on the left side. And physically, it's just the extra little bit of magnetic field has broken the symmetry between ups and bounds and has biased it. And that little bit of biasing is enough that just the tiniest little bit of H goes plus. Tiniest little bit of H down goes minus. Now, you can ask what the jump is in the magnetization going across here. Remember, this is the average magnetization. It's not zero over one. It's not one or minus one. It can be zero or it can be two-thirds or it can be one-quarter. It can be any number between zero, sorry, between minus one and one. Well, at zero temperature, way down here, it doesn't matter what the magnetic field is. At zero temperature, all these guys line up and they line up in the down direction. So all along here, sigma bar is equal to minus one, and all along here is equal to plus one. That's just the fact that at zero temperature, even they just line up, choose the lowest energy. Now, what about right at the critical point? Imagine going up and down from negative temperature to positive temperature just at the critical point. Well, just at the critical point, that's this place where the red line is just barely at 45 degrees. Just barely critical value. So, at the critical point, the magnetic position is clearly zero. If you decrease the temperature just a tiny bit, the magnetization changes a tiny bit. If you decrease the temperature just a tiny bit, that means lower in the slope here, don't turn on the magnetic field, but there are two branches, one just to the right, one just to the left, and the magnetization is tiny if the temperature is tiny. So as we move vertically here, at the critical temperature, magnetization increases but continues with it. Just above here, it's very small, but positive, just below here, it's very small, but negative. So as you come into this critical point, the magnetization tends to zero from every direction, no matter how direction, including this one, the magnetization tends to zero at the critical point, but once you pass the critical point, there's a jump in the magnetization. So, for example, if you have your magnet turned on and you slowly vary the magnetic field, if you slowly vary the magnetic field above the critical temperature, then the magnetization gets smaller and smaller and smaller and smaller, goes to zero continuously, and then gets bigger and bigger and bigger, completely smoothly, just as you might expect. When the magnet is pointing down, it makes everything tilt down, when the magnet is pointing up, it tilts up up, but completely continuously as you turn off the magnetic field, it goes to zero, and as you turn it back on, it becomes positive. We can draw a curve for that, I don't know. Question? Yeah. I'm still not clear what's happening on the horizontal axis. It jumps, pardon? It jumps. From OTC, what's it doing? It jumps. It jumps when you go across it. But what's it doing, Kamek? Oh, Armit does more than this. Okay, so it can't set the points of the minus. Now, it does one thing or the other, but of course the answer is you could never really turn off all the magnetic field, something is going to bias it a little bit. Okay, but it's, in particular, it's not continuous there. It's not continuous. That's the key to what a phase transition is. So, or this is technically called a second-order phase transition. We don't need to distinguish them, because I'm not going to get a chance to talk about first-order phase transitions. Okay, so as the experimental physicist manipulates his magnetic field at fixed temperature, the magnetic field varies. At first, the magnetization is nice and smooth until it gets to this point, when it gets over to here, it's not zero. It's negative. And then, a little tiny change in the magnetic field flips it over. Okay, flips it over and the magnetization changes discontinuously across there. That discontinuous change is a, well, it's a symptom that in a sense there are two phases. Supposing you didn't know anything about this up here, let's suppose T-critical is too high a temperature for our experimental physicists to ever have studied. They would say there are two phases. There's a phase of negative magnetization, there's a phase of positive magnetization, and as you vary the parameters, it jumps. It's not a hold down to all up, it's an average down and an average up. Because it would be the least reasons that it was called the log. No. We're not going to have time to talk about this the least. All right, but there's another way you can move around in space. You can say, can I go from here to here without having a sharp change? And of course you can. You can go from here to here going around the critical point. If you go around the critical point, you experience no jump, but you do get from negative to positive. So on the one hand, experimentalists down here who never got up to T-critical would say there are two phases. And they're distinct, and not only are there two phases, but there's a sudden jump as you vary the magnetic field. On the other hand, somebody who had more experience with this system would say, oh, well, no, there's really not a sharp difference between here and here. Let me show you how to go from one to the other without the jump. And both would be right. Both are correct statements. There's a jump if you stay at fixed temperature. If you allow yourself to explore the whole region here, you'll discover there are ways to get from here to here which don't involve a jump. Smooth. Another question? Yeah. On the part below the critical where you're going from negative to positive and there's that jump. As you're getting closer and closer to the horizontal axis, say from below, are you getting closer and closer to zero? Is the average getting closer and closer to zero? No. No, I mean it's not zero on the thing, but is it, are the actual values getting less and less or not? Here? Yeah. No. The average is staying negative. I know it's negative, but is it very near zero? No, it's not near zero. Oh, yeah. No, no, no, no, no, over here it's less. Go back to this curve. Yeah. And now draw. Now, essentially it's zero magnetic field. Right. So I don't need to shift the curve. Yeah, okay. So I have these two solutions and the two solutions don't have why near zero. Well, as you get closer and closer to one to the critical, that's here. That's here. Oh, yeah, that's here. Actually, that's here. So as you approach from here, it goes to zero. Okay. Right. But over here, that's when the temperature is... Yeah, okay, got it. Good. Okay, excellent. Okay, so now we have everything we need to know about magnets. But I'm interested in the liquid gas phase transition. What does that have to do with this? There's a way of thinking about exactly the same system as being the transition between liquid and gas. It's not good for liquid to solid. That's the center. Liquid solid phase transition is very different. So let's talk about what this possibly can have to do with a bunch of particles. Okay, let's talk about box of particles. There are box of particles, about less than a fluid. And the fluid is in... I'll tell you what this drawing means in a minute. It's a potential energy block. The edges here are the edges of the walls of a box. Here we're inside the box. Here we're outside the box. As it turns out, the box really corresponds to some... some... might correspond to material or whatever happens to correspond to. But it's not comparable when it's inside the box is lower than the energy of the particle that's outside the box. Now, we could manufacture this just by having the inside of the box and the outside of the box being in slightly different elevations, right? We could have a flat box and a drop. But for whatever reason, we're going to imagine that a particle and less inside the box has less energy, negative energy relative to what it has outside the box. That difference in energy has a name. It's called the chemical potential. It's usually written in U. It's the amount of energy difference. It's actually the amount of energy that it takes to remove the particle from the box. How much energy does it take to remove the particle from the box? Now, we're not going to ask what is it. In this particular case, it's not important what makes the difference. Why does it differ from energy? We'll just take it that the energy is lower inside the box. But first of all, there are particles outside the box and inside the box. Lots of them all over the place. And we can ask what the density is inside the box and outside the box. Oh, one more thing. The walls of the box are permeable. Particles can come in and go out. They can cross the boundaries of the box. So this could just be... You know, this could be a bunch of molecules. You dig a hole in the ground. You dig a hole in the ground. And the ground... And now you just deposit some particles. If the particles are in the hole, they have less potential energy than if they're out of the hole. Where do you think the density of particles is likely to be larger? In the hole or out of the hole? We're thinking thermally equilibrium now. We're thinking thermally equilibrium. In the hole. The energy is less in the hole. And so it's natural that the Boltzmann distribution always favors lower energy. Always favors lower energy. Remember, the Boltzmann distribution is e to the minus beta times the energy. So it's always a function which decreases its energy. It always prefers lower energy. All right, so the answer will be that the system will adjust itself so that the density in here is in a certain kind of equilibrium. And the equilibrium will depend on the energy cost of taking the particle outside the box. Now, we can set this up in a very definite way. We can say we can do ordinary statistical mechanics. And we can just say there's a potential energy which depends on position. But it depends on position in a very special way. It's flat completely over the box of interest. And maybe jumps when you're outside the box. But we can focus on inside the box. And we can ask what is the density of the fluid as a function of the depth of this potential? Question? Yeah. Is the depth as new? The chemical potential. It's called the chemical potential because in general it can be different for different chemical molecules. So you could imagine a situation where one kind of molecule prefers to be in the box and the other kind of molecule doesn't prefer to be in the box. And then turning on chemical potentials can change chemical compositions in various ways. But we're not interested in the chemistry here. We're interested just in this idea that you can tune the density by changing the chemical potential. By changing the chemical potential you can change the density. All the chemical potential is is a term in the energy which depends on the number of particles in the box. New for each particle. If there are n particles in the box then n mu. If there are n minus 1 particles in the box then n minus 1 mu. Now in this problem the number of particles is a variable. It's a variable because particles can come into the box and out of the box. That means a configuration involves not only the positions and momentum of every particle but also the number of particles. Now the number of particles is free to change because they can come into the system and out of the system. And part of determining thermal equilibrium is determining how many particles on the average are inside the box. So I'm not going to go through this in detail. I'm going to tell you what to do. You write the usual formula for the Cartesian function. e to the minus theta times the energy. You sum over all the configurations that means the momenta and the position of all of the particles. But you add another term here. Plus the number of particles times the chemical potential times theta. Why is that there? Well this is just the potential energy of the particles inside the box. If there are n of them this is the energy stored inside just in the particles of beam there. And you also sum over the number of particles. We're not going to do that. We don't have time. But I'm just indicating to you what means the chemical potential. It's the thing that you adjust if you want to adjust the density of the fluid. You can do another thing. You can seal the box and not let the particles go in and out. And then just put in the number of particles if you want. But you may be interested in the problem of particles where they can come into the box and out of the box. And then you change the density by changing the chemical potential. By changing the chemical potential you can change the average number of particles in the box. What in this language what is a liquid gas transition? And I will tell you what happens in a liquid gas transition is you take your box, you keep it at a fixed temperature and you start varying the chemical potential. The more you vary the chemical potential, the more it wants to pull by, depending on the size of course, you lower this energy here, the more it wants to suck particles into the box. And so there's a density which is a natural function of the chemical potential. And what is it that happens in a transition? What happens in a transition is as you vary the chemical potential all of a sudden, you hit a point where the density of the fluid suddenly changes. It changes from being a gas to being a liquid. So if you were to fill up this, a box of course here might mean a region of three-dimensional space. I imagine it was a hole in the ground that was in two-dimensional space. But imagine a box, a real box of three-dimensions, particles going in, you get some extra energy out of it because there's a chemical potential. Particles go out and cause energy. The extra energy that you save by bringing particles into the box will tend to increase the density inside the box. You keep lowering the energy inside the box, it keeps going to suck particles in, and then all of a sudden, even though the atmosphere around the box might be at a temperature, at whatever else, a temperature and density that corresponds to vapor, all of a sudden you hit a transition point where it becomes liquid. The difference between liquid and vapor is a sudden discontinuity in the density. Liquid being denser than vapor. So if you could do this experiment for the box and vary the potential energy of a particle inside the box, lower the potential energy, outside conditions are such that you have steam, or vapor, or vapor, and you lower the energy, particles will congregate inside the box as you lower their energy, and then all of a sudden at some point it will jump to the fluid things, to the liquid things. That transition is very much like this transition. I'm going to spell it out and spell out exactly what the relationship is. Well, first of all, what are the conditions that you need to have a liquid gas phase transition? You need two things. You need to have a hard core retulsion between molecules. Well, molecules always have hard core retulsions. You cannot stick two molecules onto the same site and repellent for a number of reasons. So it's natural to have a heart that molecules behave like little billiard walls. Little billiard walls, they don't want to interpenetrate, so roughly speaking you can't put two of them on the same site. The next thing that's important is that they be a little bit of... and that's a form of really hard core repulsion. Just like two billiard walls, really you'd tell when you try to push them together. You need something else though. You need a little bit of attraction when they're not quite touching. You need a little bit of attraction when they're not quite touching. So you need a potential energy which is big and repulsive and positive when they're trying to get into each other. And then when you separate them out a little bit, all of a sudden they want to attract a little bit. That's an extremely common feature of molecular interactions. Hard core repulsions and very short range attractions. And that's what you need in order to have the standard liquid gas transition. So I'm going to show you how the magnet produces exactly that situation. Let's start with the magnet with all of its little elementary magnets down. When they're all down, that's the ground state, and let's call that empty space. With all the magnets down, in fact what we're going to say is something like this. We're going to say on each side of the lattice, you can have a particle or not a particle. This is our game. We're going to make all of the liquid gas problem, we're going to make a lattice version of it. The particles and molecules live on a lattice. You can have no particles at a site. Let's call that down. Let's call that sigma equals down. Sigma equals minus one means no particle on that site. What about sigma equals plus one? We're going to take that model, guess particle on site. What about two particles on site? Well, the magnet doesn't allow that. Sigma can't be two. It can only be minus one or one. It can't be four. It can't be anything else. So immediately from the start, it forbids the possibility of two particles on site. But then again, that's exactly what we're trying to do. We're trying to model a gas where you can't put two particles on the same site because they have a hard core. So it's built in to this system of particles. If they're really described by the same kind of variable, sigma equals plus one and minus one, it's automatically built in that they have an infinite hard core repulsive barrier when they try to get out on the same site. So we satisfied condition number one. What about condition number two? Do they attract? Now here we're going to use for the energy exactly the icing model energy plus a little magnetic field. I'm going to show you what that does in the language of these fake particles. Well, I'm not sure which are fake. Particles fake or magnetized fake. The point is it's the same mathematical system. The particles are magnetized. Let's start with this term here. Minus J times sigma i times sigma j, where these are neighboring sites on the last sum over all the links. How much energy is there if there are no particles, whatever? Well, that's the situation where all signals are down. Then there is an energy minus j for each link. So there's total grand state energy, which is minus j times the number of links. What do we do with that? Well, they don't do anything with it. That's just the energy of the system when there's no particles at all. Does it matter for anything? No, it doesn't. It's always there. If the grand state energy, we can throw it away. And we can throw it away just by subtracting minus the number of links. I think this times 2j. Now that I've thrown it away, let's really throw it away and ignore it. Because it plays no role in anything. It's just a number. And energy differences are the important things. Alright, so we have this term in the energy. No particles, that gives us certain energy. Let's call it zero. Let's make the zero of energy when there are no particles. Now what happens if I put in one particle? That means flip one of the spins over. It means all the spins are down. If we throw a little down, we take one and flip it up. That's mathematically equivalent to putting one particle somewhere. That's the energy that we get. Let's calculate. The energy that we get, how many bonds have we broken? Depends on the dimensionality. Let's do two dimensions because it's easier to visualize. If we flip this spin over, then we break four bonds. So we get a total amount of energy equal to 2j for each bond. 8j, 8j. Yes, 8j. So one particle has energy 8j. That's how much energy we get at a cost to create a particle at that point. What about two particles? If one particle is over here, then we get a total amount of energy. That's the total amount of energy that we get at a point. That's the total amount of energy that we get at a point. What about two particles if one particle is over here and one particle is over here? 16j. 16j. Two particles, 16j. But incidentally, the same is true if the two particles are over here. Let's just check that. Let's see how many bonds we have to break if they're on the corners of the diagonal. One, two, three, four, five, six, seven, eight. Yes, so even if they're on diagonals, you still pay a price of 16j. What if they're even closer? What if they're even closer like that? Then how many bonds do you break? One less. Two less. This bond is not broken. Why not? Because they're both close. That bond is not broken, so we broken six bonds. So two particles close together. Two close particles, and close now means within one bond length away. Where they have, they have 12j. So let me suppose I plotted the energy as a function of difference. I would find, yes, just the mere act of putting particles in cost me 16j. But that's all right. That energy is always there no matter where I move the particles around. But if I move them within a bond length, the energy decreases. The energy decreases, that's like a potential energy which decreases when they get very close together. So there's a potential energy, an energy which depends on the position of the particles. And when the particles get within one bond length of each other, the energy decreases by 2j. 2j and 4j. 4j, or the energy decreases by 4j. So let's look at having a short range attractive energy where when the particles are close together, the energy is negative relative to what it would be in their far apart. Negative because, negative relative to what they would be when they're far apart. So the particles attract. They like to be close together. How close together? One bond length away. But it's not an overwhelming attraction. It doesn't say the energy is infinitely negative if you put them next to each other. This is the modest amount of attraction, a modest saving of energy of putting them close together. And it's very much like the molecular attraction of a pair of molecules when they get close together. So we have a system now which is mathematically isomorphic to a system of particles on a lattice which have an attractive force between them where you can have any number of particles. The particle number is something that can change. So it's like the system of molecules where molecules can come into and out of the system. It doesn't have a definite number of molecules. The number of molecules itself is a variable. Short range potential, short range attraction, excuse me. Very short range, infinite repulsion. You cannot put two of them on the same site. But when they get a little bit apart, they're slightly attractive. Exactly what you need for a liquid gas phase transition. Now let's add something else. Does this system have a chemical potential, incidentally? A chemical potential is the energy stored in just having one particle. That's what it is. A chemical potential is just the energy in having a particle present. Just by virtue of having a particle, if there's an energy which wouldn't be there if the particle weren't there, that is called the chemical potential. Well, yes, just having a particle with no other particles around you gives you an energy, a genuine. So yes, there is a chemical potential. But I want to be able to vary that chemical potential. I would like to be able to vary that without varying J, incidentally. I'd like to hold J fixed. I want to be able to vary the chemical potential separately from J. That's easy to do. Put a magnetic field here. Now this is sum over the science. Sum over the science of sigma i. How much does this give me when I add a particle to the brew? Sigma starts down when there's no particle. When I put a particle in a sigma, it becomes plus. So it jumps two units. It gives me two units worth of energy for every spin which is flipped out. For every particle that you put in, it gives you twice h. So you can add for each isolated particle, you can add plus two h. So now we have everything we need. We have a system that is equivalent to a collection of particles on lattice. It has a variable chemical potential which notice how we vary it. We vary it by varying the magnetic field. Magnetic field gave us an energy per particle. And we have a short range attraction. The short range attraction, not the hard core billiard ball potential, but then a short range attraction, the coefficient of which is 2J. 4J. So I want to keep the molecular properties fixed. So I'm not going to play with J. That has to do with the molecular properties that design the cellular potential energy between them. What I can vary is the chemical potential. And mathematically that seems to be the same thing as varying h. Very h. Very h. So this problem is exactly the same as the erasured, the erasured. No, there it is. It is exactly this problem here. Precisely. There's a temperature that we can vary. Particles are at some temperature. There is a chemical potential we can vary. Now h is not exactly the chemical potential. There's an offset by an amount of 8J. What about the particle density? Let's talk about the particle density. We haven't discussed the particle density. Okay. How many particles are there on the average at a point? At a particular point. Well, I say the answer is somewhere, first of all, between zero and one. I say the number of particles at a point. Of course, it seems zero or one. But we're talking about the average. The average number of particles. The average number of particles is one plus sigma divided by two. Let's see why that's true. If there is a particle there, that means sigma is one, right? So one plus one is two divided by two is one. What if there's no particle there? Zero. That's a good candidate for the number of particles at a point. It is truly the average number of particles at a point is equal to one half plus the average of sigma. The average number of particles at a point is one plus the average of sigma. That tells us that the density of particles, let's call it rho, is proportionately from one plus sigma divided by two. That's the density of particles. In other words, the number of particles per lattice cell. The average number of particles per lattice cell is one or sigma bar divided by two. The average number of particles is not sigma divided by two. It's not sigma bar. It's one plus sigma bar divided by two. That's because of this offset that sigma bar being down means no particles, not minus one. It doesn't say it's minus one particle. So we offset it by half here. Excuse me. Didn't you say sigma bar could be a value between minus one and one? Yes. Which says rho goes between zero and one. Right? It's not more than one particle on a cell because they have sharp elbows and they push each other out of the way. We can't have less than one particle because that doesn't mean anything. So it goes between zero and one. And so the density is just one half plus the average magnetization. What does that say? Half plus half the average magnetization. Half plus half the average magnetization is the density of this fluid. So now we're all set to say what happens if we were to vary the chemical potential, vary the H, keeping the temperature fixed. If we're above the critical temperature, the magnetization varies uniformly and continuously how jump? That says the density doesn't jump. The density of particles. So above the critical temperature, as we vary the chemical potential in a box of gas, lower the energy, put it down, nothing very exciting happens when we pass this horizontal axis. In other words, the horizontal axis is when the chemical potential plus the clever H plus this is equal. The horizontal axis is when H is equal to zero, corresponds to the chemical potential of Hj plus 2h, whatever. No sudden jump at anything above the critical point. Below the critical point by contrast, as you vary the chemical potential H, all of a sudden at this point here the magnetization jumps. It jumps from negative. When it's negative, that means the density is low. The density is low when this is negative. So it jumps and it jumps from low density to high density. Experimentals didn't have temperatures available as big as key critical, which was key critical for water. No. No, no, no, no. No. That's a... I shouldn't have asked that question. That's pressure dependent, which means chemical potential dependent. No, the critical temperature would be a higher than that. And it's a good deal higher than that. You see, going across here is the phenomenon of boiling. Or going down this way is the phenomenon of boiling. Here's where boiling happens. Boiling doesn't happen out of here. It's just a smooth, continuous transition of the density when you're at high temperature, high enough temperature above the critical point, below the critical point. You're here. Are you referring to superheated steam above that? You could. Yeah. Question, please. How do you vary the chemical potential? We understand magnetic, but in the other case, how do you do it? Really, by varying the density. You can do it. You can either vary the density. You see, the density jumps across here, but how do you vary the chemical potential? Pressure? Yeah, well, that's one way. So, we have a chemical potential for water molecules. A little tricky. Whatever takes the very density. So, I like to think about it in a thought experiment by doing what I stress, which is to create a box. And then, literally pull the molecules into the box. Literally pull the molecules into the box, and pull the molecules into the box by creating a potential energy inside the box, which is what I have to think about how you would do that in practice. It's not actually what you do. What you do instead is very good pressure. And a varying chemical potential is the simplest thing to do. What you do is, you suddenly find a sudden jump in the density of the fluid. And that is what happens. With a different gas phase transition, down here is gas, low density, up here is liquid. But notice that you can go from gas to liquid without any jump by going around the critical point, way above the critical temperature, and varying whatever you have to vary the density. You could vary the density by varying the pressure. That's actually the easiest way. You vary the pressure, you increase the temperature, vary the pressure, increase the temperature and the pressure, making sure you go above the critical point, the critical temperature, and then lower the temperature back, and you'll go from gas to liquid without any jump. Question? Yeah. When you say you're changing the chemical potential by changing the density, you're basically changing your effective density for a fabulous wall potential? No, no, no. The property of the molecule is a stress. But the density says the random wall potential is not constant. Oh, it's changing the position now. So when you change the density, you're changing the distance between molecules? So in that sense you're changing the potential energy. Right. So when you change the density enough, it suddenly makes a transition, and realizes that the potential, that the negative potential energy is there, and tries to pull all the molecules together. What's the relationship between, in this case, the gas-like transition between the critical temperature and the boiling point? Well, the boiling point depends on the pressure, which in turn means it depends on the density. Let's see, how do we say this? The problem with this is I haven't worked out the pressure. What we could do, the pressure is a function of temperature and chemical potential. And if I were to have worked out the pressure, we could have varied the pressure instead of the count of the potential. That's why I'm having a problem, because I don't want to go through the effort now of calculating the pressure. But just in a simple mind of mine, when I'm boiling water, what's my path along there from liquid to gas or gas to liquids? What do I say over here? We're going from liquid to gas or gas to liquid. That doesn't matter. You go across here, but the point at which you go across there depends on the pressure. Sure, I understand. So the point at which you cross does depend on the pressure. So there's another variable in here. It's not entertaining. OK. One of the very fascinating things is the properties of the system near the critical point there. There's a whole theory of the behavior of the clear. A whole theory of lots of experiments as about what happens as you approach the critical temperature. As you approach the critical temperature, the critical point, as you approach the critical point, a number of fascinating things happen, but they're all characterized by what are called critical exponents. Every quantity that you can think of that's interesting, and its dependence on temperature, will generally go as T minus T critical to some power. And those powers, they're not ones and twos. There are various kinds of irrational numbers of things, transcendental numbers. They're called the critical exponents. And of course they vary from one kind of phase transition to another, but they're exactly the same for the magnetic transitions and the liquid gas transitions. They fall into exactly the same class. The behaviors near these critical points are rather insensitive to the details, and they depend on features that don't care whether you put them on a lattice, they don't care whether it's the nearest neighbor or a second nearest neighbor, they all behave the same way. And the magnets and the fluid transitions are in the same class. Okay, I think we're finished. Do they form a continuum or is it a discrete, countable, even finite? Oh, the set of possible critical points? Yeah. Because they're discrete. Finan? No, they're probably not finite. I don't think they're finite. The number of critical points is not finite, but they're all discrete. Question? First of all, how do you define a gas number? Since you go around and have no choice, they seem to be the same. Is it density? There's no sharply defined... The density is obviously different here than it is here. And if you try to go across here, the density jumps. If you go around the critical point, the density varies continuously. But it's the density then that defines the gas number. Is that right? The density is like the magnetization. Or the density here, this is the connection. If you go to low temperature, the density jumps. If you go to high temperature, the density doesn't jump. Then you can go from here to here going around the critical point. So they coexist. Liquid and gas coexist if you go around. Well, they're not distinguishable. They're continuously... They're continuous transition from one to the other. What do you need to put into the theory to get a gas liquid in your face? A gas... To have a gas liquid in your face, you have some gravitational potential energy or something to be separated. It's not surface tension. It's not surface tension. It could be surface tension. Most of this kind of analysis is what you think about for an infinite volume of fluid. So surface tension is not really the thing here. But you could have an infinite volume of fluid laid out on an infinite plane in a gravitational field, and then you would create a liquid gas transition in some point. That could be worked out. Basically, the altitude is related to the chemical potential. So varying the altitude varies the chemical potential. And a sudden transition at some height, which is connected to this transition. In this model, you get a heat of vaporization as you go across that transition. In general, yes. No doubt. I would have to think about what that means in this model. The energy down here is exactly the same as the energy just above. I have to think about that. So... Is there a heat release at that point? Total energy is conserved, of course. Yeah, but there's a certain magnetic problem. The energy above and the energy below is just exactly the same. It's a good question. I don't know if the answer is okay. I have to think about it. That's getting me... Does this have implications for superconducting magnets? I suppose it does, but I'm not sure what. What's the question? You talked about the Germanian detectors at Soudre and Main for the dark matter detectors. And how the physical interaction creates a point where it kicks over. Where the transition happens between superconductor and ordinary. Right. Yes. Oh, you're asking... Is that a phase transition like that? It's a phase transition. It has similarities. And the point there is if you can adjust your superconductor to be very near the transition, or very near the phase transition, let's say across here, by a tiny change in subparameter, you can have it jump from superconducting to non-superconducting. And that gives you the possibility. Some energy is deposited in the detector, and even a small amount of energy can create a little local pocket of a changed phase. So it can be useful as a very sensitive kind of detector. But I don't know a lot about superconductor detectors. If you're very close to the phase transition, and something really jumps, then the presence of a phase transition can be a very, very good detector of very small changes. Any phase transition? Very good transition. Okay, we're finished with statistical mechanics. We've done our duty to... Both of us. I think. I think. I keep making the same mistake about I think. I went back and looked at my notes, and I realized with my notes, I had an apology for the students, and they're wrong thing about I think exactly the same wrong thing. I'll probably do it again. Okay, I said I would talk about the improper principle. What do you do with this question and answer? If you want to ask me questions about the improper principle, perhaps you will get me going and I will answer them. But I've sort of run out of steam. So... No cookies. No cookies. It's not my fault. There's no coffee. But who asked me about the improper principle again? That's good. Well, go ahead, shoot the question. I wish I turned it off the email. It was just basically about the generalities of it. The applications or misapplications, the implications, the quality. Well, I asked you a preceding question. Could you give a clear, if there is such a thing, the crisp statement of the improper principle? Exactly what does it say? Can I give a crisp? No, it means many things to many people. But I can tell you how I think a rational use of it looks. I don't say it's right, but I think it can, in some circumstances, be a rational explanation of something. First, you have to understand what fine tuning means. Okay, let's talk about fine tuning. There's a small number of physics. The classic case of an extreme fine tuning is, of course, the cosmological constant. The cosmological constant is very small. In natural units, in punk units, it's 153 dB. It's small. What's the difference between being small and being fine tuned? They sound like the same thing, but they're not the same thing. So I'll give you an example. It's an example that I cropped up to explain this once to a bunch of condensed matter physicists who didn't know what fine tuning meant. As soon as I told them what it meant, they said, oh, we know that. But let me explain it in terms of an example. This is a silly example. This is the only example I've ever made up like this. This doesn't have fish in it. But it does have submersible submarines. And also balloons. Okay. There's a difference. Let's suppose that you were a certain kind of creature that could only exist at a certain altitude where the air was just the right temperature, not just temperature density and so forth. So you find yourself at that temperature density and so forth. And a balloon. A balloon in the ground of a dirigible. And a zeppelin. A zeppelin of dirigible. Branding. A zeppelin, I think, is a big bag of gas. A dirigible, I think, has some supporting structure. A zeppelin of dirigible. So a zeppelin is a brand name. So there you are, you're talking about air in your balloon. And you notice you're neither rising nor falling. You see that wonderful? You're neither rising nor falling. You're not that lucky. You're at some stationary height. And that stationary height you've been there for several billion years. Long enough to be devolved and you're devolved in just the right way to be able to live at that temperature and at that behavior and so forth. You see, isn't it very lucky that this blimp happens to be exceptionally light? Light enough that it floats. Okay. Yes, you're lucky. The blimp happened to be light enough that it floats. The small number was the density of the material inside the blimp. And that number is small. In this case, it's not horribly small, but it's small. It might be the density of hydrogen. It might be the density of helium. But notice there's a factor of what? Four difference in the mass of helium and the mass of... So the density could be quite different. And still you float somewhere. Your helium will support you, your hydrogen will support you, you're okay. And you'll come to equilibrium at some position. The density of the air varies and you'll find some equilibrium and you'll just hover at that equilibrium. And you stay there forever and ever, long enough to evolve your species. And that's an example of a small number. A small number in this case being the luck that your system happens to have a low density. In fact, lower than air. Smaller. No fine tuning. It didn't matter if it was helium or hydrogen. And it wouldn't matter if it was an equal mixture of helium or hydrogen. You could have changed the parameters of the gas quite a lot. But no hundred percent you could have changed the... as long as it stays lighter than air. So there you're in luck that there was a small number, but there was no exceptional fine tuning of anything. You didn't have to fine tune in any detail percentage of hydrogen and helium. Okay. So that's a small number. Now let's talk about another situation. This object is now not a blimp, it is a submarine. Yellow submarine. We all live in the yellow submarine. Blue. Submarine has no motor, it explodes. Down here, pressure is much too great for us to survive. As it happens, the atmosphere is highly poisonous and we get the float to the top we're dead. We have to stay in the water. Maybe it doesn't matter so much how high we are in the water, but we've got to stay there. Our submarine has existed in that configuration long enough for us to evolve. In other words, for a billion years it has been hovering in the water without falling or without rising. We know that because we're here. Or these people who live in a submarine are here and they know that if there was any significant tendency to fall, they'd be dead, if there was any significant tendency to rise, they'd be dead. Let's assume for simplicity that the water has the same density from the bottom of the sea to the top of the sea, which is very possible to be true. What do these people in the yellow submarine conclude? They conclude that whatever the submarine is made out of, the density of that submarine is finely tuned. It's made out of some iron together with some brass and maybe a little bit of concrete and a few other things. Plus it's hollowed out in the interior. Its average density is such that its average density is very, very close to that of water. If there was any density of water, it would sink. If there was any lighter than water, it would flow. Once more, it's been sitting there at that compromised position for a billion years. From that you can conclude that whatever the chemical composition is, it is very, very finely tuned. If you would change the ratio of iron to concrete by one part in a thousand, or probably one part in a million, remember this thing has been sitting there for a billion years, or let's say four billion years, or ten billion years. It's been sitting there for ten billion years and it's right somewhere in the middle of the zone. That's extraordinary because there's no feedback mechanism whereby it could start to sink. It would rise, it would start to rise, it would sink. No feedback mechanism, it's just a little bit too heavy. It would just slowly sink to the bottom, it's a little too light, it will slowly rise. So somebody very, very fortunately has tuned the chemical composition and the amount of hollowing out and the weight of everything that's inside it to one part, and I don't know how much, we can try to work it out, to one part in a fairly large number to make sure that that submarine hovers. Now the submarineers are curious about this, they want to know why this is true. They have no explanation for it at all. Their first reaction is this must be something like the dirigible, the Zeppelin. The Zeppelin had a feedback. If the Zeppelin goes to high, the air density goes down and it sinks back. It goes to low, it rises up again. But they do a few calculations and they realize there's no question of that kind of feedback mechanism. The density of water is too constant over this range and there's no possibility of using that against the stability mechanism. They try all kinds of mathematical tricks, symmetry principles, whatever you have, nothing works. The reason nothing works is because this is one of these very fine-tuned things, if you just change the chemical composition a tiny bit this way or a tiny bit that way, the whole thing breaks down. What kind of conclusion could they come to? They do a little more calculation using whatever laws they go and they discover something very interesting about the nature of the fluid that they're in. The nature of the fluid that they're in is that it has an instability. It makes bubbles and the form of these bubbles is submarines. Submarine is in the fluid. Bubbles. And when the bubbles nucleate, they morph into submarines. But the trouble is the submarines that they can nucleate into are generally not of the right kind. In fact, as it turns out, the submarines can nucleate with different chemical compositions. Some of them have a little more brass, some of them have a little more iron, a few of them have more hollowed out. There's a whole huge variety of different species of submarine that can nucleate. They do it. They make these calculations and they discover this is a property of the fluid that they live in. They say, ah, they can know the answer. The answer must be that there must be zillions of different kinds of submarines, huge, huge numbers of different kinds, and moreover, they keep nucleating and nucleating and nucleating this fluid. So the fluid just keeps producing more and more and more of them of every conceivable type, 10% iron, 90% brass, and all possible things in between. And most of them either sink to the bottom, they lie and dead at the bottom, or they rise to the top and they poison the anatomy, the proto-habitants, that I had time to, there's a tiny fraction of them, some incredibly tiny fraction of them were nucleated with the right properties. They were nucleated with the right properties just because everything that could happen happened. And so they are not particularly lucky, they are just the ones that work in the kind of submarines that nucleated with exactly the right density to hover. If all of that really happened, and these people really had a genuine scientific reason to believe in submarine nucleation in their city, and they could see that the submarines could nucleate with a vast variety of different types, they would have what I would say was a reasonable explanation of what's going on. There's just a lot of stuff, the sea is very big, the sea is very big, a long time has passed, every conceivable kind of thing is nucleated many, many times, and they are just in the kind of where survival is possible. That is the logic that I think the anthropic principle, when it's rational, is making use of. That fine-tuning is a consequence of many, many, many possibilities, and many actual events taking place, creating the submarine that we live in. So many so that some fraction of them will be livable. Then it becomes a non-mystery about why we live in a fine-tuned universe. That's a logic that I find personally, and I find acceptable. There are other possible answers. The other possible answer is that I was a benevolence that created the submarine with exactly the right properties. There are some people who would like to believe that. Most of my scientific friends don't think that is an acceptable solution. So what are the other possibilities? At this time, right now, here and now, there are no other plausible possibilities for why certain things are fine-tuned. Now what is it that is playing the role of this density of the submarine? What's playing the role of the density of the submarine is the cosmological constant. If the cosmological constant is slightly negative, the universe would re-collapse. Now, if it's sufficiently small, that's like the density of the submarine being only a tiny bit different than water. If the cosmological constant is very small, then it will take a long time for it to re-collapse. In the same sense, a very, very similar way to the way it would take a long time for the submarine to fall to the bottom of the sea. If the difference between water density and submarine density was small. So if the cosmological constant is small enough, the universe lasts long enough, assuming it's negative, it lasts long enough without re-collapsing for us to be here and to survive and so forth. On the other hand, if the cosmological constant could be positive, in which case the universe doesn't crunch, it expands and it expands in an exponential way. But if it expands too fast, it expands too rapidly, then it prevents structures from forming. If the universe was accelerating very rapidly and accelerated expansion, then the material that's in it would carry the bone with the acceleration and simply be incapable of contracting into galaxies, stars, and so forth. So the sum window, which is analogous to the window of densities, of submarine densities, where either positive or negative, sum in a window where the submarine could last for a billion years. Here, it's a window between collapse and destruction on a time scale that would not allow our existence, on the one hand, and in the other direction, lack of galaxies, stars, planets, structures would form. And that window is very, very small. The window is small because time scales are so long. The same reason the window is small for the submarine. We have to be able to survive in this universe for a period of time, which is very long. And that means that the imbalances in the cosmological constant have to be very, very small. Now, there's another sense in which the cosmological constant is like the density of the submarine. The density of the submarine is made up out of some composite of the densities of different materials that make it up in various proportions. It's not just that some number is small, it's that some number is very, very finely adjusted. The same is true of the cosmological constant in the quantum field theory. It gets contributions from many, many different sources. Basically, it gets a contribution from every quantum field that exists. Bosons give positive, fermions give negative. The mass of the particles that are associated with those quantum fields shift them a little bit. And the cosmological constant that results from it shifts a little bit. And the whole thing, all of the constants, all of the particles, all of the contributions to the cosmological constant have to balance very finely, exactly the same way as the density of the submarine has to balance. For the little too much iron, you're dead. For the little too much brass, you go up. Same thing with the cosmological constant. Put in an extra fermion species in the theory and the cosmological constant gets too negative and you implode. Put in an extra boson, cosmological constant gets too big and you're out of luck and a tiny fraction of a second. So... Question please. What is the meaning of making a change but that compensates? Meaning it doesn't... it's just a different submarine but still floating. Yeah. There are parameters in physics such as various masses, coupling constants. You could change one thing one way a little bit which might increase the cosmological constant but change something else another way a little bit which might decrease the cosmological constant and you might be able to... It's not just that there's only one chemical composition of the submarine that can survive. It's just a very thin surface in some space of different configurations. That's the same thing here. You could change things but you have to change them in a very, very fine tuned way. You have to change one thing by this amount, you have to change the other thing by that amount and it has to be adjusted to 123 fescal places. So that's... So will this water that produces every kind of submarine will make various submarines that have the same density but look different than this? So these are alternate universes? Yeah. Well they're all alternate universes but some of them are capable of sustaining themselves for a long enough time. Yeah. People, others are not. Yeah. You know, we know very little about this. Our knowledge of space of possibilities is extremely limited. We don't know very much about it. String theory seems to say that there are endless huge numbers of possibilities analogous to every possible chemical composition. So that's a plus. I don't know whether to regard that as a plus for string theory or a plus for the enthrotic principle or a plus for cosmology. But it does seem to fit together with the idea. And the fact is there is no other known explanation. There's nothing out there that many years now that people have looked for explanations. Oh, there's one more thing. Just a moment. There's one more thing about the submarine. Yeah. These submarines might, as I said, they might start looking for mathematical explanations. Some symmetry of nature, some symmetry of their equations, which makes the submarine density exactly equal to the surrounding water. They think they have this theory called Roke theory or something. I don't know. Roke theory seems, they can't quite prove it. But they have some idea that they only have the right idea. The Roke theory would say that the submarines always come out with exactly the right density of water. Some symmetry between water and other things. We've got it. We don't quite understand it, but we've got it. We know there's some theory there. We could only understand that theory better. It would tell us why the cosmological constant or the density of submarines is exactly the right thing. But then, that their horror and dismay, they discover that in their own submarine it's not exactly the right thing. In fact, the submarine is slowly sinking. It really is slowly sinking. How long does it take for it to sink? More than 10 billion years. But their finest measurements tell them, no, it's not that things are perfectly adjusted. We'll have to explain why they're perfectly adjusted. They're not perfectly adjusted. They're just adjusted to 123 decimal places. And at 124th decimal place, they're sinking. Excuse me. Isn't this precision violating the uncertainty principle? One thing can be as accurate as you like. Oh, yes. Question? When you gave this explanation, you said this is an explanation of the anthropic principle that I'm comfortable with. I wouldn't say it's an explanation. I would say it's a use of the anthropic principle that I'm comfortable with. What are some other uses that you are not comfortable with? Look, I have no idea if there was an intelligence benevolent or not that created the universe. I do not know and I don't pretend to know. So are you saying that others sort of give that spin on things a little bit? I suppose not many scientists do. Yeah. At least at one time. Okay, there are other uses of it which I find much less compelling. There are all sorts of modest coincidences. If any of them were not satisfied, they might be enough to destroy life on Earth. If we were to make a list of the elementary particles, we would find that almost all of the known ones, almost all of them are necessary for our own existence. On the other hand, our understanding of particle physics would allow just about any one of them to not be there or other ones to be there. For example, I mean, stop thinking about the particles you know about. Photon is the lightest. Where would it be without photon? There would be no light. There would be heat from the sun. The worst than that, there would be no photons jumping back and forth between the atomic nucleus and the electron to hold it together. No chemistry. It wouldn't be here. What about no neutrinos? No neutrinos, there would be no nuclear reactions of a kind which created the elements from the particle of the Big Bang. So the existence of carbon and all these other elements that make us up wouldn't be here without neutrinos. What about no electrons? Well, no electrons is a clear disaster. There's no question of that. What about no quarks? No quarks, no protons, neutrons? What if you change the parameters a little bit? What if you...it's a...it was a puzzle for many years and still is a puzzle. Why the proton is lighter than the neutron? In the early days of nuclear physics, this was a great puzzle because the electron is charged. You would have thought its electrostatic energy would make it a little bit heavier than the neutron. Neutrons and protons are twin...twins. They are almost exactly the same with the exception that the proton has a little bit of positive electric charge. Ordinarily, you would say that increases its energy. It takes energy to pull together a charge. It's an increased energy. Protons should be a little bit heavier than a neutron. Or it would be a disaster because if the proton was heavier than the neutron, the proton would decay into neutrons. Instead of the neutron decaying into protons. If the proton decayed into neutrons, there would be no hydrogen. There would be nothing. We wouldn't be here. So, a little tiny difference between proton and neutron is small, which happens to go the wrong way from expectations is absolutely essential to our own existence. All sorts of other things. What would happen if the electromagnetic coupling constant, the fine structure constant, which is well over 137, what would happen if it was one over 10 instead of one over 137? Well, then the atom would be very, very tightly bound. And ordinary chemistry would probably go awry. And what would happen if the electron were heavier than the proton? It would be very deep inside the center of the atom instead of having a valence structure. So, you can look at physics as we know it, which means largely particle physics. And you can say, you know, there's so many different things that you could have changed by 10, 20%, or even 1% here and there, and any one of them would have destroyed our existence. I can easily imagine 1% accidents. I can even imagine 10% 1% accidents, although it gets a little harder. Are those accidents sufficient to really push us in the anthropic direction? I'm not going to answer that. I don't have an answer for that. That's a matter of taste of it. But the accidents of one part in 10 to the 123, or in some other case, one part in 10 to the 60, or something else having to do with the Higgs boson, 10 to the 30, I'm sorry, one part in 10 to the 30, not the kind of accidents that you can imagine are accidental. One part in 10 to the 20. So, I think if it wasn't for the cosmological constant and the other fine tuning, comparable fine tuning, I think we would not be, you know, most serious physicists would be rejecting the anthropic explanations. Maybe not completely, but not the same support that exists. That doesn't mean that every physicist approves of it. They just don't have another explanation. Spine, heart, and some other guys in April had a sort of inflationary paradigm and trope-reactor plot in 2013. Have you seen what they said or have anything to say about it? They said basically most of the inflationary models don't work. Who said that? Paul Jay, Spine, heart, Abraham, Lou. Who? Abraham, Lou, Elo, and the... Okay, these are counter-areas. The vast, vast, vast, overwhelming majority of cosmologists and physicists believe in inflation. Now, the other point of the point, they said that in the plot of 2013, there's ulcer in and they talk down most of the models. Well, they didn't model or work. Sure, people make up a lot of wild models, and then the good data comes in and knocks out most of the wild models and leaves the sensible ones. Nothing is done. Right, you have 500 models. The time is the standard, the gold standard model, and data comes in and knocks out 499. Unfortunately, it's not that good. Yeah, no, this is... No, no, no, no, there are people with axes that are with the ground. Physics is not a highly politicized subject. It does have people in it who... they're very highly politicized about their own ideas. And going around saying things like that, I think that's irresponsible, frankly. I would consider that irresponsible. I do consider it irresponsible. All this time, the... is not the most responsible cosmologist. Sorry for that. That's true. Would it also be possible to develop a logical concept of variance sometimes, perhaps, over time, so instead of having like... There's no good theory in which it varies gradually. It can vary in a jump. It can vary in a jump if one of these submarines nucleates with a different value of it. Yes, it can. It can jump. There's no good theory in which it varies smoothly over such a long time scale, slow enough to be consistent with data. That was another theory, a very, very slowly varying cosmological constant, but doesn't seem to fit with anything we know. In any case, it would require... it turns out that to have a slowly varying version of it with a second stand to zero requires a fine tuning of several parameters, not just one. Not just the vacuum potential energy, but several of its derivatives have to be fine tuned. So it's not a better explanation. Look, we don't know what... we don't know where this is going to go. Nobody knows at present where it's going to go. There are contrairings. There are people who have political agendas. I don't. I don't really. I don't... I'm not any better at inflation. I don't have a big stake in it. I do not want to know the price if it's confirmed. It is confirmed. It's highly confirmed. Somebody is talking to me about the price. The thing will be me. The logic of it to me is extremely compelling, and I honestly cannot understand these people who say that Plunk data is ruling out inflation. I think they are off base. But, you know, why trust me? Does that imply... is the anthropoconst... principle implied in the creation? It fits together with it very well. Now, you could go back to the submarines. You could ask... you could say, Well, the anthropoconstable itself, all that says is we live in a submarine, which is of the right type for us to live in. And which is cause and which is effect. There aren't cause and effect. Do we explain the fact that we live in the right kind of submarine by the fact that we're here? Or do we explain the fact that we're here by the fact that we live in the right kind of submarine? I prefer the latter. That we can understand why we're here as a consequence of there being enough right kinds of submarines, rather than to say the opposite, that we can explain the properties of submarines by the fact that we are here. That seems backward. Scientifically backward to put us at the center of the laws of nature to that extent. Yeah, I guess I think of this use of it is when you're trying to start explaining why the laws of nature have to be the way they are because we are here. It seems to me much less egregious to say that the laws of nature are highly variable, which theories tend to suggest anyway. Highly variable environments are highly variable. And that we live in a temperate zone for the same reason that we all live in Antarctica. It's just to be in cold there. And the real question of the real question is part from the fact that it's night and clock. The real question is how are we going to find out? The agenda of those who hate the idea and who love the idea should be exactly the same. Take it seriously enough to either kill it or make it into science. And the fact is that it looks very, very hard to do either. It looks extremely hard to find signatures of this kind of thing in data, which would either kill it or confirm it. That's the real problem. The problem is not that it's bad philosophy or that it's religious or that it... One of my friends seems to think it's a terrible thing to propose because young people will be... their minds will be deformed by it and they will stop thinking about other explanations. These are political reasons. These are... oh, other people think it's a bad idea because it enables the religious right to say, ah, the world was created though just the right properties. None of these are science. The scientific question is, is it true or is it not true? And if it is true or not true, how do you find confirming evidence? In other words, you want to try to observe other submarines in the cleaning. Right. So if you're very lucky or unlucky in the case of a bee, you might bump into another submarine. That could be the lucky or not lucky. It's lucky if it doesn't kill you and you find out that there are submarines out there and you confirm and you win a Nobel Prize. Or it could be bad, the submarine could make a hole in your submarine and then you sink to the bottom. It's still a no. Right. In principle, there could be collisions between regions of space with different properties. It's expected to be extremely rare. Computation seems to say that it's extremely rare. Basically, computation seems to say that the ocean is exceedingly big, but the nucleation of submarines is very rare. That doesn't happen very often and very frequently. And so most submarines will last a long time without being able to know. So we found no evidence in the cosmic microwave background for a collision like this. There's no evidence. It's probably not going to be there. So the only question is then what other possible kinds of signatures could there be for the physics that goes into this kind of theory? It's hard. Nobody has a really good idea. That's the worst thing about this kind of idea that just looks overwhelmingly hard to find convincing data for. Okay. We are finished. Thank you. | {
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"text": [
" Stanford University.",
" You all know about the water at the steam.",
" And it has been understood qualitatively for more than 100 years.",
" When was Vandervals?",
" Vandervals was the birth of India in the 1970s.",
" Somebody will have quit Vandervals.",
" My cue is usually the one to go through for that.",
" Okay, it doesn't matter.",
" It's been understood qualitatively from both mining statistical mechanics",
" for a well over a century, I think.",
" And we'd like to work with a little bit.",
" I'm going to show you how the liquid gas phase transition works,",
" not in the standard way that Vandervals approached it.",
" So the way Vandervals approached it is very elegant, very nice,",
" and is equivalent to what I'm going to show you.",
" What I'm going to show you is based on a system that we've already studied, magnets.",
" So the Ising magnet.",
" What I'm going to show you is in what way the Ising magnet is a model for the liquid gas phase transition.",
" But before we do it, let me just remind you what we learned last time",
" by studying the mean field approximation.",
" To the, not the liquid gas, but to the Ising model.",
" We also know that it doesn't work very well in one dimension.",
" In fact, it gives the wrong answer in one dimension.",
" Mean field approximation always tells you that it's a phase transition,",
" and this happens not to be true in one dimension,",
" but it is true in two dimensions and higher.",
" And by the time you get to three dimensions, every particle in the lattice",
" has enough nearest neighbors, basically six of them, that the mean field approximation becomes very good.",
" Not accurate in the sense of one percent, but probably accurate in the sense of ten percent, I would say.",
" And so the mean field approximation is a good way to qualitatively understand",
" how phase transitions take place.",
" Now, it has to do with these little magnets, which can either be up or down.",
" They would describe replacing at each point on a lattice a variable, what we call sigma.",
" And sigma could easily be the last one in which case we thought of it as pointing up",
" or minus one in which case we thought it was pointing down.",
" And we talked about several, well, at least two different kinds of models to magnets.",
" The Ising model was the more difficult of the models.",
" It said there's an energy not for every particle, every site.",
" Not for every site, but for every pair of neighboring sites.",
" That the energy was stored in the relationship between neighboring sites.",
" And in particular, energy was lower if the sites are parallel and higher if they're anti-parallel.",
" You can think of the energy as being located in some sense in the links between the sites.",
" If two neighbors are parallel, we'll call them an unbroken bond.",
" If they're parallel, then the energy is low.",
" If they're anti-parallel, we'll call that a broken bond.",
" If the bond is broken, then the energy is higher.",
" So we wrote an expression to represent that.",
" And the representation was that the energy is the sum not over the sites of the lattice,",
" but the bonds, the links, they're usually called links.",
" Links of the product of sigma at one end of the link, let's call it sigma i and sigma j.",
" Let's not try to make a fancy notation.",
" i and j just mean the two opposite ends of a link.",
" We put in here a minus sign, whatever you want to admit, and a constant j.",
" A constant j is just associated with how tightly bound, how tightly, how hard it is to break apart.",
" How much energy does it cost to break a bond?",
" Well, if we start with two spins to sign us a line, then sigma dot sigma is plus one.",
" If they're anti-aligned, it's minus one, so the difference in this expression here is two units.",
" Not one unit, but two units to go from minus one to plus one.",
" And so that means every broken bond costs you an energy plus j.",
" Two j.",
" Two j.",
" Every broken bond costs an energy.",
" That's how much energy it takes to flip it.",
" So first thing we introduced, and the second thing was another possible term in the energy",
" that would correspond to a uniform magnetic field, which is just a term which acts not on the neighbor,",
" not on the links, but acts on the individual sites.",
" It's an extra energy if a site is up or down.",
" It doesn't depend on what its neighbors are doing.",
" And so let's put that in there as something proportional to a magnetic field.",
" That just means a number here.",
" I think I probably called it capital H last time.",
" I'm going to use standard notation, and then there's a notation that's in the statistical mechanics.",
" Note that I'm going to call it lil H.",
" You called it b last time.",
" I called it b, you're right.",
" You're right.",
" I called it b.",
" But I'm not going to now because in the notes it's called h.",
" But h looks a lot like b.",
" It just looks like a family drawn slide.",
" So let's not belabor the difference.",
" Yeah.",
" So let's say if h is positive, then this term is lower if sigma is negative.",
" So if this was all there was, which was incidentally the first model that we studied, only h,",
" then this would favor spin.",
" This would favor sigma being negative or being down.",
" That would lower the energy.",
" Lower the energy by this speed.",
" Down configuration, in other words the minus one.",
" But of course thermal fluctuations would cause it to fluctuate up sometimes,",
" and a very, very high energy would be a very high temperature.",
" It would be up as much as down.",
" And so there's a pattern according to energy.",
" So this term here tends to want to make things lie in the same direction,",
" and this one wants to make them lie down.",
" OK.",
" Since you're summing over links, shouldn't there be a h-sigma j as well?",
" No, that coefficient is called j.",
" Yeah.",
" This one?",
" No, no, no.",
" You're summing over links.",
" Yes.",
" So shouldn't there be an h-sigma j as well?",
" OK, so I'm going to have a science.",
" So h-sigma j. Sorry.",
" I'm missing your point.",
" If you're summing over the links, here's something on the links.",
" Right, and there's something over it.",
" Here's something on the sides.",
" Quick question, please.",
" It seems that the larger j when sigma is one,",
" I mean the product, it's the lower energy.",
" When they're parallel, it's the lower energy.",
" Not only when they're parallel, but also when j is large.",
" Yes, j will.",
" Yes, but j is not a variable, the problem.",
" j is just a number.",
" I understand.",
" But it's a number that varies from one substance to another.",
" Yes, yes, it's a number that varies from one substance to another.",
" So does that mean that it is harder to break a bond when the energy is lower?",
" Is there physical meaning to that?",
" Do you mean when j is lower?",
" When j is bigger, it's a lower energy.",
" And at the same time, when j is bigger, it's harder to break.",
" Both are consequences of larger j.",
" I don't think the zero energy matters.",
" It's the energy difference between the broken and the unbroken bond.",
" We are perfectly free to add any constant that we want.",
" And in fact, the material may tell us what constant to add.",
" For example, this energy stored in mass, you know, little v equals mc squared.",
" A lump of uranium would have a bigger number here than a lump of frozen hydrogen.",
" But the point is, this is just a numerical constant and an additive constant in the energy.",
" That doesn't matter.",
" And also, the ground state energy here, how much energy you have,",
" let's calculate how much energy you have.",
" Let's forget this for the moment.",
" It has how much energy you have if all the spins are aligned.",
" That's the lowest possible energy.",
" If they're all aligned, then it's just a number of links times minus j.",
" Minus j times the number of links.",
" And, Sinect, how many links are there in a, let's take a three-dimensional lattice.",
" If the number of sites is in, how many links are there?",
" Everybody know?",
" Sixth and seventh.",
" Sixth and seventh.",
" How many people say six?",
" How many people say three?",
" Three is the way.",
" I'll show you why, but two, because I can't do it for you so well.",
" Is that a sum of infinite?",
" Yeah, yeah, a sum of infinite.",
" And it's a word of infinite.",
" So each site is connected to four links.",
" So you might think the answer is four times the number of sites,",
" but that's not the way it works.",
" So the way to draw it is to draw, for each site, draw two links.",
" Now, the next slide here also draws two links.",
" For the site over here, draw two links.",
" For the site over here, draw two links.",
" Numbers will fill up all the links.",
" So all you have to do is put two links at each site,",
" and that fills up everything.",
" Three dimensions, it's three links at each site,",
" and that covers everything.",
" The reason, of course, is that the link connects two sites.",
" So it's only half.",
" The half of what we might have thought.",
" Okay, but in any case, you'll have in here an energy",
" that's associated with the ground state,",
" which means when everybody is aligned.",
" Well, that's a constant.",
" We could subtract it out.",
" We could hand it over, subtract it in such a way that",
" the ground state has zero energy,",
" and group it together with this constant here.",
" It wouldn't make any difference.",
" Only energy differences are important",
" with statistical mechanics.",
" Or, in almost everything, energy differences.",
" So the ground state energy is not so important.",
" What is important is the energy difference",
" between the up spin and the down spin,",
" or a bond and a broken bond.",
" Let's just remember what we found out from mean field theory.",
" Excuse me, because I need to be a J by the sum over the sites.",
" No, no J there.",
" No J there.",
" Alright, so we're running out of the recalculated partition function.",
" E to the minus beta times the E.",
" That means every time you see a J,",
" it'll come together with a beta.",
" And also every time you see an H,",
" it'll come together with a beta.",
" They'll always be there.",
" Alright, we calculate the partition function.",
" We go through the manipulations that we went through.",
" And if you remember, we calculated",
" in the mean field approximation,",
" just to remind you,",
" if anybody wasn't here,",
" just what the mean field approximation was.",
" So here's a site.",
" It's surrounded by a bunch of neighbors.",
" The number of neighbors across this pool.",
" The neighbors have an average sigma.",
" That average sigma might be zero,",
" or I have a net average value",
" when averaged over the neighboring sites.",
" The more neighboring sites that you have,",
" the better approximation it is",
" to just say each spin here,",
" each neighbor has a spin",
" which is equal to the value equal to the average.",
" So on a high enough dimension,",
" which turns out to only mean two,",
" you can make a reasonable approximation",
" by saying each spin here sees around it",
" a collection of spins,",
" each one of which has the average spin.",
" Alright, then we'll say, alright, good.",
" Then what is the energy of the spin on the average?",
" It is minus j",
" times the average spin of the neighbors",
" times the spin itself of that particular particle.",
" Sigma bar means the average,",
" minus j, sigma bar, sigma.",
" And then what about this time over here?",
" Well, this time doesn't care about the neighbors,",
" and so it's just,",
" oh, and this one here, what about this?",
" The number of neighbors.",
" That's where twice the dimension came from.",
" Twice the dimension, whatever the dimension is,",
" minus twice the dimension of space",
" and dimension of the wireless,",
" j,",
" average of the spins, whatever happens to be",
" times the spin itself,",
" and then another term from here,",
" which is just h times the spin itself.",
" And we pretended that the average here was just a number",
" and did the statistical mechanical problem of one spin",
" with an energy given by this expression.",
" And of course, this is just a number,",
" times the spin.",
" Okay, so that's a calculation that we did",
" when we first looked at these magnets,",
" and we found out that the average spin,",
" the average spin that we get,",
" I'm going to call it for the moment sigma double bar,",
" it's the average of the spin that we get.",
" If all of the neighbors happen to be frozen",
" and had the value of sigma bar,",
" then if you remember, that's equal to the hyperbolic tangent",
" of the coefficient of being in the bracket,",
" minus 2 dj sigma bar,",
" plus h times beta times the inverse temperature,",
" and that was it.",
" Yes, that's it.",
" Now, to complicate the expression,",
" we started a little bit,",
" but the important thing was we now make the mean field assumption",
" that sigma bar, the average of this spin",
" is just equal to the average of all of its neighbors.",
" This, of course, would be true",
" if we had a big enough sample",
" and we would deepen the interior of it.",
" It wouldn't matter which one of the spins we looked at,",
" they all have the same average,",
" so we now do the self-consistency trick of saying,",
" that's our equation that determines sigma bar.",
" 2 dj and h are all just numbers,",
" so this is now an equation for sigma bar.",
" Just remind ourselves what the solution looked like.",
" What we did, we did a trick.",
" We defined a new variable.",
" We redefined this thing by calling it y.",
" We wound up with an equation that looked like y",
" divided by 2 dj,",
" which is, let's say, y, or this bigger,",
" equals tanh of y.",
" That was our equation.",
" Let's forget the rest of it.",
" Let's just start right there",
" and remind ourselves what we learned.",
" Oh, and y plus h.",
" Yes, y plus beta h, I think.",
" Aren't you missing a minus sign on this?",
" Yeah, y minus beta h, I think.",
" Y minus beta h.",
" What was y equal to 2 dj beta or minus 2 dj beta?",
" Yes, plus, if you had a minus sign,",
" the thing to worry about is tan minus 2 dj.",
" That's what I'm saying.",
" That's why the minus sign is here, I think.",
" No.",
" Oh, OK.",
" There's a relative minus sign between that.",
" OK, all right.",
" OK, so this was our equation,",
" our mean field equation, our consistency equation,",
" which tells us what y is.",
" Y is just the average spin divided by 2 dj.",
" One more element, one more thing.",
" Let's just remember that beta is 1 over the temperature,",
" so let's put the temperature over here,",
" the temperature over here.",
" First exercise, what happens if h is equal to 0?",
" Then it's just y times t over 2 dj is equal to tanh of y.",
" We can get this, and now we can just, by inspection,",
" get a pretty good idea of what things look like.",
" h function looks like this.",
" h of y, y on this axis,",
" and on the vertical axis, the hyperbolic tangent.",
" On the left side, you have a linear function of y,",
" and the linear function of y passes through the origin,",
" and it has a slope.",
" It has a slope, which is the temperature divided by 2 dj.",
" So in particular, when the temperature is very high,",
" the slope is steep.",
" And the only place where these curves intersect",
" is right in the origin,",
" and therefore, for large temperature,",
" there is no magnetization.",
" y is equal to 0.",
" y is proportional to sigma bar,",
" and therefore, sigma bar is equal to 0 when the temperature is high,",
" as expected.",
" Now we start lowering the temperature.",
" The slope of this curve decreases,",
" and at some point, it becomes exactly tangent",
" for the hyperbolic tangent.",
" Tangent, two different uses of the term tangent.",
" Tangent being hugged closely.",
" Hyperbolic tangent means, yes,",
" what is the slope of the hyperbolic tangent function",
" at the origin? Anybody know?",
" It's exactly one.",
" Easy to differentiate.",
" You differentiate the hyperbolic tangent",
" and you do it yourself,",
" and you discover that the slope of the hyperbolic tangent",
" at the origin is exactly equal to 0.",
" That means when the slope of the left-hand side",
" is adjusted to 1,",
" that's the magic point",
" where the two curves lie on top of each other.",
" So past that, decrease the temperature even more.",
" You decrease the slope of this line even more,",
" and all of a sudden, you'll have a solution over here",
" and a solution over here.",
" Those solutions are magnetized.",
" Those solutions have non-zero y,",
" and therefore non-zero sigma bar over 2dj, I believe.",
" That's a phase transition.",
" At that point, the system doesn't quite know what to do.",
" Should it go here? It doesn't go there.",
" It doesn't go there. Should it go here?",
" Or should it go here?",
" In other words, there's an ambiguity.",
" It seems like the spins want to have an average value.",
" What is that average value up, or is it down?",
" That's the question.",
" The average value is up, sigma bar is positive,",
" here it's down,",
" and the problem is there's a complete symmetry in the problem.",
" If you take all the spins and turn them over,",
" whatever they are, turn everyone over,",
" it's the same problem back again.",
" So there's no way to decide whether it does this or it does that.",
" To decide, we add a tiny, tiny, tiny little bit of h.",
" And the effect of adding a tiny little bit of h",
" is to shift this curve, even to the left or the right.",
" But once you shift it, that point of the solution is longer there,",
" and only one of these two will remain.",
" In that case, that's what happens.",
" That's magnetization.",
" One of the wins, and the one that wins, of course,",
" is if you put a little magnetic field",
" which is trying to push the spins to be up,",
" then upward-away.",
" If you put this magnetic field such that they want to be down,",
" then downward-away.",
" So we can now draw a graph of what the system looks like",
" with the function of two variables.",
" The two variables are the temperature and the external magnetic field.",
" Those are the two things we can vary in the problem.",
" We can turn up the temperature or down the temperature.",
" We do that by turning up the heat bath,",
" by turning it down, putting a little fire under it.",
" Or we can change the magnetic field by turning on the current",
" through a coil and making the magnetic field larger or smaller.",
" Those are the two things about disposal.",
" What about J?",
" J is also something we can change,",
" but in order to change it, what we usually have to do",
" is change the material.",
" Now, in fact, you can also change it by changing the pressure",
" and other things, but we're going to take J to be a fixed number.",
" Fixed chemical composition, fixed pressure,",
" fixed distance between the atoms.",
" J is a well-defined number that you can look up in there.",
" I can't put the physics, chemistry, or whatever.",
" But beta and J are the things that you...",
" sorry, beta and H are the things that are easily under your disposal.",
" So you might want to ask,",
" is an experimental fact,",
" what does the magnetization look like",
" as a function of temperature and magnetic field?",
" So first of all, let's turn off H altogether.",
" No H.",
" No H and go to large temperature.",
" Oh, oh, oh, let me come back a second.",
" One second.",
" Let's go back to this formula.",
" The question is, at what temperature",
" does the transition happen?",
" The transition from having a solution like this",
" to not having a solution like that.",
" So the transition happens when the slope of the left side",
" is equal to one.",
" When the slope of the left side is equal to one,",
" that's where you're hugging the two curves, huh?",
" Okay, so the slope of the left side is one",
" when the temperature is equal to 2DJ.",
" That's called the critical temperature.",
" It's not exact, this is a mean field approximation,",
" it becomes exact when the D becomes large.",
" And I'm sorry, I forgot to look up what the correct answer is",
" for D equals 2 and 3.",
" But for D equals 2, it's off by a significant amount.",
" For D equals 3, it's fairly good.",
" But it captures qualitatively what it does really happen.",
" Okay, so at temperature 2DJ,",
" that's called the critical temperature.",
" And let's write it as C R I T, the critical temperature",
" below which the system is spontaneously magnetized.",
" Everybody wants to line up and the lining up tendency",
" wins and everybody lines up.",
" Might absolutely.",
" They're not 100% lined up, they fluctuate",
" but there's a net tendency for everybody to line up.",
" Above that temperature, thermal fluctuations win",
" and it doesn't magnetize.",
" Okay, so now let's draw a picture here.",
" Is that the Curie temperature?",
" What's that?",
" The Curie temperature?",
" Curie temperature.",
" Yeah, yeah, yeah, same thing.",
" Curie temperature is a term that's used for magnets,",
" that's strictly.",
" If we're talking about the general phenomena of phase transitions,",
" this would just be called critical temperature.",
" So Curie temperature is a special case of a critical thing.",
" A lambda point in liquid helium is another one",
" and they're very similar to each other.",
" In fact, they fall into a class of phase transitions",
" which are almost identical in many ways.",
" I'm not sure why they're also related",
" or at least how this is related to another",
" interesting transition.",
" Okay, so some critical temperature over here, T critical.",
" Now, what is the magnetization?",
" Let's think about the value of the magnetization on this plane.",
" First of all, if you're at higher temperatures than T critical",
" and there's no magnetic field,",
" then magnetization is just zero.",
" There's no bias loop in the magnetic field",
" but at higher temperatures, the statistics of ups and bounds",
" being equal will win.",
" And so over here, let's call it average of sigma bar",
" is equal to zero along this line.",
" Along the line, the average of sigma bar is equal to zero.",
" Now you start lowering the temperature",
" and you come to this critical point.",
" When you get to the critical point,",
" it just doesn't know what to do.",
" Should it go up or should it go down?",
" But the tiniest little bit of stray magnetic field H",
" will tell you which way to go.",
" If H is positive, the magnetization will be positive",
" and right above this line, it won't be zero.",
" And right below the line, it won't be zero.",
" Spontaneous magnetization has happened",
" and so all along here, sigma is positive,",
" plus, plus, plus, plus, plus, plus,",
" on the average, the average is positive.",
" And down below is negative.",
" Corresponds to the two possible roots",
" where the red line either intersected on the right side",
" or did the second one on the left side.",
" And physically, it's just the extra little bit of magnetic field",
" has broken the symmetry between ups and bounds",
" and has biased it.",
" And that little bit of biasing is enough",
" that just the tiniest little bit of H goes plus.",
" Tiniest little bit of H down goes minus.",
" Now, you can ask what the jump is",
" in the magnetization going across here.",
" Remember, this is the average magnetization.",
" It's not zero over one.",
" It's not one or minus one.",
" It can be zero or it can be two-thirds or it can be one-quarter.",
" It can be any number between zero,",
" sorry, between minus one and one.",
" Well, at zero temperature, way down here,",
" it doesn't matter what the magnetic field is.",
" At zero temperature, all these guys line up",
" and they line up in the down direction.",
" So all along here,",
" sigma bar is equal to minus one,",
" and all along here is equal to plus one.",
" That's just the fact that at zero temperature,",
" even they just line up, choose the lowest energy.",
" Now, what about right at the critical point?",
" Imagine going up and down from negative temperature",
" to positive temperature just at the critical point.",
" Well, just at the critical point,",
" that's this place where the red line",
" is just barely at 45 degrees.",
" Just barely critical value.",
" So, at the critical point,",
" the magnetic position is clearly zero.",
" If you decrease the temperature just a tiny bit,",
" the magnetization changes a tiny bit.",
" If you decrease the temperature just a tiny bit,",
" that means lower in the slope here,",
" don't turn on the magnetic field,",
" but there are two branches, one just to the right,",
" one just to the left,",
" and the magnetization is tiny if the temperature is tiny.",
" So as we move vertically here,",
" at the critical temperature,",
" magnetization increases but continues with it.",
" Just above here, it's very small,",
" but positive, just below here, it's very small, but negative.",
" So as you come into this critical point,",
" the magnetization tends to zero from every direction,",
" no matter how direction, including this one,",
" the magnetization tends to zero at the critical point,",
" but once you pass the critical point,",
" there's a jump in the magnetization.",
" So, for example, if you have your magnet turned on",
" and you slowly vary the magnetic field,",
" if you slowly vary the magnetic field above the critical temperature,",
" then the magnetization gets smaller and smaller and smaller",
" and smaller, goes to zero continuously,",
" and then gets bigger and bigger and bigger,",
" completely smoothly, just as you might expect.",
" When the magnet is pointing down,",
" it makes everything tilt down,",
" when the magnet is pointing up, it tilts up up,",
" but completely continuously as you turn off the magnetic field,",
" it goes to zero, and as you turn it back on, it becomes positive.",
" We can draw a curve for that, I don't know.",
" Question?",
" Yeah.",
" I'm still not clear what's happening on the horizontal axis.",
" It jumps, pardon?",
" It jumps.",
" From OTC, what's it doing?",
" It jumps.",
" It jumps when you go across it.",
" But what's it doing, Kamek?",
" Oh, Armit does more than this.",
" Okay, so it can't set the points of the minus.",
" Now, it does one thing or the other,",
" but of course the answer is you could never really turn off all the magnetic field,",
" something is going to bias it a little bit.",
" Okay, but it's, in particular, it's not continuous there.",
" It's not continuous.",
" That's the key to what a phase transition is.",
" So, or this is technically called a second-order phase transition.",
" We don't need to distinguish them,",
" because I'm not going to get a chance to talk about first-order phase transitions.",
" Okay, so as the experimental physicist manipulates his magnetic field at fixed temperature,",
" the magnetic field varies.",
" At first, the magnetization is nice and smooth until it gets to this point,",
" when it gets over to here, it's not zero.",
" It's negative.",
" And then, a little tiny change in the magnetic field flips it over.",
" Okay, flips it over and the magnetization changes discontinuously across there.",
" That discontinuous change is a, well, it's a symptom that in a sense there are two phases.",
" Supposing you didn't know anything about this up here, let's suppose T-critical is too high a temperature",
" for our experimental physicists to ever have studied.",
" They would say there are two phases.",
" There's a phase of negative magnetization, there's a phase of positive magnetization,",
" and as you vary the parameters, it jumps.",
" It's not a hold down to all up, it's an average down and an average up.",
" Because it would be the least reasons that it was called the log.",
" No.",
" We're not going to have time to talk about this the least.",
" All right, but there's another way you can move around in space.",
" You can say, can I go from here to here without having a sharp change?",
" And of course you can. You can go from here to here going around the critical point.",
" If you go around the critical point, you experience no jump, but you do get from negative to positive.",
" So on the one hand, experimentalists down here who never got up to T-critical would say there are two phases.",
" And they're distinct, and not only are there two phases, but there's a sudden jump as you vary the magnetic field.",
" On the other hand, somebody who had more experience with this system would say, oh, well, no, there's really not a sharp difference between here and here.",
" Let me show you how to go from one to the other without the jump.",
" And both would be right. Both are correct statements.",
" There's a jump if you stay at fixed temperature.",
" If you allow yourself to explore the whole region here, you'll discover there are ways to get from here to here which don't involve a jump.",
" Smooth.",
" Another question?",
" Yeah.",
" On the part below the critical where you're going from negative to positive and there's that jump.",
" As you're getting closer and closer to the horizontal axis, say from below, are you getting closer and closer to zero?",
" Is the average getting closer and closer to zero?",
" No.",
" No, I mean it's not zero on the thing, but is it, are the actual values getting less and less or not?",
" Here?",
" Yeah.",
" No.",
" The average is staying negative.",
" I know it's negative, but is it very near zero?",
" No, it's not near zero.",
" Oh, yeah.",
" No, no, no, no, no, over here it's less.",
" Go back to this curve.",
" Yeah.",
" And now draw.",
" Now, essentially it's zero magnetic field.",
" Right.",
" So I don't need to shift the curve.",
" Yeah, okay.",
" So I have these two solutions and the two solutions don't have why near zero.",
" Well, as you get closer and closer to one to the critical, that's here.",
" That's here.",
" Oh, yeah, that's here.",
" Actually, that's here.",
" So as you approach from here, it goes to zero.",
" Okay.",
" Right.",
" But over here, that's when the temperature is...",
" Yeah, okay, got it.",
" Good.",
" Okay, excellent.",
" Okay, so now we have everything we need to know about magnets.",
" But I'm interested in the liquid gas phase transition.",
" What does that have to do with this?",
" There's a way of thinking about exactly the same system as being the transition between liquid and gas.",
" It's not good for liquid to solid.",
" That's the center.",
" Liquid solid phase transition is very different.",
" So let's talk about what this possibly can have to do with a bunch of particles.",
" Okay, let's talk about box of particles.",
" There are box of particles, about less than a fluid.",
" And the fluid is in...",
" I'll tell you what this drawing means in a minute.",
" It's a potential energy block.",
" The edges here are the edges of the walls of a box.",
" Here we're inside the box.",
" Here we're outside the box.",
" As it turns out, the box really corresponds to some...",
" some...",
" might correspond to material or whatever happens to correspond to.",
" But it's not comparable when it's inside the box is lower than the energy of the particle that's outside the box.",
" Now, we could manufacture this just by having the inside of the box and the outside of the box being in slightly different elevations, right?",
" We could have a flat box and a drop.",
" But for whatever reason, we're going to imagine that a particle and less inside the box has less energy,",
" negative energy relative to what it has outside the box.",
" That difference in energy has a name.",
" It's called the chemical potential.",
" It's usually written in U.",
" It's the amount of energy difference.",
" It's actually the amount of energy that it takes to remove the particle from the box.",
" How much energy does it take to remove the particle from the box?",
" Now, we're not going to ask what is it.",
" In this particular case, it's not important what makes the difference.",
" Why does it differ from energy?",
" We'll just take it that the energy is lower inside the box.",
" But first of all, there are particles outside the box and inside the box.",
" Lots of them all over the place.",
" And we can ask what the density is inside the box and outside the box.",
" Oh, one more thing.",
" The walls of the box are permeable.",
" Particles can come in and go out.",
" They can cross the boundaries of the box.",
" So this could just be...",
" You know, this could be a bunch of molecules.",
" You dig a hole in the ground.",
" You dig a hole in the ground.",
" And the ground...",
" And now you just deposit some particles.",
" If the particles are in the hole, they have less potential energy than if they're out of the hole.",
" Where do you think the density of particles is likely to be larger?",
" In the hole or out of the hole?",
" We're thinking thermally equilibrium now.",
" We're thinking thermally equilibrium.",
" In the hole.",
" The energy is less in the hole.",
" And so it's natural that the Boltzmann distribution always favors lower energy.",
" Always favors lower energy.",
" Remember, the Boltzmann distribution is e to the minus beta times the energy.",
" So it's always a function which decreases its energy.",
" It always prefers lower energy.",
" All right, so the answer will be that the system will adjust itself",
" so that the density in here is in a certain kind of equilibrium.",
" And the equilibrium will depend on the energy cost of taking the particle outside the box.",
" Now, we can set this up in a very definite way.",
" We can say we can do ordinary statistical mechanics.",
" And we can just say there's a potential energy which depends on position.",
" But it depends on position in a very special way.",
" It's flat completely over the box of interest.",
" And maybe jumps when you're outside the box.",
" But we can focus on inside the box.",
" And we can ask what is the density of the fluid as a function of the depth of this potential?",
" Question?",
" Yeah.",
" Is the depth as new?",
" The chemical potential.",
" It's called the chemical potential because in general it can be different",
" for different chemical molecules.",
" So you could imagine a situation where one kind of molecule prefers to be in the box",
" and the other kind of molecule doesn't prefer to be in the box.",
" And then turning on chemical potentials can change chemical compositions in various ways.",
" But we're not interested in the chemistry here.",
" We're interested just in this idea that you can tune the density by changing the chemical potential.",
" By changing the chemical potential you can change the density.",
" All the chemical potential is is a term in the energy which depends on the number of particles in the box.",
" New for each particle.",
" If there are n particles in the box then n mu.",
" If there are n minus 1 particles in the box then n minus 1 mu.",
" Now in this problem the number of particles is a variable.",
" It's a variable because particles can come into the box and out of the box.",
" That means a configuration involves not only the positions and momentum of every particle",
" but also the number of particles.",
" Now the number of particles is free to change because they can come into the system and out of the system.",
" And part of determining thermal equilibrium is determining how many particles on the average are inside the box.",
" So I'm not going to go through this in detail.",
" I'm going to tell you what to do.",
" You write the usual formula for the Cartesian function.",
" e to the minus theta times the energy.",
" You sum over all the configurations that means the momenta and the position of all of the particles.",
" But you add another term here.",
" Plus the number of particles times the chemical potential times theta.",
" Why is that there?",
" Well this is just the potential energy of the particles inside the box.",
" If there are n of them this is the energy stored inside just in the particles of beam there.",
" And you also sum over the number of particles.",
" We're not going to do that.",
" We don't have time.",
" But I'm just indicating to you what means the chemical potential.",
" It's the thing that you adjust if you want to adjust the density of the fluid.",
" You can do another thing.",
" You can seal the box and not let the particles go in and out.",
" And then just put in the number of particles if you want.",
" But you may be interested in the problem of particles where they can come into the box and out of the box.",
" And then you change the density by changing the chemical potential.",
" By changing the chemical potential you can change the average number of particles in the box.",
" What in this language what is a liquid gas transition?",
" And I will tell you what happens in a liquid gas transition is you take your box,",
" you keep it at a fixed temperature and you start varying the chemical potential.",
" The more you vary the chemical potential,",
" the more it wants to pull by, depending on the size of course,",
" you lower this energy here, the more it wants to suck particles into the box.",
" And so there's a density which is a natural function of the chemical potential.",
" And what is it that happens in a transition?",
" What happens in a transition is as you vary the chemical potential all of a sudden,",
" you hit a point where the density of the fluid suddenly changes.",
" It changes from being a gas to being a liquid.",
" So if you were to fill up this, a box of course here might mean a region of three-dimensional space.",
" I imagine it was a hole in the ground that was in two-dimensional space.",
" But imagine a box, a real box of three-dimensions, particles going in,",
" you get some extra energy out of it because there's a chemical potential.",
" Particles go out and cause energy.",
" The extra energy that you save by bringing particles into the box",
" will tend to increase the density inside the box.",
" You keep lowering the energy inside the box,",
" it keeps going to suck particles in,",
" and then all of a sudden, even though the atmosphere around the box might be at a temperature,",
" at whatever else, a temperature and density that corresponds to vapor,",
" all of a sudden you hit a transition point where it becomes liquid.",
" The difference between liquid and vapor is a sudden discontinuity in the density.",
" Liquid being denser than vapor.",
" So if you could do this experiment for the box and vary the potential energy of a particle inside the box,",
" lower the potential energy, outside conditions are such that you have steam,",
" or vapor, or vapor, and you lower the energy, particles will congregate inside the box",
" as you lower their energy, and then all of a sudden at some point",
" it will jump to the fluid things, to the liquid things.",
" That transition is very much like this transition.",
" I'm going to spell it out and spell out exactly what the relationship is.",
" Well, first of all, what are the conditions that you need to have a liquid gas phase transition?",
" You need two things.",
" You need to have a hard core retulsion between molecules.",
" Well, molecules always have hard core retulsions.",
" You cannot stick two molecules onto the same site and repellent for a number of reasons.",
" So it's natural to have a heart that molecules behave like little billiard walls.",
" Little billiard walls, they don't want to interpenetrate,",
" so roughly speaking you can't put two of them on the same site.",
" The next thing that's important is that they be a little bit of...",
" and that's a form of really hard core repulsion.",
" Just like two billiard walls, really you'd tell when you try to push them together.",
" You need something else though.",
" You need a little bit of attraction when they're not quite touching.",
" You need a little bit of attraction when they're not quite touching.",
" So you need a potential energy which is big and repulsive and positive when they're trying to get into each other.",
" And then when you separate them out a little bit, all of a sudden they want to attract a little bit.",
" That's an extremely common feature of molecular interactions.",
" Hard core repulsions and very short range attractions.",
" And that's what you need in order to have the standard liquid gas transition.",
" So I'm going to show you how the magnet produces exactly that situation.",
" Let's start with the magnet with all of its little elementary magnets down.",
" When they're all down, that's the ground state, and let's call that empty space.",
" With all the magnets down, in fact what we're going to say is something like this.",
" We're going to say on each side of the lattice, you can have a particle or not a particle.",
" This is our game. We're going to make all of the liquid gas problem, we're going to make a lattice version of it.",
" The particles and molecules live on a lattice.",
" You can have no particles at a site. Let's call that down.",
" Let's call that sigma equals down. Sigma equals minus one means no particle on that site.",
" What about sigma equals plus one?",
" We're going to take that model, guess particle on site.",
" What about two particles on site?",
" Well, the magnet doesn't allow that. Sigma can't be two.",
" It can only be minus one or one. It can't be four. It can't be anything else.",
" So immediately from the start, it forbids the possibility of two particles on site.",
" But then again, that's exactly what we're trying to do.",
" We're trying to model a gas where you can't put two particles on the same site because they have a hard core.",
" So it's built in to this system of particles.",
" If they're really described by the same kind of variable, sigma equals plus one and minus one,",
" it's automatically built in that they have an infinite hard core repulsive barrier when they try to get out on the same site.",
" So we satisfied condition number one. What about condition number two? Do they attract?",
" Now here we're going to use for the energy exactly the icing model energy plus a little magnetic field.",
" I'm going to show you what that does in the language of these fake particles.",
" Well, I'm not sure which are fake. Particles fake or magnetized fake.",
" The point is it's the same mathematical system. The particles are magnetized.",
" Let's start with this term here. Minus J times sigma i times sigma j,",
" where these are neighboring sites on the last sum over all the links.",
" How much energy is there if there are no particles, whatever?",
" Well, that's the situation where all signals are down.",
" Then there is an energy minus j for each link.",
" So there's total grand state energy, which is minus j times the number of links.",
" What do we do with that? Well, they don't do anything with it.",
" That's just the energy of the system when there's no particles at all.",
" Does it matter for anything? No, it doesn't. It's always there.",
" If the grand state energy, we can throw it away.",
" And we can throw it away just by subtracting minus the number of links.",
" I think this times 2j.",
" Now that I've thrown it away, let's really throw it away and ignore it.",
" Because it plays no role in anything. It's just a number.",
" And energy differences are the important things.",
" Alright, so we have this term in the energy.",
" No particles, that gives us certain energy. Let's call it zero.",
" Let's make the zero of energy when there are no particles.",
" Now what happens if I put in one particle?",
" That means flip one of the spins over.",
" It means all the spins are down.",
" If we throw a little down, we take one and flip it up.",
" That's mathematically equivalent to putting one particle somewhere.",
" That's the energy that we get. Let's calculate.",
" The energy that we get, how many bonds have we broken?",
" Depends on the dimensionality.",
" Let's do two dimensions because it's easier to visualize.",
" If we flip this spin over, then we break four bonds.",
" So we get a total amount of energy equal to 2j for each bond.",
" 8j, 8j.",
" Yes, 8j.",
" So one particle has energy 8j.",
" That's how much energy we get at a cost to create a particle at that point.",
" What about two particles?",
" If one particle is over here, then we get a total amount of energy.",
" That's the total amount of energy that we get at a point.",
" That's the total amount of energy that we get at a point.",
" What about two particles if one particle is over here and one particle is over here?",
" 16j.",
" 16j.",
" Two particles, 16j.",
" But incidentally, the same is true if the two particles are over here.",
" Let's just check that.",
" Let's see how many bonds we have to break if they're on the corners of the diagonal.",
" One, two, three, four, five, six, seven, eight.",
" Yes, so even if they're on diagonals, you still pay a price of 16j.",
" What if they're even closer?",
" What if they're even closer like that?",
" Then how many bonds do you break?",
" One less.",
" Two less.",
" This bond is not broken.",
" Why not?",
" Because they're both close.",
" That bond is not broken, so we broken six bonds.",
" So two particles close together.",
" Two close particles, and close now means within one bond length away.",
" Where they have, they have 12j.",
" So let me suppose I plotted the energy as a function of difference.",
" I would find, yes, just the mere act of putting particles in cost me 16j.",
" But that's all right. That energy is always there no matter where I move the particles around.",
" But if I move them within a bond length, the energy decreases.",
" The energy decreases, that's like a potential energy which decreases when they get very close together.",
" So there's a potential energy, an energy which depends on the position of the particles.",
" And when the particles get within one bond length of each other, the energy decreases by 2j.",
" 2j and 4j.",
" 4j, or the energy decreases by 4j.",
" So let's look at having a short range attractive energy where when the particles are close together,",
" the energy is negative relative to what it would be in their far apart.",
" Negative because, negative relative to what they would be when they're far apart.",
" So the particles attract. They like to be close together.",
" How close together? One bond length away.",
" But it's not an overwhelming attraction.",
" It doesn't say the energy is infinitely negative if you put them next to each other.",
" This is the modest amount of attraction, a modest saving of energy of putting them close together.",
" And it's very much like the molecular attraction of a pair of molecules when they get close together.",
" So we have a system now which is mathematically isomorphic to a system of particles on a lattice",
" which have an attractive force between them where you can have any number of particles.",
" The particle number is something that can change.",
" So it's like the system of molecules where molecules can come into and out of the system.",
" It doesn't have a definite number of molecules.",
" The number of molecules itself is a variable.",
" Short range potential, short range attraction, excuse me.",
" Very short range, infinite repulsion.",
" You cannot put two of them on the same site.",
" But when they get a little bit apart, they're slightly attractive.",
" Exactly what you need for a liquid gas phase transition.",
" Now let's add something else.",
" Does this system have a chemical potential, incidentally?",
" A chemical potential is the energy stored in just having one particle.",
" That's what it is.",
" A chemical potential is just the energy in having a particle present.",
" Just by virtue of having a particle, if there's an energy which wouldn't be there if the particle weren't there,",
" that is called the chemical potential.",
" Well, yes, just having a particle with no other particles around you gives you an energy, a genuine.",
" So yes, there is a chemical potential.",
" But I want to be able to vary that chemical potential.",
" I would like to be able to vary that without varying J, incidentally.",
" I'd like to hold J fixed.",
" I want to be able to vary the chemical potential separately from J.",
" That's easy to do.",
" Put a magnetic field here.",
" Now this is sum over the science.",
" Sum over the science of sigma i.",
" How much does this give me when I add a particle to the brew?",
" Sigma starts down when there's no particle.",
" When I put a particle in a sigma, it becomes plus.",
" So it jumps two units.",
" It gives me two units worth of energy for every spin which is flipped out.",
" For every particle that you put in, it gives you twice h.",
" So you can add for each isolated particle, you can add plus two h.",
" So now we have everything we need.",
" We have a system that is equivalent to a collection of particles on lattice.",
" It has a variable chemical potential which notice how we vary it.",
" We vary it by varying the magnetic field.",
" Magnetic field gave us an energy per particle.",
" And we have a short range attraction.",
" The short range attraction, not the hard core billiard ball potential,",
" but then a short range attraction, the coefficient of which is 2J.",
" 4J.",
" So I want to keep the molecular properties fixed.",
" So I'm not going to play with J.",
" That has to do with the molecular properties that design the cellular potential energy between them.",
" What I can vary is the chemical potential.",
" And mathematically that seems to be the same thing as varying h.",
" Very h.",
" Very h.",
" So this problem is exactly the same as the erasured, the erasured.",
" No, there it is.",
" It is exactly this problem here.",
" Precisely.",
" There's a temperature that we can vary.",
" Particles are at some temperature.",
" There is a chemical potential we can vary.",
" Now h is not exactly the chemical potential.",
" There's an offset by an amount of 8J.",
" What about the particle density?",
" Let's talk about the particle density.",
" We haven't discussed the particle density.",
" Okay.",
" How many particles are there on the average at a point?",
" At a particular point.",
" Well, I say the answer is somewhere, first of all, between zero and one.",
" I say the number of particles at a point.",
" Of course, it seems zero or one.",
" But we're talking about the average.",
" The average number of particles.",
" The average number of particles is one plus sigma divided by two.",
" Let's see why that's true.",
" If there is a particle there, that means sigma is one, right?",
" So one plus one is two divided by two is one.",
" What if there's no particle there?",
" Zero.",
" That's a good candidate for the number of particles at a point.",
" It is truly the average number of particles at a point",
" is equal to one half plus the average of sigma.",
" The average number of particles at a point is one plus the average of sigma.",
" That tells us that the density of particles, let's call it rho,",
" is proportionately from one plus sigma divided by two.",
" That's the density of particles.",
" In other words, the number of particles per lattice cell.",
" The average number of particles per lattice cell is one or sigma bar divided by two.",
" The average number of particles is not sigma divided by two.",
" It's not sigma bar.",
" It's one plus sigma bar divided by two.",
" That's because of this offset that sigma bar being down means no particles,",
" not minus one.",
" It doesn't say it's minus one particle.",
" So we offset it by half here.",
" Excuse me.",
" Didn't you say sigma bar could be a value between minus one and one?",
" Yes.",
" Which says rho goes between zero and one.",
" Right?",
" It's not more than one particle on a cell because they have sharp elbows",
" and they push each other out of the way.",
" We can't have less than one particle because that doesn't mean anything.",
" So it goes between zero and one.",
" And so the density is just one half plus the average magnetization.",
" What does that say? Half plus half the average magnetization.",
" Half plus half the average magnetization is the density of this fluid.",
" So now we're all set to say what happens if we were to vary the chemical potential,",
" vary the H, keeping the temperature fixed.",
" If we're above the critical temperature, the magnetization varies uniformly",
" and continuously how jump?",
" That says the density doesn't jump.",
" The density of particles.",
" So above the critical temperature, as we vary the chemical potential",
" in a box of gas, lower the energy, put it down,",
" nothing very exciting happens when we pass this horizontal axis.",
" In other words, the horizontal axis is when the chemical potential",
" plus the clever H plus this is equal.",
" The horizontal axis is when H is equal to zero,",
" corresponds to the chemical potential of Hj plus 2h, whatever.",
" No sudden jump at anything above the critical point.",
" Below the critical point by contrast, as you vary the chemical potential H,",
" all of a sudden at this point here the magnetization jumps.",
" It jumps from negative.",
" When it's negative, that means the density is low.",
" The density is low when this is negative.",
" So it jumps and it jumps from low density to high density.",
" Experimentals didn't have temperatures available as big as key critical,",
" which was key critical for water.",
" No.",
" No, no, no, no.",
" No.",
" That's a... I shouldn't have asked that question.",
" That's pressure dependent, which means chemical potential dependent.",
" No, the critical temperature would be a higher than that.",
" And it's a good deal higher than that.",
" You see, going across here is the phenomenon of boiling.",
" Or going down this way is the phenomenon of boiling.",
" Here's where boiling happens.",
" Boiling doesn't happen out of here.",
" It's just a smooth, continuous transition of the density when you're at high temperature,",
" high enough temperature above the critical point, below the critical point.",
" You're here.",
" Are you referring to superheated steam above that?",
" You could.",
" Yeah.",
" Question, please. How do you vary the chemical potential?",
" We understand magnetic, but in the other case, how do you do it?",
" Really, by varying the density.",
" You can do it. You can either vary the density.",
" You see, the density jumps across here, but how do you vary the chemical potential?",
" Pressure?",
" Yeah, well, that's one way.",
" So, we have a chemical potential for water molecules.",
" A little tricky.",
" Whatever takes the very density.",
" So, I like to think about it in a thought experiment by doing what I stress,",
" which is to create a box.",
" And then, literally pull the molecules into the box.",
" Literally pull the molecules into the box, and pull the molecules into the box",
" by creating a potential energy inside the box, which is what I have to think about how you would do that in practice.",
" It's not actually what you do. What you do instead is very good pressure.",
" And a varying chemical potential is the simplest thing to do.",
" What you do is, you suddenly find a sudden jump in the density of the fluid.",
" And that is what happens.",
" With a different gas phase transition, down here is gas, low density, up here is liquid.",
" But notice that you can go from gas to liquid without any jump by going around the critical point,",
" way above the critical temperature, and varying whatever you have to vary the density.",
" You could vary the density by varying the pressure. That's actually the easiest way.",
" You vary the pressure, you increase the temperature, vary the pressure,",
" increase the temperature and the pressure, making sure you go above the critical point, the critical temperature,",
" and then lower the temperature back, and you'll go from gas to liquid without any jump.",
" Question?",
" Yeah.",
" When you say you're changing the chemical potential by changing the density,",
" you're basically changing your effective density for a fabulous wall potential?",
" No, no, no. The property of the molecule is a stress.",
" But the density says the random wall potential is not constant.",
" Oh, it's changing the position now.",
" So when you change the density, you're changing the distance between molecules?",
" So in that sense you're changing the potential energy.",
" Right.",
" So when you change the density enough, it suddenly makes a transition,",
" and realizes that the potential, that the negative potential energy is there,",
" and tries to pull all the molecules together.",
" What's the relationship between, in this case, the gas-like transition",
" between the critical temperature and the boiling point?",
" Well, the boiling point depends on the pressure,",
" which in turn means it depends on the density.",
" Let's see, how do we say this?",
" The problem with this is I haven't worked out the pressure.",
" What we could do, the pressure is a function of temperature and chemical potential.",
" And if I were to have worked out the pressure, we could have varied the pressure",
" instead of the count of the potential.",
" That's why I'm having a problem, because I don't want to go through the effort",
" now of calculating the pressure.",
" But just in a simple mind of mine, when I'm boiling water,",
" what's my path along there from liquid to gas or gas to liquids?",
" What do I say over here?",
" We're going from liquid to gas or gas to liquid.",
" That doesn't matter.",
" You go across here, but the point at which you go across there depends on the pressure.",
" Sure, I understand.",
" So the point at which you cross does depend on the pressure.",
" So there's another variable in here.",
" It's not entertaining.",
" OK.",
" One of the very fascinating things is the properties of the system",
" near the critical point there.",
" There's a whole theory of the behavior of the clear.",
" A whole theory of lots of experiments as about what happens",
" as you approach the critical temperature.",
" As you approach the critical temperature, the critical point,",
" as you approach the critical point,",
" a number of fascinating things happen,",
" but they're all characterized by what are called critical exponents.",
" Every quantity that you can think of that's interesting,",
" and its dependence on temperature,",
" will generally go as T minus T critical",
" to some power.",
" And those powers, they're not ones and twos.",
" There are various kinds of irrational numbers of things,",
" transcendental numbers.",
" They're called the critical exponents.",
" And of course they vary from one kind of phase transition to another,",
" but they're exactly the same for the magnetic transitions",
" and the liquid gas transitions.",
" They fall into exactly the same class.",
" The behaviors near these critical points are rather insensitive",
" to the details, and they depend on features that don't care",
" whether you put them on a lattice,",
" they don't care whether it's the nearest neighbor",
" or a second nearest neighbor, they all behave the same way.",
" And the magnets and the fluid transitions are in the same class.",
" Okay, I think we're finished.",
" Do they form a continuum or is it a discrete,",
" countable, even finite?",
" Oh, the set of possible critical points?",
" Yeah.",
" Because they're discrete.",
" Finan?",
" No, they're probably not finite.",
" I don't think they're finite.",
" The number of critical points is not finite,",
" but they're all discrete.",
" Question?",
" First of all, how do you define a gas number?",
" Since you go around and have no choice,",
" they seem to be the same.",
" Is it density?",
" There's no sharply defined...",
" The density is obviously different here than it is here.",
" And if you try to go across here, the density jumps.",
" If you go around the critical point, the density varies continuously.",
" But it's the density then that defines the gas number.",
" Is that right?",
" The density is like the magnetization.",
" Or the density here, this is the connection.",
" If you go to low temperature, the density jumps.",
" If you go to high temperature, the density doesn't jump.",
" Then you can go from here to here going around the critical point.",
" So they coexist.",
" Liquid and gas coexist if you go around.",
" Well, they're not distinguishable.",
" They're continuously...",
" They're continuous transition from one to the other.",
" What do you need to put into the theory to get a gas liquid in your face?",
" A gas...",
" To have a gas liquid in your face,",
" you have some gravitational potential energy or something to be separated.",
" It's not surface tension.",
" It's not surface tension.",
" It could be surface tension.",
" Most of this kind of analysis is what you think about for an infinite volume of fluid.",
" So surface tension is not really the thing here.",
" But you could have an infinite volume of fluid laid out on an infinite plane in a gravitational field,",
" and then you would create a liquid gas transition in some point.",
" That could be worked out.",
" Basically, the altitude is related to the chemical potential.",
" So varying the altitude varies the chemical potential.",
" And a sudden transition at some height, which is connected to this transition.",
" In this model, you get a heat of vaporization as you go across that transition.",
" In general, yes. No doubt.",
" I would have to think about what that means in this model.",
" The energy down here is exactly the same as the energy just above.",
" I have to think about that.",
" So...",
" Is there a heat release at that point?",
" Total energy is conserved, of course.",
" Yeah, but there's a certain magnetic problem.",
" The energy above and the energy below is just exactly the same.",
" It's a good question.",
" I don't know if the answer is okay.",
" I have to think about it.",
" That's getting me...",
" Does this have implications for superconducting magnets?",
" I suppose it does, but I'm not sure what.",
" What's the question?",
" You talked about the Germanian detectors at Soudre and Main for the dark matter detectors.",
" And how the physical interaction creates a point where it kicks over.",
" Where the transition happens between superconductor and ordinary.",
" Right. Yes.",
" Oh, you're asking...",
" Is that a phase transition like that?",
" It's a phase transition.",
" It has similarities.",
" And the point there is if you can adjust your superconductor to be very near the transition,",
" or very near the phase transition, let's say across here,",
" by a tiny change in subparameter, you can have it jump from superconducting to non-superconducting.",
" And that gives you the possibility.",
" Some energy is deposited in the detector,",
" and even a small amount of energy can create a little local pocket of a changed phase.",
" So it can be useful as a very sensitive kind of detector.",
" But I don't know a lot about superconductor detectors.",
" If you're very close to the phase transition,",
" and something really jumps,",
" then the presence of a phase transition can be a very, very good detector of very small changes.",
" Any phase transition?",
" Very good transition.",
" Okay, we're finished with statistical mechanics.",
" We've done our duty to...",
" Both of us.",
" I think.",
" I think.",
" I keep making the same mistake about I think.",
" I went back and looked at my notes, and I realized with my notes,",
" I had an apology for the students,",
" and they're wrong thing about I think exactly the same wrong thing.",
" I'll probably do it again.",
" Okay, I said I would talk about the improper principle.",
" What do you do with this question and answer?",
" If you want to ask me questions about the improper principle,",
" perhaps you will get me going and I will answer them.",
" But I've sort of run out of steam.",
" So...",
" No cookies.",
" No cookies. It's not my fault.",
" There's no coffee.",
" But who asked me about the improper principle again?",
" That's good.",
" Well, go ahead, shoot the question.",
" I wish I turned it off the email.",
" It was just basically about the generalities of it.",
" The applications or misapplications, the implications, the quality.",
" Well, I asked you a preceding question.",
" Could you give a clear, if there is such a thing,",
" the crisp statement of the improper principle?",
" Exactly what does it say?",
" Can I give a crisp?",
" No, it means many things to many people.",
" But I can tell you how I think a rational use of it looks.",
" I don't say it's right, but I think it can, in some circumstances,",
" be a rational explanation of something.",
" First, you have to understand what fine tuning means.",
" Okay, let's talk about fine tuning.",
" There's a small number of physics.",
" The classic case of an extreme fine tuning is, of course, the cosmological constant.",
" The cosmological constant is very small.",
" In natural units, in punk units, it's 153 dB.",
" It's small.",
" What's the difference between being small and being fine tuned?",
" They sound like the same thing, but they're not the same thing.",
" So I'll give you an example.",
" It's an example that I cropped up to explain this once to a bunch of condensed matter physicists",
" who didn't know what fine tuning meant.",
" As soon as I told them what it meant, they said, oh, we know that.",
" But let me explain it in terms of an example.",
" This is a silly example.",
" This is the only example I've ever made up like this.",
" This doesn't have fish in it.",
" But it does have submersible submarines.",
" And also balloons.",
" Okay.",
" There's a difference.",
" Let's suppose that you were a certain kind of creature that could only exist at a certain altitude where the air was just the right temperature,",
" not just temperature density and so forth.",
" So you find yourself at that temperature density and so forth.",
" And a balloon.",
" A balloon in the ground of a dirigible.",
" And a zeppelin.",
" A zeppelin of dirigible.",
" Branding.",
" A zeppelin, I think, is a big bag of gas.",
" A dirigible, I think, has some supporting structure.",
" A zeppelin of dirigible.",
" So a zeppelin is a brand name.",
" So there you are, you're talking about air in your balloon.",
" And you notice you're neither rising nor falling.",
" You see that wonderful? You're neither rising nor falling.",
" You're not that lucky.",
" You're at some stationary height.",
" And that stationary height you've been there for several billion years.",
" Long enough to be devolved and you're devolved in just the right way to be able to live at that temperature and at that behavior and so forth.",
" You see, isn't it very lucky that this blimp happens to be exceptionally light?",
" Light enough that it floats.",
" Okay.",
" Yes, you're lucky.",
" The blimp happened to be light enough that it floats.",
" The small number was the density of the material inside the blimp.",
" And that number is small.",
" In this case, it's not horribly small, but it's small.",
" It might be the density of hydrogen.",
" It might be the density of helium.",
" But notice there's a factor of what?",
" Four difference in the mass of helium and the mass of...",
" So the density could be quite different.",
" And still you float somewhere.",
" Your helium will support you, your hydrogen will support you, you're okay.",
" And you'll come to equilibrium at some position.",
" The density of the air varies and you'll find some equilibrium and you'll just hover at that equilibrium.",
" And you stay there forever and ever, long enough to evolve your species.",
" And that's an example of a small number.",
" A small number in this case being the luck that your system happens to have a low density.",
" In fact, lower than air.",
" Smaller.",
" No fine tuning.",
" It didn't matter if it was helium or hydrogen.",
" And it wouldn't matter if it was an equal mixture of helium or hydrogen.",
" You could have changed the parameters of the gas quite a lot.",
" But no hundred percent you could have changed the... as long as it stays lighter than air.",
" So there you're in luck that there was a small number, but there was no exceptional fine tuning of anything.",
" You didn't have to fine tune in any detail percentage of hydrogen and helium.",
" Okay.",
" So that's a small number.",
" Now let's talk about another situation.",
" This object is now not a blimp, it is a submarine.",
" Yellow submarine.",
" We all live in the yellow submarine.",
" Blue.",
" Submarine has no motor, it explodes.",
" Down here, pressure is much too great for us to survive.",
" As it happens, the atmosphere is highly poisonous and we get the float to the top we're dead.",
" We have to stay in the water.",
" Maybe it doesn't matter so much how high we are in the water, but we've got to stay there.",
" Our submarine has existed in that configuration long enough for us to evolve.",
" In other words, for a billion years it has been hovering in the water without falling or without rising.",
" We know that because we're here.",
" Or these people who live in a submarine are here and they know that if there was any significant tendency to fall,",
" they'd be dead, if there was any significant tendency to rise, they'd be dead.",
" Let's assume for simplicity that the water has the same density from the bottom of the sea to the top of the sea,",
" which is very possible to be true.",
" What do these people in the yellow submarine conclude?",
" They conclude that whatever the submarine is made out of, the density of that submarine is finely tuned.",
" It's made out of some iron together with some brass and maybe a little bit of concrete and a few other things.",
" Plus it's hollowed out in the interior.",
" Its average density is such that its average density is very, very close to that of water.",
" If there was any density of water, it would sink.",
" If there was any lighter than water, it would flow.",
" Once more, it's been sitting there at that compromised position for a billion years.",
" From that you can conclude that whatever the chemical composition is, it is very, very finely tuned.",
" If you would change the ratio of iron to concrete by one part in a thousand, or probably one part in a million,",
" remember this thing has been sitting there for a billion years, or let's say four billion years, or ten billion years.",
" It's been sitting there for ten billion years and it's right somewhere in the middle of the zone.",
" That's extraordinary because there's no feedback mechanism whereby it could start to sink.",
" It would rise, it would start to rise, it would sink.",
" No feedback mechanism, it's just a little bit too heavy.",
" It would just slowly sink to the bottom, it's a little too light, it will slowly rise.",
" So somebody very, very fortunately has tuned the chemical composition and the amount of hollowing out",
" and the weight of everything that's inside it to one part, and I don't know how much, we can try to work it out,",
" to one part in a fairly large number to make sure that that submarine hovers.",
" Now the submarineers are curious about this, they want to know why this is true.",
" They have no explanation for it at all.",
" Their first reaction is this must be something like the dirigible, the Zeppelin.",
" The Zeppelin had a feedback.",
" If the Zeppelin goes to high, the air density goes down and it sinks back.",
" It goes to low, it rises up again.",
" But they do a few calculations and they realize there's no question of that kind of feedback mechanism.",
" The density of water is too constant over this range and there's no possibility of using that against the stability mechanism.",
" They try all kinds of mathematical tricks, symmetry principles, whatever you have, nothing works.",
" The reason nothing works is because this is one of these very fine-tuned things, if you just change the chemical composition a tiny bit this way",
" or a tiny bit that way, the whole thing breaks down.",
" What kind of conclusion could they come to?",
" They do a little more calculation using whatever laws they go and they discover something very interesting about the nature of the fluid that they're in.",
" The nature of the fluid that they're in is that it has an instability.",
" It makes bubbles and the form of these bubbles is submarines.",
" Submarine is in the fluid.",
" Bubbles.",
" And when the bubbles nucleate, they morph into submarines.",
" But the trouble is the submarines that they can nucleate into are generally not of the right kind.",
" In fact, as it turns out, the submarines can nucleate with different chemical compositions.",
" Some of them have a little more brass, some of them have a little more iron, a few of them have more hollowed out.",
" There's a whole huge variety of different species of submarine that can nucleate.",
" They do it. They make these calculations and they discover this is a property of the fluid that they live in.",
" They say, ah, they can know the answer.",
" The answer must be that there must be zillions of different kinds of submarines, huge, huge numbers of different kinds,",
" and moreover, they keep nucleating and nucleating and nucleating this fluid.",
" So the fluid just keeps producing more and more and more of them of every conceivable type,",
" 10% iron, 90% brass, and all possible things in between.",
" And most of them either sink to the bottom, they lie and dead at the bottom,",
" or they rise to the top and they poison the anatomy, the proto-habitants,",
" that I had time to, there's a tiny fraction of them, some incredibly tiny fraction of them were nucleated with the right properties.",
" They were nucleated with the right properties just because everything that could happen happened.",
" And so they are not particularly lucky, they are just the ones that work in the kind of submarines that nucleated with exactly the right density to hover.",
" If all of that really happened, and these people really had a genuine scientific reason to believe in submarine nucleation in their city,",
" and they could see that the submarines could nucleate with a vast variety of different types,",
" they would have what I would say was a reasonable explanation of what's going on.",
" There's just a lot of stuff, the sea is very big, the sea is very big, a long time has passed,",
" every conceivable kind of thing is nucleated many, many times, and they are just in the kind of where survival is possible.",
" That is the logic that I think the anthropic principle, when it's rational, is making use of.",
" That fine-tuning is a consequence of many, many, many possibilities, and many actual events taking place, creating the submarine that we live in.",
" So many so that some fraction of them will be livable.",
" Then it becomes a non-mystery about why we live in a fine-tuned universe.",
" That's a logic that I find personally, and I find acceptable.",
" There are other possible answers.",
" The other possible answer is that I was a benevolence that created the submarine with exactly the right properties.",
" There are some people who would like to believe that.",
" Most of my scientific friends don't think that is an acceptable solution.",
" So what are the other possibilities?",
" At this time, right now, here and now, there are no other plausible possibilities for why certain things are fine-tuned.",
" Now what is it that is playing the role of this density of the submarine?",
" What's playing the role of the density of the submarine is the cosmological constant.",
" If the cosmological constant is slightly negative, the universe would re-collapse.",
" Now, if it's sufficiently small, that's like the density of the submarine being only a tiny bit different than water.",
" If the cosmological constant is very small, then it will take a long time for it to re-collapse.",
" In the same sense, a very, very similar way to the way it would take a long time for the submarine to fall to the bottom of the sea.",
" If the difference between water density and submarine density was small.",
" So if the cosmological constant is small enough, the universe lasts long enough, assuming it's negative,",
" it lasts long enough without re-collapsing for us to be here and to survive and so forth.",
" On the other hand, if the cosmological constant could be positive, in which case the universe doesn't crunch,",
" it expands and it expands in an exponential way.",
" But if it expands too fast, it expands too rapidly, then it prevents structures from forming.",
" If the universe was accelerating very rapidly and accelerated expansion, then the material that's in it would carry the bone with the acceleration",
" and simply be incapable of contracting into galaxies, stars, and so forth.",
" So the sum window, which is analogous to the window of densities, of submarine densities,",
" where either positive or negative, sum in a window where the submarine could last for a billion years.",
" Here, it's a window between collapse and destruction on a time scale that would not allow our existence, on the one hand,",
" and in the other direction, lack of galaxies, stars, planets, structures would form.",
" And that window is very, very small.",
" The window is small because time scales are so long.",
" The same reason the window is small for the submarine.",
" We have to be able to survive in this universe for a period of time, which is very long.",
" And that means that the imbalances in the cosmological constant have to be very, very small.",
" Now, there's another sense in which the cosmological constant is like the density of the submarine.",
" The density of the submarine is made up out of some composite of the densities of different materials that make it up in various proportions.",
" It's not just that some number is small, it's that some number is very, very finely adjusted.",
" The same is true of the cosmological constant in the quantum field theory.",
" It gets contributions from many, many different sources.",
" Basically, it gets a contribution from every quantum field that exists.",
" Bosons give positive, fermions give negative.",
" The mass of the particles that are associated with those quantum fields shift them a little bit.",
" And the cosmological constant that results from it shifts a little bit.",
" And the whole thing, all of the constants, all of the particles, all of the contributions to the cosmological constant",
" have to balance very finely, exactly the same way as the density of the submarine has to balance.",
" For the little too much iron, you're dead. For the little too much brass, you go up.",
" Same thing with the cosmological constant.",
" Put in an extra fermion species in the theory and the cosmological constant gets too negative and you implode.",
" Put in an extra boson, cosmological constant gets too big and you're out of luck and a tiny fraction of a second.",
" So...",
" Question please. What is the meaning of making a change but that compensates?",
" Meaning it doesn't... it's just a different submarine but still floating.",
" Yeah.",
" There are parameters in physics such as various masses, coupling constants.",
" You could change one thing one way a little bit which might increase the cosmological constant",
" but change something else another way a little bit which might decrease the cosmological constant",
" and you might be able to...",
" It's not just that there's only one chemical composition of the submarine that can survive.",
" It's just a very thin surface in some space of different configurations.",
" That's the same thing here. You could change things but you have to change them in a very, very fine tuned way.",
" You have to change one thing by this amount, you have to change the other thing by that amount",
" and it has to be adjusted to 123 fescal places.",
" So that's...",
" So will this water that produces every kind of submarine will make various submarines that have the same density",
" but look different than this? So these are alternate universes?",
" Yeah. Well they're all alternate universes but some of them are capable of sustaining themselves for a long enough time.",
" Yeah.",
" People, others are not. Yeah.",
" You know, we know very little about this. Our knowledge of space of possibilities is extremely limited.",
" We don't know very much about it.",
" String theory seems to say that there are endless huge numbers of possibilities analogous to every possible chemical composition.",
" So that's a plus.",
" I don't know whether to regard that as a plus for string theory or a plus for the enthrotic principle or a plus for cosmology.",
" But it does seem to fit together with the idea.",
" And the fact is there is no other known explanation.",
" There's nothing out there that many years now that people have looked for explanations.",
" Oh, there's one more thing. Just a moment. There's one more thing about the submarine.",
" Yeah. These submarines might, as I said, they might start looking for mathematical explanations.",
" Some symmetry of nature, some symmetry of their equations, which makes the submarine density exactly equal to the surrounding water.",
" They think they have this theory called Roke theory or something.",
" I don't know.",
" Roke theory seems, they can't quite prove it.",
" But they have some idea that they only have the right idea. The Roke theory would say that the submarines always come out with exactly the right density of water.",
" Some symmetry between water and other things.",
" We've got it. We don't quite understand it, but we've got it. We know there's some theory there.",
" We could only understand that theory better. It would tell us why the cosmological constant or the density of submarines is exactly the right thing.",
" But then, that their horror and dismay, they discover that in their own submarine it's not exactly the right thing.",
" In fact, the submarine is slowly sinking.",
" It really is slowly sinking.",
" How long does it take for it to sink? More than 10 billion years.",
" But their finest measurements tell them, no, it's not that things are perfectly adjusted.",
" We'll have to explain why they're perfectly adjusted.",
" They're not perfectly adjusted. They're just adjusted to 123 decimal places.",
" And at 124th decimal place, they're sinking.",
" Excuse me. Isn't this precision violating the uncertainty principle?",
" One thing can be as accurate as you like.",
" Oh, yes.",
" Question?",
" When you gave this explanation, you said this is an explanation of the anthropic principle that I'm comfortable with.",
" I wouldn't say it's an explanation. I would say it's a use of the anthropic principle that I'm comfortable with.",
" What are some other uses that you are not comfortable with?",
" Look, I have no idea if there was an intelligence benevolent or not that created the universe.",
" I do not know and I don't pretend to know.",
" So are you saying that others sort of give that spin on things a little bit?",
" I suppose not many scientists do.",
" Yeah.",
" At least at one time.",
" Okay, there are other uses of it which I find much less compelling.",
" There are all sorts of modest coincidences.",
" If any of them were not satisfied, they might be enough to destroy life on Earth.",
" If we were to make a list of the elementary particles, we would find that almost all of the known ones,",
" almost all of them are necessary for our own existence.",
" On the other hand, our understanding of particle physics would allow just about any one of them to not be there",
" or other ones to be there.",
" For example, I mean, stop thinking about the particles you know about.",
" Photon is the lightest. Where would it be without photon?",
" There would be no light.",
" There would be heat from the sun.",
" The worst than that, there would be no photons jumping back and forth between the atomic nucleus and the electron to hold it together.",
" No chemistry.",
" It wouldn't be here.",
" What about no neutrinos?",
" No neutrinos, there would be no nuclear reactions of a kind which created the elements from the particle of the Big Bang.",
" So the existence of carbon and all these other elements that make us up wouldn't be here without neutrinos.",
" What about no electrons?",
" Well, no electrons is a clear disaster.",
" There's no question of that.",
" What about no quarks?",
" No quarks, no protons, neutrons?",
" What if you change the parameters a little bit?",
" What if you...it's a...it was a puzzle for many years and still is a puzzle.",
" Why the proton is lighter than the neutron?",
" In the early days of nuclear physics, this was a great puzzle because the electron is charged.",
" You would have thought its electrostatic energy would make it a little bit heavier than the neutron.",
" Neutrons and protons are twin...twins.",
" They are almost exactly the same with the exception that the proton has a little bit of positive electric charge.",
" Ordinarily, you would say that increases its energy.",
" It takes energy to pull together a charge.",
" It's an increased energy.",
" Protons should be a little bit heavier than a neutron.",
" Or it would be a disaster because if the proton was heavier than the neutron, the proton would decay into neutrons.",
" Instead of the neutron decaying into protons.",
" If the proton decayed into neutrons, there would be no hydrogen.",
" There would be nothing.",
" We wouldn't be here.",
" So, a little tiny difference between proton and neutron is small,",
" which happens to go the wrong way from expectations is absolutely essential to our own existence.",
" All sorts of other things.",
" What would happen if the electromagnetic coupling constant, the fine structure constant, which is well over 137,",
" what would happen if it was one over 10 instead of one over 137?",
" Well, then the atom would be very, very tightly bound.",
" And ordinary chemistry would probably go awry.",
" And what would happen if the electron were heavier than the proton?",
" It would be very deep inside the center of the atom instead of having a valence structure.",
" So, you can look at physics as we know it, which means largely particle physics.",
" And you can say, you know, there's so many different things that you could have changed by 10, 20%,",
" or even 1% here and there, and any one of them would have destroyed our existence.",
" I can easily imagine 1% accidents.",
" I can even imagine 10% 1% accidents, although it gets a little harder.",
" Are those accidents sufficient to really push us in the anthropic direction?",
" I'm not going to answer that. I don't have an answer for that.",
" That's a matter of taste of it.",
" But the accidents of one part in 10 to the 123, or in some other case, one part in 10 to the 60,",
" or something else having to do with the Higgs boson, 10 to the 30, I'm sorry, one part in 10 to the 30,",
" not the kind of accidents that you can imagine are accidental.",
" One part in 10 to the 20.",
" So, I think if it wasn't for the cosmological constant and the other fine tuning, comparable fine tuning,",
" I think we would not be, you know, most serious physicists would be rejecting the anthropic explanations.",
" Maybe not completely, but not the same support that exists.",
" That doesn't mean that every physicist approves of it.",
" They just don't have another explanation.",
" Spine, heart, and some other guys in April had a sort of inflationary paradigm and trope-reactor plot in 2013.",
" Have you seen what they said or have anything to say about it?",
" They said basically most of the inflationary models don't work.",
" Who said that?",
" Paul Jay, Spine, heart, Abraham, Lou.",
" Who?",
" Abraham, Lou, Elo, and the...",
" Okay, these are counter-areas.",
" The vast, vast, vast, overwhelming majority of cosmologists and physicists believe in inflation.",
" Now, the other point of the point, they said that in the plot of 2013,",
" there's ulcer in and they talk down most of the models.",
" Well, they didn't model or work.",
" Sure, people make up a lot of wild models, and then the good data comes in and knocks out most of the wild models",
" and leaves the sensible ones.",
" Nothing is done.",
" Right, you have 500 models.",
" The time is the standard, the gold standard model, and data comes in and knocks out 499.",
" Unfortunately, it's not that good.",
" Yeah, no, this is...",
" No, no, no, no, there are people with axes that are with the ground.",
" Physics is not a highly politicized subject.",
" It does have people in it who... they're very highly politicized about their own ideas.",
" And going around saying things like that, I think that's irresponsible, frankly.",
" I would consider that irresponsible.",
" I do consider it irresponsible.",
" All this time, the... is not the most responsible cosmologist.",
" Sorry for that.",
" That's true.",
" Would it also be possible to develop a logical concept of variance sometimes, perhaps, over time, so instead of having like...",
" There's no good theory in which it varies gradually.",
" It can vary in a jump.",
" It can vary in a jump if one of these submarines nucleates with a different value of it. Yes, it can. It can jump.",
" There's no good theory in which it varies smoothly over such a long time scale, slow enough to be consistent with data.",
" That was another theory, a very, very slowly varying cosmological constant, but doesn't seem to fit with anything we know.",
" In any case, it would require... it turns out that to have a slowly varying version of it with a second stand to zero",
" requires a fine tuning of several parameters, not just one.",
" Not just the vacuum potential energy, but several of its derivatives have to be fine tuned.",
" So it's not a better explanation.",
" Look, we don't know what... we don't know where this is going to go.",
" Nobody knows at present where it's going to go.",
" There are contrairings. There are people who have political agendas.",
" I don't. I don't really. I don't... I'm not any better at inflation.",
" I don't have a big stake in it. I do not want to know the price if it's confirmed.",
" It is confirmed. It's highly confirmed.",
" Somebody is talking to me about the price. The thing will be me.",
" The logic of it to me is extremely compelling, and I honestly cannot understand these people who say that",
" Plunk data is ruling out inflation. I think they are off base.",
" But, you know, why trust me?",
" Does that imply... is the anthropoconst... principle implied in the creation?",
" It fits together with it very well.",
" Now, you could go back to the submarines. You could ask... you could say,",
" Well, the anthropoconstable itself, all that says is we live in a submarine,",
" which is of the right type for us to live in.",
" And which is cause and which is effect. There aren't cause and effect.",
" Do we explain the fact that we live in the right kind of submarine by the fact that we're here?",
" Or do we explain the fact that we're here by the fact that we live in the right kind of submarine?",
" I prefer the latter.",
" That we can understand why we're here as a consequence of there being enough right kinds of submarines,",
" rather than to say the opposite, that we can explain the properties of submarines by the fact that we are here.",
" That seems backward. Scientifically backward to put us at the center of the laws of nature to that extent.",
" Yeah, I guess I think of this use of it is when you're trying to start explaining why the laws of nature",
" have to be the way they are because we are here.",
" It seems to me much less egregious to say that the laws of nature are highly variable,",
" which theories tend to suggest anyway.",
" Highly variable environments are highly variable.",
" And that we live in a temperate zone for the same reason that we all live in Antarctica.",
" It's just to be in cold there.",
" And the real question of the real question is part from the fact that it's night and clock.",
" The real question is how are we going to find out?",
" The agenda of those who hate the idea and who love the idea should be exactly the same.",
" Take it seriously enough to either kill it or make it into science.",
" And the fact is that it looks very, very hard to do either.",
" It looks extremely hard to find signatures of this kind of thing in data,",
" which would either kill it or confirm it.",
" That's the real problem.",
" The problem is not that it's bad philosophy or that it's religious or that it...",
" One of my friends seems to think it's a terrible thing to propose",
" because young people will be... their minds will be deformed by it",
" and they will stop thinking about other explanations.",
" These are political reasons.",
" These are... oh, other people think it's a bad idea",
" because it enables the religious right to say,",
" ah, the world was created though just the right properties.",
" None of these are science.",
" The scientific question is, is it true or is it not true?",
" And if it is true or not true, how do you find confirming evidence?",
" In other words, you want to try to observe other submarines in the cleaning.",
" Right.",
" So if you're very lucky or unlucky in the case of a bee,",
" you might bump into another submarine.",
" That could be the lucky or not lucky.",
" It's lucky if it doesn't kill you and you find out that there are submarines out there",
" and you confirm and you win a Nobel Prize.",
" Or it could be bad, the submarine could make a hole in your submarine",
" and then you sink to the bottom.",
" It's still a no.",
" Right.",
" In principle, there could be collisions between regions of space with different properties.",
" It's expected to be extremely rare.",
" Computation seems to say that it's extremely rare.",
" Basically, computation seems to say that the ocean is exceedingly big,",
" but the nucleation of submarines is very rare.",
" That doesn't happen very often and very frequently.",
" And so most submarines will last a long time without being able to know.",
" So we found no evidence in the cosmic microwave background for a collision like this.",
" There's no evidence.",
" It's probably not going to be there.",
" So the only question is then what other possible kinds of signatures could there be",
" for the physics that goes into this kind of theory?",
" It's hard.",
" Nobody has a really good idea.",
" That's the worst thing about this kind of idea",
" that just looks overwhelmingly hard to find convincing data for.",
" Okay.",
" We are finished.",
" Thank you."
],
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10.5446/14934 (DOI) | Statistical Mechanics Lecture 1 | https://av.tib.eu/media/14934 | https://tib.flowcenter.de/mfc/medialink/3/de5c30134741ce6d779f08394784a64c91f492f23f326f61ce962f798057709d1dc7/StatisticalMechanics_Lecture1_720p__flash9_1_1.mp4 | CC Attribution 3.0 Germany:
You are free to use, adapt and copy, distribute and transmit the work or content in adapted or unchanged form for any legal purpose as long as the work is attributed to the author in the manner specified by the author or licensor. | Physics | Lecture | 2013 | Susskind, Leonard | null | Leonard Susskind introduces statistical mechanics as one of the most universal disciplines in modern physics. He begins with a brief review of probability theory, and then presents the concepts of entropy and conservation of information. | Okay, if there are no more questions, tonight we begin the study of statistical mechanics. Now statistical mechanics is not really modern physics. It's pre-modern physics. It's modern physics. And I assure you it will be post-modern physics. It's probably, the second law of thermodynamics will probably outlast anything that comes up time and time again. The second law of thermodynamics has been sort of our guidepost, our guiding light, if you like, to know what we're talking about and to make sure we're making sense. Statistical mechanics and thermodynamics may not be as sexy as the Higgs boson, but I assure you it is at least as deep. And it's a lot deeper. My particle physics friends shouldn't disown me. It's a lot deeper. It's a lot more general. And it covers a lot more ground than explaining the world as we know it. And in fact, without statistical mechanics, we probably would not know about the Higgs boson. All right, so with that little starting point, what are statistical mechanics about? Well, let's go back a step. The laws of physics, the basic laws of physics, Newton's laws, principles of classical physics, quantum mechanics, the things that were in the classical mechanics course, quantum mechanics, and so forth, those things are all about predictability, perfect predictability. Now you say, well, in quantum mechanics, you can't predict perfectly. And that's true, but there are some things you can predict perfectly. And those things are the predictables of quantum mechanics. Again, as in classical mechanics, you can make your predictions with maximal, let's call it maximal precision or maximal, whatever it is, predictability, if you know two things. If you know the starting point, which is what we call initial conditions, and if you know the laws of evolution of a system. If you can measure the, or if you know for whatever reason, the initial starting point of a closed system, a closed system means one which is either everything or it is sufficiently isolated from everything else that the other things in the system don't influence it. If you have a closed system, if you know the initial conditions exactly, or at least with whatever precision is necessary, and you know the laws of evolution of the system, you have complete predictability and that's all there is to say. Now of course, in many cases that complete predictability would be totally useless, having a list of the position and velocities of every particle in this room would not be very useful to us. The list would be too long and subject to rather quick change as a matter of fact. So you can see while the basic laws of physics are very, very powerful in their predictability, they also in many cases can be totally useless for actually analyzing what's really going on. Statistical mechanics is what you use, basically probability theory. Statistical mechanics, let me say first of all, it is just basic probability theory. Statistical as applied to physical systems. When is it applicable? It's applicable when you don't know the initial conditions with complete perfection. It's applicable when you, it may even be applicable if you don't know the laws of motion with infinite precision. And it's applicable when the system you're investigating is not a closed system, whether it's interacting with other things on the outside. In other words, in just those situations where ideal predictability is impossible, then what do you resort to? You resort to probabilities. But because the number of molecules in this room is so large, and probabilities tend to become very, very precise predictors when the laws of large numbers are applicable, statistical mechanics itself can be highly predictable, but not for everything. As an illustration, you have a box of gas. The box of gas might even be an isolated closed box of gas. It has some energy in it. The particles rattle around. If you know some things about that box of gas, you can predict other things with great precision. If you know the temperature, you can predict the energy in the box of gas. You can predict the pressure. These things are highly predictable, but there are some things you can't predict. You can't predict the position of every molecule. You can't predict when there might be a fluctuation. A fluctuation, which, you know, fluctuations are things which happen which don't really violate probability theory. They're the sort of tails of the probability distribution, things which are unlikely but not impossible. Fluctions happen from time to time in the sealed room every so often. An extra large group, an extra large density of molecules will appear in some small region bigger than the average someplace else. Molecules will be less dense. And fluctuations like that are hard to predict. You can predict the probability for a fluctuation, but you can't predict when a fluctuation is going to happen. It's exactly the same sort of thing, flipping coins. Flipping coins is a good example, probably our favorite example for thinking about probabilities. If I flip a coin a billion times, you can bet that approximately half of them will come up heads and half will come up tails within some margin of error. But there will also be fluctuations. Every now and then, if you do it enough times, a thousand heads in a row will come up. Can you predict when a thousand heads will come up? No. But can you predict how often a thousand heads will come up? Yes. Not very often. So that's what statistical mechanics is for. It's for making statistical probabilistic predictions about systems which are either too small, contain elements which are too small to see, too numerous to keep track of, usually both, too small to see by sea. I mean, you know, it's true. You can see some pretty small things, molecules, but so they may not be too small to see. But there are too many of them. There are too many of them to keep track of. And that's when we use probability theory or statistical mechanics. We're going to go through some of the basic statistical mechanics applications, not just applications, the theory, the laws of thermodynamics, the laws of statistical mechanics, and then how they apply to gases, liquids, solids, whether we will get the quantum mechanical systems or not. I don't know. But just the basic ideas. Okay. And, incidentally, another thing which is very striking is that generally speaking over the history of, certainly over my history in physics, and I'm sure this goes back to the middle of the 19th century sometime, all great physicists, all of them were masters of statistical mechanics. It may not have been the sexiest thing in the world, but they were all masters of it. Why? First of all, because it was useful, but second of all, because it is truly beautiful. It is a truly beautiful subject of physics and mathematics. And it's hard not to get caught up in it. Not to fall in love with it. The reason I teach it is not for you. It's for me. I love teaching it. I love teaching it. I teach it over and over and over again. And in a sense, my life is consisted of learning and forgetting and learning and forgetting and learning and forgetting statistical mechanics. So here's my opportunity to learn it again. Okay. Let's begin with what I usually call a mathematical interlude. In this case, it's not an interlude. It's a starting point. And I'm just going to make some extremely brief remarks which you all know. At least I think you all know them about probability. Just to have, you know, just to level the ground, what are we talking about? And what I am not going to explain, because I don't think anybody can explain it, is why probability works. Why does it work? If you ask why it works, the first answer will be it doesn't always work. You may have a probability for something and you test it out. And sometimes it doesn't work. Those are called the exceptions. So the answer to the question is why does it work? Well, it doesn't always work. It mostly works, except when it doesn't. When doesn't it? Rarely. How rarely? Every so often. But there is a calculus of probability, a mathematical theory of probabilities. And we'll talk about it a little bit. Okay. So we'll take probability to be a primitive concept, basically primitive concept. And we'll suppose that there is a space of some sort, a space of possibilities. The space of possibilities could be the space of outcomes of experiments, or it could actually be the space of states of a system. The state of a system could be the outcome of an experiment. If the experiment consists of determining the state of the system, then the state of the system is the outcome. So we have a space, and let's call that space, let's label the elements of that space with a little i. For example, if we were flipping coins, i would be either heads or tails. If we were flipping dice, you know, dice, dice of dice, i would run from one to six. If there were two dyes, then we would have enough indices to keep track of two dice and so forth. So i is the space of possibilities of outcomes, or the space of possible states of a system. And if we are ignorant, statistics always has to do with ignorance. You don't know everything, and so you assign probabilities to outcomes. And so we assign a probability p of i to the ith outcome to the answer to our question. Okay? What are the rules for p of i? What does p of i have to satisfy? And let's, for the beginning, at least in the beginning, let's imagine that i enumerates some discrete finite collection of possibilities. Later on, we can have an infinite number of possibilities, or even a continuously infinite number of possibilities. But for the time being, i might run from one to n, n possibilities. And the rules are, first of all, p sub i has to be greater than or equal to zero. Negative probabilities, we don't like them. Don't know what they mean. Okay? Next, the summation over i of p sub i, p of i, should be one. That means that the total probability, when you add everything up, all possibilities should be one. You certainly should get some result. Okay? Next, now this is a kind of hypothesis. This is the law of large numbers that if you either make many replicas of the same system or do the same experiment over and over a very, very large number of times, and take all of the outcomes which gave you all of the experiments which gave you the i-th outcome, that's some number, let's call it n of i, that's the number of times that the experiment turned up the i-th possibility, and you divide it by the total number of trials. Total number of trials means the sum of all i, or just the total number of trials, that the limit of this, this is a physical hypothesis. It's a physical hypothesis, it can go wrong if n is not large enough, but in the limit of large n, n goes to infinity, and the limit of very, very, of course n never goes to infinity, you never get to do an infinite number of experiments. But nevertheless we're kind of idealizing, we're assuming we can do so many experiments that the limit n goes to infinity is effectively been reached, then that is p of i. So p of i controls by assumption the ratio of the n of i's. Okay, everybody happy with that? You use this all the time I think. Well, sometimes we use it. Okay, now let's suppose that there is a quantity, let's call it f of i. It's some quantity that's associated with the i-th state. We can assign it, we can make it up. For example, if our system is heads and tails and nothing but heads and tails, we could assign f of heads and call it plus one, and f of tails and call it minus one. If our system has many, many more states, we may want to assign a much larger number of possible f's, but f is some function of the state. It's also a thing that we imagine measuring. It could be the energy of a state, or it could be the momentum of a state. Given a state of some system, it has an energy. It would be called in that case perhaps e of i, or it could be the momentum, or it could be something else. It could be whatever you happen to like to think about. Then an important quantity is the average of f of i. The average of f of i, I will use the quantum mechanical notation for it, even though we're not doing quantum mechanics. It's a nice notation. Physicists tend to use it all over the place. Mathematicians hate it. Just put a pair of brackets around it. It means the average. The average value of the quantity averaged over the probability distribution. It has a definition. Its definition is that it's the sum of i of f of i weighted with the probability. For example, and incidentally, the average of f of i does not have to be any of the possible values that f can take on. For example, in this case, where f of heads is plus one and f of tails is minus one, and you flip a million times, and the probability is a half of heads and a half of tails, the average of f will be zero. So it's not a possible outcome to the experiment. There's no rule why the average should be one of the possible experimental outputs, but it is the average. This is its definition. Each value of f is weighted with the probability for that value of f. You can write it another way. You can write it as a sum over i of f of i times the number of times that you measure i divided by the total number of measurements. That's what p of i is in the limit that there are a large number of measurements. That's defined to be the average. That's our mathematical preliminary for today. That's all I wanted to level the playing field by making sure everybody knows what the probability is and what an average is. We'll use it over and over. Okay, let's start with coin flips. I always start with coin. I start every single class with coin flips, even when I'm teaching about the Higgs boson. Okay. If I flip a coin a lot of times, or whether I flip a coin a lot of times or not, the probability for heads is usually deemed to be one-half and the probability for tails is usually also deemed to be one-half. Why do we do that? Why is it a half and a half? What's the logic there? What logic tells us that? In this case, it's symmetry. It's the symmetry of the coin. Of course, no coin is perfectly symmetric and even making a little mark on it to distinguish the heads and tails, bias is it a little bit, but apart from that tiny, tiny bias of marking the coin with maybe just a tiny little scratch, the coin is symmetric. Higgs and tails are symmetric with respect to each other and therefore there is no reason, no rationale for when you flip a coin for it to turn up heads more often than tails. It's symmetry quite often. I might even say always in some deeper sense, but at least in many cases, symmetry is the thing which dictates probabilities. Probabilities are usually taken to be equal for configurations which are related to each other by some symmetry. Symmetry means if you act with a symmetry, you reflect everything, you turn everything over that the system behaves the same way. Okay, another example besides coin flipping would be dice flipping. Dice flipping instead of having two states has six states, one die, and we can imagine coloring them. We color the faces, red, yellow, blue, and then on the back green, purple, and orange. Okay, that's our die and it's been colored. We don't have to keep track of numbers, we can keep track of colors. And what is the probability that when we flip the die, flip it into the air, hits the ground, what's the probability that it turns up red? No, it's one-sixth, right? There are six possibilities. They're all symmetric with respect to each other. We use the principle of symmetry to tell us that the P of each i, they're all equal and they're all equal to one-sixth. But what if there is no symmetry? What if really the die is not symmetric? For example, what if it's weighted in some unfair way? Or what if it's been cut with faces that are not nice and parallel cubes? Then what's the answer? The answer is symmetry won't tell you. You may be able to use some deeper underlying theory and to use some concept of symmetry from the deeper underlying theory, but in the absence of something else, there is no answer. The answer is experiment. Do this experiment a billion times, keep track of the numbers, assume that things have converged and that way you measure the probabilities. You measure the probabilities and thereafter you can use them. You can use them if you keep a table of them and then you can use them in the next round of experiments. Or you may have some theory, some deep underlying theory which tells you well, like quantum mechanics or statistical mechanics. Statistical mechanics tends to rely mostly on symmetry, as we'll see. So if there's no symmetry to guide you or to guide your implementation of probabilities, then it's experiment. Now there's another answer. There's another possible answer. This answer is often frequently invoked and it's a correct answer under other circumstances. It can have to do with the evolution of a system, the way a system changes with time. So let me give you some examples of what it might have to do. Let's take our six sided cube and assume that our six sided cube is not symmetric. It's not symmetric but we know a rule. We know that if we put that cube down on the table, it's not a cube, when we put that die down on the table and we stand back, this thing has this habit of jumping to another state and jumping to another state and jumping to another state. It's called the law of motion of the system. The law of motion of the system is that whatever it is at one instant, at some next instant, it will be something else according to a definite rule. The instance could be seconds, it could be microseconds or whatever, but imagine a discrete sequence and let's suppose there's a law, a genuine law that tells us how this cube moves around. For example, if it's red, now we've done this over and over many times in different contexts but it is so important that I feel a need to emphasize it again. This is what a law of motion is. It's a rule telling you what the next configuration will be given, it's a rule of updating, of updating configurations. Red goes to blue, blue goes to yellow, yellow goes to green, green goes to orange, orange goes to purple and purple goes back to red. Given the configuration at any time, you know what it will be next and you know what it will continue to do. Of course, you may not know the law. Maybe all you know is that there is a law of this type. You know what I'm going to do next, what am I going to do next? I'm going to draw this law as a diagram. You've all seen me do this in other contexts. Let's do it. We have red, too hard to draw squares. Red, blue, green, orange, what happened? Yellow, yellow, orange, purple. A law like this can be just represented by a set of lines connecting a set of arrows. Red goes to blue, blue goes to green, green goes to yellow, yellow goes to orange, orange goes to purple, purple goes back to red. Given the assumption now that there's a discrete time interval between such events, I am not assuming that the cube has any symmetry to it anymore. The cube may not be symmetric at all. It may have points, you know, one edge, one face, maybe tiny, another face, but if this is the rule to go from one configuration to another and each step takes, let's say, a microsecond, I might have no idea where I begin, but I can still tell you if I, let's say it's a microsecond, a microsecond, and my job is to catch it at a particular instant and ask what the color is. I don't know where it started, okay? But I can still tell you the probability for each one of these is one-sixth. It doesn't have to do with symmetry. Well, maybe it does have to do with some symmetry, but in this case, it wouldn't be the symmetry of the structure of a die. It would just be the fact that as it passes through these sequence of states, it spends one-sixth of its time red, one-sixth of its time blue, one-sixth of a time green, and if I don't know where it starts and I just take a flash, you know, a flash shot of it, my probability will be one-sixth. Now that one-sixth did not really depend on knowing the detailed law. For example, the law could have been different. Let's make up a new law. Red goes to green, green goes to orange, orange goes to yellow, yellow goes to purple, purple goes to blue, and blue goes back to red. This shares with the previous law that there's a closed cycle of events in which you pass through each color once before you cycle around. You may not know which law of nature is for this system, but you can tell me again that the probability will be one-sixth for each one of them. So this prediction of one-sixth doesn't depend on knowing the starting point and doesn't depend on knowing the law of physics. It's just important to know that there is a particular kind of law. Are there possible laws for the system which will not give you one-sixth? Yes. Let's write another law. Red, blue, green, yellow, orange, purple. This rule says that if you start with red, you go to blue. If you start with blue, you go to green, and if you get to green, you go back to red. Or if you start with purple, you go to yellow, yellow to orange, orange back to purple. Notice in this case, if you're on one of these two cycles, you stay there forever. If you knew you were on the upper cycle, if you knew you would start it, it doesn't matter where you start, but if you knew that you started on the upper cycle somewhere, then you would know that there was a one-third probability to be red, a one-third probability to be blue, and a one-third probability to be green, and zero probability to be red, yellow, or orange. On the other hand, you could have started with the second cycle. You could have started with purple. Might not have known where you started, but you knew that you started in the lower triangle and the lower cycle here. Then you would know the probabilities of one-third for each of these and zero for each of these. Now, what about a more general case? The more general case might be that you know with some probability that you start on the upper triangle here and with some other probability on the lower triangle. In fact, let's give these triangles names. Let's call this triangle the plus one triangle, and this one the minus one triangle. Just giving them names, attaching to them a number, a numerical value. If you hear something or other is called plus one, if you hear something or other is called minus one. All right, now you'll have to append, you'll have to start with something you've got to get from someplace else. It doesn't follow from symmetry, and it doesn't follow from cycling through the system some probability that you're either on cycle plus one or cycle minus one. Where might that come from? Flipping somebody else's coin over here, flipping a coin over here might decide which of these two. It might be a biased coin, so you will have a probability to be plus one and a probability to be minus one. These two probabilities are not probabilities for individual colors, they're probabilities for individual cycles. Okay, now what's the probability for blue? The probability for blue begins with the probability that you're on the first cycle, times the probability that if you're on the first cycle, you get blue. That's one third. So the probability for blue, red, or green is one third the probability that you're on the first cycle, and likewise the probability that you're at yellow will be, this is the probability for red, blue, or green in this case, and this times one third will be the probability for purple, yellow, or orange. Okay, so in this case you need to supply another probability that you've got to get from somebody's else. This case here is what we call having a conservation law. In this case, the conservation law would be just the conservation of this number. For red, blue, and green, we've assigned the value plus one. That plus one could be the energy, or it could be something else. I tend to call it the zilch for some reason. I call everything a zilch if there is no name for it. So anyway, let's think of it as the energy to keep things familiar. The energy of these three configurations might all be plus one, and the energy of these three configurations might all be minus one. And the point is that because the rule keeps you always on the same cycle, that quantity, energy, zilch, whatever we call it, is conserved. It doesn't change. That's what a conservation law is. A conservation law is that the configuration space, the space of possibilities, divides up into cycles like this. Now, the cycles don't have to have equal size. Here's another case. One, two, three, four. You go around this way, and then the two guys over here go into each other. So red goes to blue, goes to green, goes to purple. That's the upper cycle here, and the lower cycle is yellow goes to orange, goes to yellow goes to orange. Still, we have a conservation law here. It's just the number of states with one value of the conserved quantities, not the same as the number of states or the other value, but still, it's a conservation law. And again, somebody would have to supply for you some idea of the relative probabilities of these two. Where that comes from is part of the study of statistical mechanics. And the other part of the study has to do with saying, if I know I'm one of these tracks, how much time do I spend with each particular configuration? That's what determines probabilities of statistical mechanics. Some a priori probability from somewheres that tells you the probabilities for different conserved quantities and cycling through the system. Yeah, question? No, okay. So, so far, you're assuming that within any conservation arena, you will, the probabilities of all the states are the same? The time spent in each state is the same. Right. So, it's completely deterministic. Laws are completely deterministic. This would be classical physics. Laws completely deterministic. No real ambiguity of what the state is, except you're kind of lazy. You didn't determine the initial condition. Your timing wasn't very good. Each state only lasts for a microsecond. You're a lazy guy and you only have a resolution of a millisecond. But nevertheless, you're able to take a very quick flash picture and pick out one of the states. That's the circumstance that we're talking about. Yeah. If we take two pictures, is it reasonable to then assume that if the first picture indicated that we are in one cycle, the later one should indicate the same cycle since it couldn't get out of it? Yes, that's a good assumption. Yes. Right. So, once you determine the value of some conserved quantities, then you know it. And then you can reset the probabilities for it. Unless, all right, so let's talk about honest energy for a minute. Yes, if we have a closed system, to represent the closed system, I will just draw a box. Closed now, closed means that it's not in interaction with anything else and therefore can be thought of as a whole universe unto itself. Okay. It has an energy. The energy is some function of the state of the system, whatever determines the state of the system. Now let's suppose we have another closed system which is built out of two identical or not the identical versions of the same thing. Now, if they're both closed systems, there will be two conserved quantities. The energy of this system and the energy of this system and they'll both be separately conserved. Why? Because they don't talk to each other. They don't interact with each other. The two energies are conserved and you could have probabilities for each of those individuals. But now supposing they're connected. They're connected by a little tiny tube which allows energy to flow back and forth. Then there's only one conserved quantity, the total energy, and it's sort of split between the two of them. You can then ask, what is the probability given a total amount of energy? You could ask, what's the probability that the energy of one subsystem is one thing and the energy of the other subsystem is the other? If the two boxes are equal, you would expect on the average they have equal energy, but you can still ask, what's the probability for a given energy in this box given some overall piece of information? That's a circumstance where it may be that giving the probability for which cycle you're on, now which cycle you're on, I'm talking about the cycle of one of these systems here, may be determined by thinking about the system as part of a bigger system. And we're going to do that. That's important. But in general, you need some other ingredient besides just cycling around through the system here to tell you the relative probabilities of conserved quantities. Okay. So we're often flying with statistical mechanics. There are bad laws. By bad laws, I don't, not in the sense of DOMA or any of those kind of laws, but in the sense that the rules of physics don't allow them. You all know what they are. The laws that violate the conservation of information. The most primitive and basic rule of physics, the conservation of information. Conservation of information is not a standard conservation law like this. It's the rule that you keep, that you can keep track. You can keep track both going forward and backward. So let's just mention that again. It's all work, it's all described in the classical mechanics book. I'm just reviewing it now. But let's take a bad law. It's a possible law. By bad, I mean one, two, three, four, one, two, three, four, five, six. So these are the faces of a die again. But the rule is wherever you are, this is red, wherever you are, you go to red. Even if you are red, you go to red. Okay. We'll discuss in a moment what's wrong with this law. But this law has one of the features that it has, is it's not reversible. It's not reversible in the sense that you can go from blue to red, but you cannot go from red back to blue. So in that sense, it's not reversible. You can predict the future wherever you are. The future is very simple for this particular law, wherever you are, you'll next be at red. You can make it more complicated. You could make a few, you can make it more complicated. But this law always winds up with red. It's a bad law because it loses track of where you started. Whereas these laws don't lose track. If you know that you've gone through 56, 56 and a half cycles, then you know that if you started at red, you'll come back and you can tell exactly where you'll be. And you can also tell where you came from. You can tell not only where you'll be, but exactly where you came from. Well, this law, you can't say where you came from. This is a law that loses information. And it's exactly the kind of thing that classical physics does not allow. Classical physics also doesn't allow the quantum mechanical version of it. So the rule that this type of rule, that this type of law is unallowed, I give a name to. It's as I said many times, there is no name for it because it's just so basically primitive that everybody always forgets about it. It's so basic. I call it the minus first law of physics. And I wish it would catch on. People should start using it. I mean, it is really the most basic law of physics that information is never lost, that distinctions or differences between states propagate with time and you never lose track. In principle, if you have the capacity to follow the system, because you may be too lazy to follow the system, that's your problem. But nature doesn't have that problem. Nature allows, in principle, that you can reconstruct where you came from. All right, so that's a bad law. How do you tell the good laws from the bad laws? Just by diagrammatics here, it's very simple. Good laws, every state has one incoming arrow and one outgoing arrow. An arrow to tell you where you came from and an arrow to tell you where you're going. So those are good laws. In classical mechanics, continuum classical mechanics, there is a version of this same law. Anybody know the name of that version? Of the theorem that goes with the conservation of information? It's called Leaville's theorem. We studied it in classical mechanics. But let me give you a counter example to Leaville's theorem. Friction is an apparent contradiction. Wherever you start, you come to rest. It's sort of like saying wherever you start, you come to red. Wherever you start, you come to rest. Well, you may not know exactly where you are, but you always come to rest. That seems like a violation of the laws that tell you that distinctions have to be preserved. But of course, it's not really true. What's really going on is that when you run the eraser through here, it's heating up the surface here. And if you could keep track of every molecule, you would find out that the distinctions between starting points is recorded. But let's imagine now that there was a fundamental law of physics. By fundamental law, I mean a rock bottom fundamental law for a series of particles, for a collection of particles, and the equations of motion for the particles were this. d second x, that's the position of the particle, somewhere by dt squared, that's called acceleration. We could put a mass in, but the mass is not doing anything. There's a lot of particles, so I'll label them i. Oh, we've used i to label states. I should not do that. Let's call it n, little n. The nth particle, and what is that equal to? It's equal to minus some number gamma, we've seen that number before in another context, times dxn by dt. Anybody remember what this formula represents? Friction. Viscous drag. Again, it has the property that if you start with a moving particle, it will very quickly come almost to rest. It'll exponentially come to rest pretty quickly. And so if all particles in a gas, for example, satisfied this law of physics, it's perfectly deterministic. It tells you what happens next, but it has the unfortunate consequence that every particle just comes to rest. That sounds odd. It sounds like no matter what temperature you start the room, it will quickly come to zero temperature. That doesn't happen. This is a perfectly good differential equation, but there's something wrong with it from the point of view of conservation of energy. There's something wrong with it from the point of view of thermodynamics. If you start a closed system and you start it running, you start with a lot of kinetic energy, temperature we usually call it. It doesn't run to zero temperature. That's not what happens. In fact, this is not only a violation of energy conservation. It looks like a violation of the second law of thermodynamics. It says things get simpler. You start with a random bunch of particles moving in random directions, and you let it run and they all come to rest. What you end up with is simpler and requires less information to describe than what you started with. That's very, very much like everything going to red. Among other things, it violates the second law of thermodynamics, which generally says things get worse. Things get more complicated, not less complicated. Okay. But there's another way to say, another important way to say this rule, that every state has to have one arrow in and one arrow out. The thing that I called either the minus first law or the conservation of information. Supposing we have a collection of states and we assign to them probabilities. P of state one, P of state two, P of state three, and so forth. For some subset of the states, not all of them, some subset of them. All the others, we say have probability zero. Okay. So, for example, we can take our die and assign red, yellow, and blue probability a third, and green, orange, and pink, or whatever it was, probability zero. Where we got that from, doesn't matter. We got it from somewhere. Somebody secretly told us in our ear, it's either red, yellow, or blue, and I'm not going to tell you which. All right. And now you follow the system. You follow it as it evolves. Whatever kind of law of physics, as long as it's an allowable law of physics, after a while, and you're following it in detail, you're not constrained by your laziness in this case. You are capable of following in detail. And what is the probability, what are the probabilities at a later time? Well, if you don't know which the laws of physics are, you can't say, of course. But you can say one thing. You can say there are three states with probability one-third and three states with probability zero. They may get reshuffled, which ones were probable and which ones were improbable, but after a certain time, there will be those same three, not the same three states, but there will continue to be three states which have probability and the rest don't. So in general, you could characterize these information-conserving theories by saying, supposing you assign some subset of the states, let's say, N out of N states, let's say there are N states altogether, that's the total number of states, and now we look at some M where M is less than N, and we say for those M states, the probability for those M states is one over M for these states and zero for all the others. You understand why I say one over M? If there are M states equally probable, then each one has probability one over M and all the remaining have probability zero. Then the number of states which have non-zero probability will remain constant and the probabilities will remain equal to one over M. Is that clear? Is that obvious? That should be obvious. The states may reshuffle, but the number with non-zero probability will remain fixed. That's a characterization, a different characterization of the information-conserving laws. For the information-non-conserving laws, everybody goes to red. You may start with a probability distribution that's one over five for red, green, purple, orange, and yellow, and then a little bit later, there's only one state that has a probability and that's red. This is another way to describe information conservation. We can quantify that. We can quantify that by saying let M be the number of states which all have equal, under the assumption that they all have equal probability, let M be the number, let's give it a name, occupied states, states which have non-zero probability with equal probability, and then M, what is M characterizing? M is characterizing your ignorance. The bigger M is, if M is equal to N, that means equal probability for everything. Maximum ignorance. If M is equal to one-half N, that means you know that the system is in one out of half the states. You're still pretty ignorant, but you're not that ignorant. You're less ignorant. What's the maximum, what's the minimum amount of ignorance you can have? That you know precisely what state it's in, in which case M is what? M is one. You know that it's in one particular state. All right, so M is a measure of your ignorance. Really M in relation to N is a measure of your ignorance. And associated with it is the concept of entropy. Now we come to the concept of entropy, notice entropy is coming before anything else. Entropy is coming before temperature, it's even coming before energy. Entropy is more fundamental in a certain sense than any of them, although in a certain sense it's, we'll discuss entropy in a minute, but S is a logarithm of M. Logarithm of the number of states that have an appreciable probability more or less all equal for the specific circumstance that I talked about. That entropy is conserved. All that happens is the states which are occupied reshuffle, but there will always be M of them with probability one over M. Okay, so that's where we are. And that's the conservation of entropy if we can follow the system in detail. Now, of course in reality we may be again lazy, lose track of the system, and we might have after a point lost track of the equations, and lost track of our timing device, and so forth and so on. Now we may wind up, we may have started with some, a lot of knowledge, and wound up with very little knowledge. That's because, again, not because the equations cause information to be lost, but because we just weren't careful. Perhaps we can't be careful, perhaps there are too many degrees of freedom to keep track of. So when that happens, the entropy increases, but it simply increases because our ignorance has gone up, not because anything has really happened in the system which has, if we could follow it, we would find that the entropy is conserved. Okay, that's the concept of entropy in a nutshell. We're going to expand on it. We're going to expand on it a lot. We're going to redefine it with a more careful definition. But what does it measure? It measures approximately the number of states that have a non-zero probability. Okay. The bigger it is, the less you know. What's the maximum value of s? Log in, log in. Now of course, n could be infinite. You might have an infinite number of states, and if you do, then there's no upper bound to the amount of ignorance you can have. But you know, in a world with only n states, your ignorance is bounded. So the notion of maximum entropy is a measure of how many states there are altogether. Now I said that entropy is deep and fundamental, and so it is, but there's also an aspect to it which makes it in a certain sense less fundamental. It's not just a property of a system. It's a property of a system and your state of knowledge of the system. It depends on two things. It depends on characteristics of the system, and it also depends on your state of knowledge of the system or the state of knowledge of the system. So keep that in mind. Okay. Now let's talk about continuous mechanics. Mechanics of particles moving around with continuous positions, continuous velocities. How do we describe that? How do we describe the space of states of a mechanical system, you know, a real mechanical system, particles and so forth? We describe it as points in phase space. We learned about phase space. Phase space consists of positions and momenta. Momenta in simple context, momentum is mass times velocity, so roughly speaking, it's the space of positions and velocities. Let's draw it. P is momentum. It goes that way. And this axis is a stand-in for all of the momentum degrees of freedom. If there are 10 to the 23rd particles, there are 10 to the 23rd P's, but I can't draw more than one of them. Well, I could draw two of them, but then I wouldn't have any room for the Q's, for the X's. And horizontally, the positions of the particles, which we can call X. X or P, doesn't matter. All right. A point here is a possible state of the system. If you know a point here, you know a position and a velocity, and you can predict from that through if you know the forces. Okay, let's start with the analog of a probability distribution, which is zero for some set of states and constant or the same for some other set, for some smaller set. Well, some fraction of the states all have the same probability, and the other states have zero probability. We can represent that by drawing a patch in here, a subregion in the phase space, and say in that subregion, there's equal probability that the system is at any point in here and zero probability outside. This is sort of a situation where you may know something, where you may know something about the particles that they're in some subregion here. For example, you know that all the particles in this room are in the room, right? So that puts some boundaries on what X are. You may know that all the particles have momentum which are within some range that confines them to this way. So a typical bit of knowledge about the room might be represented at least approximately by saying that there's zero probability to be outside this region and a probability equal, I won't say one, but equal probability to be in there. Okay, now what happens as the system evolves? As the system evolves, X and P change. The equations of motion say that X and P change, if you start over here, you might go to here. If you start nearby, you'll go to some nearby point and so forth. And the motion of the system with time is almost like a fluid flowing in the phase space. If you think of the points of the phase space as fluid points and let time go, the phase space moves like a fluid. In particular, this patch over here, let's call it the occupied patch, the occupied patch becomes some other patch. That other patch, after a certain amount of time, the system now is known to be in here. After a certain amount of time, we now know that the system is in here, not in here anymore, and that it has equal, in some sense, equal probability to be anywhere in there. Okay? There's a theorem that goes with this. The theorem is called Leaville's theorem. And what it says is that the volume in phase space, the amount of volume of this region in the XP space, and keep in mind, the XP space may be high dimensional. Not just two, if it were two dimensional, we would think of it as the area. Phase space is never three dimensional. It's always even dimensional. It has a P for every X. So the next more complicated system would be four dimensional. When I speak of the volume in phase space, I mean the volume in whatever dimensionality the phase space is. If you follow the phase space in this manner here, Leaville's theorem, you can go back to Leaville's theorem. It's in the classical mechanics lecture notes. It occupies, I think, a whole lecture, I think. All right, you follow? And it tells you whatever this evolves into, it evolves into something of the same volume. In other words, roughly speaking, the same number of states. It's the immediate analog of the discrete situation where if you start with M states and you follow the system according to the equations of motion, you will occupy the same number of states afterwards as you started with. There'll be different states, but you will preserve the number of them, and the probabilities will remain equal. So the rule is, not the rule, the theorem says that the volume of this occupied region will stay the same. And a little bit better, it says that if you start with a uniform probability distribution in here, it will be uniform in here. So there's a very, very close analog between the discrete case and the continuous case. And this is what prevents this kind of fundamental equation from this kind of equation, an equation where everything comes to rest, that can't happen. Okay, why not? Let's see why it can't happen. Let's just look on this blackboard and see why. Imagine that no matter where you started, you ended up with P equals zero. That would mean every point on here got mapped to the x-axis. It would mean that this entire region here would get mapped to a one-dimensional region, and one-dimensional region has zero area. So Leeville's theorem prevents that. What it says, in fact, is if the blob squeezes in one direction, it must expand in the other direction. The situation for the moving eraser is that if the phase space of the eraser gets shrunk, it means somebody else's, some other components in the phase space, the probability distribution is spread out. What are the other components in this case? It's the P's and X's of all the molecules that are in the table. So for the case of the eraser, there's really a very high-dimensional phase space, and as the eraser may come to rest, almost rest, so that the phase space squeezes this way, it spreads out in the other directions, the other directions having to do with the other hidden microscopic degrees of freedom. Okay, so there we are with information conservation, minus first law of physics, and let's pass... Let's not go to the zeroth law. Let's jump the zeroth law. We'll come back to the zeroth law. You know what the zeroth law says? Well, I'll tell you what it says. We haven't defined what thermal equilibrium is, okay? But it says whatever the hell thermal equilibrium is, if you have several systems, and system A is in thermal equilibrium with B, and B is in thermal equilibrium with C, then A is in thermal equilibrium with C. We will come back to that. Just put it out of your mind for the time being, because we haven't described what thermal equilibrium means. But we can now jump to the first law, minus one, zero, and first law. And the first law is simply energy conservation. It is simply energy conservation, nothing more. It's really simple to write down. That simplicity belies its power. It is the statement that, first of all, there is a conserved quantity, and the fact that we call that conserved quantity energy will play for the moment not such a big role right now, but let's just say there's energy conservation. What does that say? That simply says DE. Simply energy is DE by DT is equal to zero. Now this is the law of energy conservation for a closed system. If a system consists of more than one part in interaction with each other, then of course any one of the parts can have a changing energy, but the sum total of all of the parts will conserve energy. So if a system is composed, as I drew before, of two parts with a link between them, and this is called one and this is two, then this reads that DE1 by DT is equal to minus DE2 by DT. I've really written DE1 by DT plus DE2 by DT is equal to zero, but then I transposed one of them to the right hand side just to indicate, just to make graphic, that if you lose energy on one side, you gain it on the other. So that's the first law of thermodynamics, and that's all the first law of thermodynamics says, it says energy conservation. Now in this context here, there's a slightly hidden assumption. We've assumed that if a system is composed of two parts, that the energy is the sum of the two parts. That's really not generally true. If you have two systems and they interact with each other, there may be, for example, forces between the two parts, so there might be a potential energy that's a function of both of the coordinates. For example, the energy of the solar system, being very naive, I'm thinking of the solar system as two orbiting Newtonian particles. The energy consists of the kinetic energy of one particle plus the kinetic energy of the other particle plus a term which doesn't belong to either particle. It belongs to both of them in a sense, and it's the potential energy of interaction between them. In that context, you really can't say that the energy is the sum of the energy of one thing plus the energy of the other thing. Energy conservation is still true, but you can't divide the system into two parts this way. On the other hand, there are many, many contexts where the interaction energies between systems is negligible compared to the energy that the systems themselves have. If we were to divide this table top up into blocks, let's think about it, divide the table top up into blocks, how much energy is in each block? Well, the amount of energy that's in each block is more or less proportional to the volume of each block. How much energy of interaction is there between the blocks? The energy of interaction is a surface effect. They interact with each other because their surface is touch, and typically, surface area is small by comparison with volume. So many, many, we'll come back to that. We'll come back to that. In many, many contexts, the energy of interaction between two systems is negligible compared to the energy of either of them. When that happens, you can say to a good approximation, the energy can just be represented as the sum of two energies of the two parts of the system plus a teeny little thing which has to do with their interactions. Under those circumstances, the first law of thermodynamics, the top is always true. The second has that little caveat that we're talking about systems where energy is strictly additive, where you add energies. Does everybody understand why I say you don't always add energies that sometimes energies are not additive? Yeah, actually, I was thinking that we have the same possible problem with the probabilities. We assume that the outcomes were mutually exclusive. Otherwise, the sum law doesn't work. Yeah. Yeah. Okay. So in all the contexts which we've talked about, the dye, if it's yellow, it can't be red. You say orange. You say orange, orange is both yellow and red, well, we don't count that way. Yeah. So that's correct. No, that's absolutely correct. We made the assumption that what I called states, what I called states are mutually exclusive. Absolutely. Okay, let's come back to entropy. We're not finished with entropy. We've done entropy. We've done energy. We haven't gotten the temperature yet. Just the temperature comes in behind entropy, and even energy comes in behind entropy. But temperature is a highly derived quantity. By highly derived, I mean it's a, despite the fact that it's the thing you feel with your body, so it makes it really feel like it's something intuitive, it is a mathematically derived concept, less primitive and less fundamental in either energy or entropy, but we'll come to it. Let's come back to entropy. We define entropy, but only for certain special probability distributions. Let's lay out on the horizontal axis, just to be schematic. On the horizontal axis, we will put down all the different states. Here's i equals one, here's i equals two, here's i equals three. This axis just labels the various states, and of course, vertically, let me plot probability. Okay. Well, the probability, of course, is only defined on the integers here. That's not very good. It's some probability. But let's, I don't want to have to draw such a complicated thing every time I want to draw a probability distribution, let's just draw a graph. Some probability distribution, or some probability for each position. Now what we did was we defined entropy for a very special case, the special case being where some subset have equal probabilities and the rest have zero probability. For example, if our subset consists of this group over here of m, the whole group being n, then all of these have the same probability, and their probabilities have to add up to one, so the probability is one over n. We just draw that by drawing a box like that. Then we define the entropy to be the logarithm of the number of states in here. Generally speaking, we don't have probability distributions like this. Generally speaking, we have probability distributions which are more complicated. In fact, they can be anything as long as they're positive and all add up to one. So the question is how do we define entropy in a more general context where the probability distribution looks like this? I'm going to write down the formula, and then we're going to check that it really gives us this answer when it should give us this answer. For today, I'm just going to write it down and tell you this is the definition. You'll get familiar with it, and you'll start to see why it's a good definition. It's representing something about the probability distribution, and what it's representing is in some average sense, the average number of states which are importantly contained inside the probability distribution. The narrower the probability distribution, the smaller the entropy will be. The broader the probability distribution, the bigger the entropy will be. I'll write it down for you now. We'll write it down and then explore it just a little bit tonight. S for a general probability distribution is, first of all, minus. That's funny because this is positive, but nevertheless the formula begins with minus, a sum, and it's a sum over all of the states, all of the possibilities. So it's a sum over i. There's a contribution for each place here. The probability of i times the logarithm of the probability of i. Do you remember? All right, let's write something else then. Remember that the average of f is equal to the summation over i, f of i times p of i. This is actually the average of log p sub i. It's the average of log p sub i. All right, let's work this out. Let's see what this gives. In the special case where the probability distribution is 1 over m for m states altogether. It has width m, and because it has width m, it must have height 1 over m because all the probabilities have to add up to 1. All right, so let's work this out. Let's take the contribution for all the unoccupied states. All the unoccupied states, p sub i is 0. You get nothing. But log of p over i is minus infinity. Yeah, that's right. Now what's the, all right, good. So let's consider the limit of log p over p. Or let's just say the limit of, no, not that's not right, the limit of p log p as p goes to 0. You know how to calculate that? Okay, I'm going to leave it to you. It's a little calculus exercise to calculate the limit as p goes to 0 of p log p. It's 0. The point is that p goes to 0 a lot faster than log p goes to infinity. Log p as p goes to 0 goes to, is very slow. p goes to 0 fast. So this goes to 0. You're absolutely right though. That has to be, that has to be shown. But p log p in the limit that p goes to 0 is 0. So with that piece of knowledge, the contribution from states with very, very small probabilities, probability will be very, very small. And as the probability for those states goes to 0, this quantity, the contribution will go to 0. But what about the ones here which have significant probability? They all have the same piece of body. And they all have the same log piece of body. The log piece of body is logarithm of 1 over m for all of them. The piece of body is also 1 over m. Not log 1 over m, but 1 over m. So each contribution is 1 over m times log 1 over m. How many contributions are there like that? All right, so we multiply by m and get rid of the 1 over m there. There's a minus sign here, I'll carry it along. All times m because there are m such terms. So the 1 over m cancels and we just left with log 1 over m. What's log 1 over m? Minus log m, right? So that's why the minus sign was put there in the first place. There's no miracle. The minus sign was put there because probabilities are less than 1. And so the logarithms of them are always negative. So you soak up that negative with an overall negative sign and entropy is positive. But this is exactly the same answer that we are, the original definition just s equals log m. Logarithm of the number of states. But this is a definition now that makes sense even when you have a more complicated probability distribution. And it is a good and effective. It's the average of log p. For the special case where the probability distribution is constant like this, then all of the probabilities in here are 1 over m and calculating the average of log 1 over m just gives you this. All right. So this is the general definition of the entropy that's associated with a probability distribution. And notice, entropy is associated with a probability distribution. It's not a thing like energy which is a property of a system. It's not a thing like momentum. It's a thing which has to do with a specific probability distribution, probability distribution on the space of possible states. So that's why it's a little bit of a more obscure quantity from the point of view of, you know, intuitive definition. As I said, its definition has to do with both the system and your state of knowledge of the system. Let's do some examples. Let's calculate some entropy for a couple of simple systems. The first system is just going to be not a single coin but a lot of coins. So we have capital N coins, N of them, and each one can be heads or tails, et cetera, and so on. As a matter of fact, we have no idea what the state of the system is. We know nothing. The probability distribution, in other words, is the same for all states. Good ignorance, absolute ignorance. What is the entropy associated with such a configuration? There are n of these. All the probabilities are equal. Under the circumstance where all of the probabilities are equal, we just get to use logarithm of the number of possible states. The answer here is the logarithm of the number of total states. How many states are there altogether? Two to the N. Right. Two to the N. Oh, let's, I'm sorry, I'm going to change definitions for a minute for a reason that you'll see in a minute. I'm going to call this little N. Number of coins is little N. Over here, big N stood for the total number of states. So if I match terminologies, big N, the total number of states, is e to the 2n, no, 2 to the N. Sorry, I'm getting tired. 2 to the N. 2 to the N states altogether. Two states for the first coin, two states for the second coin, and so forth. 2 to the N altogether. And the total number of states is capital N. What's the entropy given that we know nothing? Two log N. Two log N. N log 2. S is equal to N log 2. That's the logarithm of 2 to the N. All right, so here we see an example of the fact that entropy is kind of additive over the system. It's proportional to the number of degrees of freedom in this case. N times log 2. And we also discover a unit of entropy. The unit of entropy is called a bit. That's what a bit is, an information theory. It's the basic unit of entropy for a system which has only two states, up or down, heads or tails or whatever. The entropy is proportional to the number of bits, or in this case, the number of coins, times the logarithm of 2. So log 2 plays a fundamental role in information theory as the unit of entropy. It does not mean that in general that entropy is an integer multiple of log 2. We'll see in a second, it doesn't mean that. Okay, so that's this case over here. Let's try another case. Let's try another state of knowledge. Here our state of knowledge was zilch. We knew nothing. This is the maximum entropy. Logarithm of capital N, or logarithm of 2 to the little n, which is n log 2. One bit of entropy for each coin, if you like. Okay, let's try something else. Let's try a state in which what we know is, oh, something else. Supposing we know the state completely. In other words, that's the case where m is equal to 1. That would be the case where we know that the probability is only nonzero for one state. Then m is 1 and s is 0, logarithm of 0. So absolute knowledge, perfect knowledge, complete knowledge corresponds to zero entropy. The more you know, the smaller the entropy, excuse me. Okay, let's take an interesting case. Let's take our heads and tails again. And here's what we know. We know that all of them are heads. Let's begin with that, all of them are heads, what's the entropy then? Zero. Except for one of them, which is tails. Now, supposing we know which one is tails, what's the entropy? Zero. But suppose we don't know which one is tails. Equal probability for all states which contain one tail and n minus one heads. What's the entropy? Indeed. Okay, so why is it log n? How many states are there with nonzero probability? The answer is little n. It could be this one, it could be this one, it could be this one. In other words, for this particular situation, capital M, the number of states that have nonzero probability is just little n. All the states have the same probability. So we're in exactly this situation except that M is just equal to little n. N possible states, this one could be tails, this one could be tails, this one could be tails. They all have equal probability. So capital M is n and the entropy now is, for this situation, the entropy is equal to logarithm of little n. Notice that that's not an integer multiple of log two in general. So in general, entropy is not an integer multiple of log two. Nevertheless, log two is a good unit, it's a basic unit of entropy, it's called a bit. Yes in this case is equal to log n. Yes? The two in the log two come from the fact that you only have a probability of heads or tail, right? Yeah. So if you had three, it'd be log three. Absolutely. Absolutely. Now, computer scientists of course like to think in terms of two for a variety of reasons. First of all, the mathematics of it is nice, but two is the smallest number which is not one. It's a small signature. Yeah, it's a small signature, not equal to one. But it's also true that the physics that goes on inside your computer is connected with switches which are either on or off. So counting in units of log two is very useful. Excuse me. Yeah? It's probably not important, but what is the base of the logarithm? Yeah, the standard, this is a definition, of course. This is definition. The definition, okay, good. The, good. It depends on who you are. If you're an information theorist, a computer scientist often, or a disciple of Shannon, then you like log to the base two. In which case, this is just one, n, and the entropy here is just n, measured in units of bits. If you're a physicist, then you usually work in base E. Okay? But the relationship is just a multiplicative, you know, log to the base E and log to the base two are just related by a numerical factor that's always the same. Okay. So, when I write log log, I mean log to the base E, but very little would change if we use some other base for the logs. Okay. So there we are. Let me just tell you what the definition is of the entropy in phase space. If we're not talking about, if we're not talking about these finite discrete systems like this, we're talking about continuously infinite systems, phase space. Now let's begin supposing the probability distribution is just some blob where the probabilities are equal inside the blob and zero outside the blob. In other words, the simple situation. Then the definition of the entropy is simply, well, you could say the logarithm of the number of states, but how many states are there in here? Clearly a continuous infinity of them. So instead, you just say it's the log of the volume of the probability distribution in phase space. S for a continuous system is just defined to be the logarithm of the volume in phase space. Now if I wrote V, you might get the sense that I'm talking about volume in space. No, I'm talking about the logarithm in phase space, the volume in the phase space. The phase space is high dimensional. Another dimensionality of the phase space is the volume is measured in those kind of in units of momentum times velocity to the power of the number of coordinates in the system. All right, but S is equal to the logarithm of the phase space volume. Let's call it P phase, V phase space. That's under the volume of the region which is occupied and has a non-zero probability distribution. This is the closest analog that we can think of to log M, where M represents the number of equally probable states of the discrete system. More generally, if we want, if we have some arbitrary probability distribution, the arbitrary probability distribution, P, what would P probability? What would it be a function of? It would be a function of all of the coordinates and all of the momenta. All of the momenta and all of the coordinates. P's and Q's or P's and X's, whichever you like, all of them. Probability for the system to be located at point of phase space P and X. Incidentally, when you have continuous variables like that, do you write that the sum is all equal to one? That wouldn't make sense. You can't sum over a continuously infinite set of variables. It becomes integral. We'll come back to this, but let's just spell it out right now. That if you have a probability distribution on a phase space, the rule is that the integral of it is equal one. P is really a probability density on phase space. It's a probability per unit cell in phase space. What would you expect the entropy to be? I'll give you a hint that starts out with minus. Where's the formula that we had here before? Let's rewrite the formula that we had before. S is equal to minus summation over I of P sub I log of P sub I. To go to the continuum, you simply replace sum by integral. There's an integral over the phase space. That's like the sum over I. Then probability of P and Q times the logarithm of the probability. In first approximation, I don't mean first approximation numerically. I mean first conceptual approximation. It's measuring the logarithm of the volume of the probability blob in phase space. Okay, oh, I could know mistake. In classical mechanics, X's and Q's are coordinates and P's are momenta. You know that. Okay, so we've now defined entropy, which, as we've seen, depends on a probability distribution. I'm going to go one more step tonight and define temperature. We could stop here. I think that's probably enough for one night. Next time we'll do temperature and then discuss the Boltzmann distribution, which is the probability distribution for thermal equilibrium. We haven't quite defined thermal equilibrium yet, but we will. You can ask some questions. I don't mind some questions now. I just, I just, I have the feeling that I've probably done enough for one night. Oh. Well, I haven't made any such assumption. I haven't made any such assumption. My only assumption in calculating the various entropies was either that you know nothing or I told you what you know. How you got to know that and what the reasoning was and whether it had to do with knowing something about the dynamics and the independence and so forth may come into your calculation of what P is. But in saying you know nothing, the implication was that all states are equally probable. Without asking how you knew that. To say that all states are equally probable is closely related to saying that there are no correlations. It does say that if you, all right, let's, good. Let's talk about correlations for a moment. To say that you know nothing means you know nothing. So in particular, if you, you, you begin knowing nothing and you measure one of the coins, what do you know about the other ones? Nothing. Nothing. You started knowing nothing about them. You measured one of them. You still know nothing about the other ones. Of course, you know about the one that you've measured. Now let's take the other, the other case that we studied. Anything. We know that all coins are heads except for one which is tails. And we now measure one of them and we find that it's tails. What can we say about some other one? It's surely heads. What if we measure that that one is heads? What do we know about the other ones? It changes the probability. It changes the probability. Just a new probability that one of the other ones is tails. Right. It's one over n, one over n minus one instead of one over, sorry, one over, yeah, one over n minus one. So that's correlation. That's correlation where when you measure something, you learn something new or the probability distribution for the other things is modified by measuring something. That's called correlation. For the complete ignorance, there is no correlation. For any other kind of configuration in general, there's very likely to be some, not necessarily, but there's very likely to be some correlation. Correlation as I said means you learn something about the probability distribution of other things by measuring the first one. Or you modify the probability distribution. Good. Okay. I don't know if that's what you asked about or not, but yeah. Okay. All right. Well, is that original system there? We had two parts. Once we measured one thing, we had to conserve quantity. What's that again? We had to conserve quantity there that once we did one measurement, we knew which side we were. Yeah. Yeah. Incidentally, entropy is additive. It's additive. It's the sum of the entropies of all the individuals. It's proportional to a number of things. It's additive whenever there is no correlation. When there's no correlation, it's additive. Now, so uncorrelated systems have additive entropies. We'll come back. That's a theme that we'll come back to. Is an interesting question. Here's what you know, you know that if you measure a coin up, then with three-quarter probability, it's too, they're laid out in a row. If you measure one of them and it's up, then the probability for its neighbors is three-quarters to be down. That's what you're given. That's all you know. All you know is that if one of them is up, if any one of them is up, its neighbors are three-quarters likely to be down. It's an interesting thing to try to calculate the entropy of such a distribution. That is correlated, of course. That is correlated because when you measure one, you immediately know something about its neighbor. Well, make up your own example like that. Make up your own example like that and compute the entropy. You can learn something from it. Any other questions before we go home? Not particularly. The formula here is due to Boltzmann. I think it's just the one that's on Boltzmann's tomb, I'm not sure. Now what's on Boltzmann's tomb? He meant this. Well he did write this. This is Boltzmann's final formula for entropy. The only difference between the Shannon entropy and the Boltzmann entropy is that Shannon used log two. Now, of course, Shannon discovered this entirely by himself. He didn't know Boltzmann's work from an entirely different direction from information theory rather than thermodynamics, but none of it would have surprised Boltzmann. Nor do I think Boltzmann's definition would have surprised Shannon. So they're really the same thing. There's no real point in comparing them because they're comparing them, they are the same. There's no real difference. Shannon may have, I don't know whether he did or not put the minus sign in here. If you don't put the minus sign in there, it's called information. If you put the minus sign, it's called lack of information or entropy. So I don't know which Shannon wrote down. Anyway. Shannon wrote down entropy. He did. Yeah. Is there a simple way to relate to Heisenberg's uncertainty principle? No, no, this is a separate issue. This doesn't have to do with quantum mechanical uncertainty. It has to do with the uncertainty implicit in mixed states, not the uncertainty implicit in pure states. Okay. Oh, oh boy. Yeah, there's a conversion factor. You know, C equals H bar equals G equals Boltzmann's constant equals one. Very natural. Boltzmann's constant was a conversion factor from temperature to energy. The natural unit for temperature is really energy. But the energy of a molecule, for example, is approximately equal to its temperature in certain units. Those units contain a conversion factor K Boltzmann. I'll remind me to talk about K Boltzmann. For more, please visit us at stanford.edu. | {
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"text": [
" Okay, if there are no more questions, tonight we begin the study of statistical mechanics.",
" Now statistical mechanics is not really modern physics.",
" It's pre-modern physics.",
" It's modern physics.",
" And I assure you it will be post-modern physics.",
" It's probably, the second law of thermodynamics will probably outlast anything that comes",
" up time and time again.",
" The second law of thermodynamics has been sort of our guidepost, our guiding light,",
" if you like, to know what we're talking about and to make sure we're making sense.",
" Statistical mechanics and thermodynamics may not be as sexy as the Higgs boson, but I",
" assure you it is at least as deep.",
" And it's a lot deeper.",
" My particle physics friends shouldn't disown me.",
" It's a lot deeper.",
" It's a lot more general.",
" And it covers a lot more ground than explaining the world as we know it.",
" And in fact, without statistical mechanics, we probably would not know about the Higgs",
" boson.",
" All right, so with that little starting point, what are statistical mechanics about?",
" Well, let's go back a step.",
" The laws of physics, the basic laws of physics, Newton's laws, principles of classical physics,",
" quantum mechanics, the things that were in the classical mechanics course, quantum mechanics,",
" and so forth, those things are all about predictability, perfect predictability.",
" Now you say, well, in quantum mechanics, you can't predict perfectly.",
" And that's true, but there are some things you can predict perfectly.",
" And those things are the predictables of quantum mechanics.",
" Again, as in classical mechanics, you can make your predictions with maximal, let's",
" call it maximal precision or maximal, whatever it is, predictability, if you know two things.",
" If you know the starting point, which is what we call initial conditions, and if you know",
" the laws of evolution of a system.",
" If you can measure the, or if you know for whatever reason, the initial starting point",
" of a closed system, a closed system means one which is either everything or it is sufficiently",
" isolated from everything else that the other things in the system don't influence it.",
" If you have a closed system, if you know the initial conditions exactly, or at least with",
" whatever precision is necessary, and you know the laws of evolution of the system, you have",
" complete predictability and that's all there is to say.",
" Now of course, in many cases that complete predictability would be totally useless,",
" having a list of the position and velocities of every particle in this room would not be",
" very useful to us.",
" The list would be too long and subject to rather quick change as a matter of fact.",
" So you can see while the basic laws of physics are very, very powerful in their predictability,",
" they also in many cases can be totally useless for actually analyzing what's really going",
" on.",
" Statistical mechanics is what you use, basically probability theory.",
" Statistical mechanics, let me say first of all, it is just basic probability theory.",
" Statistical as applied to physical systems.",
" When is it applicable?",
" It's applicable when you don't know the initial conditions with complete perfection.",
" It's applicable when you, it may even be applicable if you don't know the laws of motion with",
" infinite precision.",
" And it's applicable when the system you're investigating is not a closed system, whether",
" it's interacting with other things on the outside.",
" In other words, in just those situations where ideal predictability is impossible, then what",
" do you resort to?",
" You resort to probabilities.",
" But because the number of molecules in this room is so large, and probabilities tend to",
" become very, very precise predictors when the laws of large numbers are applicable, statistical",
" mechanics itself can be highly predictable, but not for everything.",
" As an illustration, you have a box of gas.",
" The box of gas might even be an isolated closed box of gas.",
" It has some energy in it.",
" The particles rattle around.",
" If you know some things about that box of gas, you can predict other things with great",
" precision.",
" If you know the temperature, you can predict the energy in the box of gas.",
" You can predict the pressure.",
" These things are highly predictable, but there are some things you can't predict.",
" You can't predict the position of every molecule.",
" You can't predict when there might be a fluctuation.",
" A fluctuation, which, you know, fluctuations are things which happen which don't really",
" violate probability theory.",
" They're the sort of tails of the probability distribution, things which are unlikely but",
" not impossible.",
" Fluctions happen from time to time in the sealed room every so often.",
" An extra large group, an extra large density of molecules will appear in some small region",
" bigger than the average someplace else.",
" Molecules will be less dense.",
" And fluctuations like that are hard to predict.",
" You can predict the probability for a fluctuation, but you can't predict when a fluctuation is",
" going to happen.",
" It's exactly the same sort of thing, flipping coins.",
" Flipping coins is a good example, probably our favorite example for thinking about probabilities.",
" If I flip a coin a billion times, you can bet that approximately half of them will come",
" up heads and half will come up tails within some margin of error.",
" But there will also be fluctuations.",
" Every now and then, if you do it enough times, a thousand heads in a row will come up.",
" Can you predict when a thousand heads will come up?",
" No.",
" But can you predict how often a thousand heads will come up?",
" Yes.",
" Not very often.",
" So that's what statistical mechanics is for.",
" It's for making statistical probabilistic predictions about systems which are either",
" too small, contain elements which are too small to see, too numerous to keep track of,",
" usually both, too small to see by sea.",
" I mean, you know, it's true.",
" You can see some pretty small things, molecules, but so they may not be too small to see.",
" But there are too many of them.",
" There are too many of them to keep track of.",
" And that's when we use probability theory or statistical mechanics.",
" We're going to go through some of the basic statistical mechanics applications, not just",
" applications, the theory, the laws of thermodynamics, the laws of statistical mechanics, and then",
" how they apply to gases, liquids, solids, whether we will get the quantum mechanical",
" systems or not.",
" I don't know.",
" But just the basic ideas.",
" Okay.",
" And, incidentally, another thing which is very striking is that generally speaking over",
" the history of, certainly over my history in physics, and I'm sure this goes back to",
" the middle of the 19th century sometime, all great physicists, all of them were masters",
" of statistical mechanics.",
" It may not have been the sexiest thing in the world, but they were all masters of it.",
" Why?",
" First of all, because it was useful, but second of all, because it is truly beautiful.",
" It is a truly beautiful subject of physics and mathematics.",
" And it's hard not to get caught up in it.",
" Not to fall in love with it.",
" The reason I teach it is not for you.",
" It's for me.",
" I love teaching it.",
" I love teaching it.",
" I teach it over and over and over again.",
" And in a sense, my life is consisted of learning and forgetting and learning and forgetting",
" and learning and forgetting statistical mechanics.",
" So here's my opportunity to learn it again.",
" Okay.",
" Let's begin with what I usually call a mathematical interlude.",
" In this case, it's not an interlude.",
" It's a starting point.",
" And I'm just going to make some extremely brief remarks which you all know.",
" At least I think you all know them about probability.",
" Just to have, you know, just to level the ground, what are we talking about?",
" And what I am not going to explain, because I don't think anybody can explain it, is",
" why probability works.",
" Why does it work?",
" If you ask why it works, the first answer will be it doesn't always work.",
" You may have a probability for something and you test it out.",
" And sometimes it doesn't work.",
" Those are called the exceptions.",
" So the answer to the question is why does it work?",
" Well, it doesn't always work.",
" It mostly works, except when it doesn't.",
" When doesn't it?",
" Rarely.",
" How rarely?",
" Every so often.",
" But there is a calculus of probability, a mathematical theory of probabilities.",
" And we'll talk about it a little bit.",
" Okay.",
" So we'll take probability to be a primitive concept, basically primitive concept.",
" And we'll suppose that there is a space of some sort, a space of possibilities.",
" The space of possibilities could be the space of outcomes of experiments, or it could actually",
" be the space of states of a system.",
" The state of a system could be the outcome of an experiment.",
" If the experiment consists of determining the state of the system, then the state of",
" the system is the outcome.",
" So we have a space, and let's call that space, let's label the elements of that space with",
" a little i.",
" For example, if we were flipping coins, i would be either heads or tails.",
" If we were flipping dice, you know, dice, dice of dice, i would run from one to six.",
" If there were two dyes, then we would have enough indices to keep track of two dice and",
" so forth.",
" So i is the space of possibilities of outcomes, or the space of possible states of a system.",
" And if we are ignorant, statistics always has to do with ignorance.",
" You don't know everything, and so you assign probabilities to outcomes.",
" And so we assign a probability p of i to the ith outcome to the answer to our question.",
" Okay?",
" What are the rules for p of i?",
" What does p of i have to satisfy?",
" And let's, for the beginning, at least in the beginning, let's imagine that i enumerates",
" some discrete finite collection of possibilities.",
" Later on, we can have an infinite number of possibilities, or even a continuously infinite",
" number of possibilities.",
" But for the time being, i might run from one to n, n possibilities.",
" And the rules are, first of all, p sub i has to be greater than or equal to zero.",
" Negative probabilities, we don't like them.",
" Don't know what they mean.",
" Okay?",
" Next, the summation over i of p sub i, p of i, should be one.",
" That means that the total probability, when you add everything up, all possibilities should",
" be one.",
" You certainly should get some result.",
" Okay?",
" Next, now this is a kind of hypothesis.",
" This is the law of large numbers that if you either make many replicas of the same system",
" or do the same experiment over and over a very, very large number of times, and take all of",
" the outcomes which gave you all of the experiments which gave you the i-th outcome, that's some",
" number, let's call it n of i, that's the number of times that the experiment turned up the",
" i-th possibility, and you divide it by the total number of trials.",
" Total number of trials means the sum of all i, or just the total number of trials, that",
" the limit of this, this is a physical hypothesis.",
" It's a physical hypothesis, it can go wrong if n is not large enough, but in the limit",
" of large n, n goes to infinity, and the limit of very, very, of course n never goes to infinity,",
" you never get to do an infinite number of experiments.",
" But nevertheless we're kind of idealizing, we're assuming we can do so many experiments",
" that the limit n goes to infinity is effectively been reached, then that is p of i.",
" So p of i controls by assumption the ratio of the n of i's.",
" Okay, everybody happy with that?",
" You use this all the time I think.",
" Well, sometimes we use it.",
" Okay, now let's suppose that there is a quantity, let's call it f of i.",
" It's some quantity that's associated with the i-th state.",
" We can assign it, we can make it up.",
" For example, if our system is heads and tails and nothing but heads and tails, we could assign",
" f of heads and call it plus one, and f of tails and call it minus one.",
" If our system has many, many more states, we may want to assign a much larger number",
" of possible f's, but f is some function of the state.",
" It's also a thing that we imagine measuring.",
" It could be the energy of a state, or it could be the momentum of a state.",
" Given a state of some system, it has an energy.",
" It would be called in that case perhaps e of i, or it could be the momentum, or it",
" could be something else.",
" It could be whatever you happen to like to think about.",
" Then an important quantity is the average of f of i.",
" The average of f of i, I will use the quantum mechanical notation for it, even though we're",
" not doing quantum mechanics.",
" It's a nice notation.",
" Physicists tend to use it all over the place.",
" Mathematicians hate it.",
" Just put a pair of brackets around it.",
" It means the average.",
" The average value of the quantity averaged over the probability distribution.",
" It has a definition.",
" Its definition is that it's the sum of i of f of i weighted with the probability.",
" For example, and incidentally, the average of f of i does not have to be any of the possible",
" values that f can take on.",
" For example, in this case, where f of heads is plus one and f of tails is minus one, and",
" you flip a million times, and the probability is a half of heads and a half of tails, the",
" average of f will be zero.",
" So it's not a possible outcome to the experiment.",
" There's no rule why the average should be one of the possible experimental outputs, but",
" it is the average.",
" This is its definition.",
" Each value of f is weighted with the probability for that value of f.",
" You can write it another way.",
" You can write it as a sum over i of f of i times the number of times that you measure",
" i divided by the total number of measurements.",
" That's what p of i is in the limit that there are a large number of measurements.",
" That's defined to be the average.",
" That's our mathematical preliminary for today.",
" That's all I wanted to level the playing field by making sure everybody knows what the probability",
" is and what an average is.",
" We'll use it over and over.",
" Okay, let's start with coin flips.",
" I always start with coin.",
" I start every single class with coin flips, even when I'm teaching about the Higgs boson.",
" Okay.",
" If I flip a coin a lot of times, or whether I flip a coin a lot of times or not, the probability",
" for heads is usually deemed to be one-half and the probability for tails is usually also",
" deemed to be one-half.",
" Why do we do that?",
" Why is it a half and a half?",
" What's the logic there?",
" What logic tells us that?",
" In this case, it's symmetry.",
" It's the symmetry of the coin.",
" Of course, no coin is perfectly symmetric and even making a little mark on it to distinguish",
" the heads and tails, bias is it a little bit, but apart from that tiny, tiny bias of marking",
" the coin with maybe just a tiny little scratch, the coin is symmetric.",
" Higgs and tails are symmetric with respect to each other and therefore there is no reason,",
" no rationale for when you flip a coin for it to turn up heads more often than tails.",
" It's symmetry quite often.",
" I might even say always in some deeper sense, but at least in many cases, symmetry is the",
" thing which dictates probabilities.",
" Probabilities are usually taken to be equal for configurations which are related to each",
" other by some symmetry.",
" Symmetry means if you act with a symmetry, you reflect everything, you turn everything",
" over that the system behaves the same way.",
" Okay, another example besides coin flipping would be dice flipping.",
" Dice flipping instead of having two states has six states, one die, and we can imagine",
" coloring them.",
" We color the faces, red, yellow, blue, and then on the back green, purple, and orange.",
" Okay, that's our die and it's been colored.",
" We don't have to keep track of numbers, we can keep track of colors.",
" And what is the probability that when we flip the die, flip it into the air, hits the ground,",
" what's the probability that it turns up red?",
" No, it's one-sixth, right?",
" There are six possibilities.",
" They're all symmetric with respect to each other.",
" We use the principle of symmetry to tell us that the P of each i, they're all equal and",
" they're all equal to one-sixth.",
" But what if there is no symmetry?",
" What if really the die is not symmetric?",
" For example, what if it's weighted in some unfair way?",
" Or what if it's been cut with faces that are not nice and parallel cubes?",
" Then what's the answer?",
" The answer is symmetry won't tell you.",
" You may be able to use some deeper underlying theory and to use some concept of symmetry",
" from the deeper underlying theory, but in the absence of something else, there is no",
" answer.",
" The answer is experiment.",
" Do this experiment a billion times, keep track of the numbers, assume that things have converged",
" and that way you measure the probabilities.",
" You measure the probabilities and thereafter you can use them.",
" You can use them if you keep a table of them and then you can use them in the next round",
" of experiments.",
" Or you may have some theory, some deep underlying theory which tells you well, like quantum",
" mechanics or statistical mechanics.",
" Statistical mechanics tends to rely mostly on symmetry, as we'll see.",
" So if there's no symmetry to guide you or to guide your implementation of probabilities,",
" then it's experiment.",
" Now there's another answer.",
" There's another possible answer.",
" This answer is often frequently invoked and it's a correct answer under other circumstances.",
" It can have to do with the evolution of a system, the way a system changes with time.",
" So let me give you some examples of what it might have to do.",
" Let's take our six sided cube and assume that our six sided cube is not symmetric.",
" It's not symmetric but we know a rule.",
" We know that if we put that cube down on the table, it's not a cube, when we put that die",
" down on the table and we stand back, this thing has this habit of jumping to another",
" state and jumping to another state and jumping to another state.",
" It's called the law of motion of the system.",
" The law of motion of the system is that whatever it is at one instant, at some next instant,",
" it will be something else according to a definite rule.",
" The instance could be seconds, it could be microseconds or whatever, but imagine a discrete",
" sequence and let's suppose there's a law, a genuine law that tells us how this cube moves",
" around.",
" For example, if it's red, now we've done this over and over many times in different contexts",
" but it is so important that I feel a need to emphasize it again.",
" This is what a law of motion is.",
" It's a rule telling you what the next configuration will be given, it's a rule of updating, of",
" updating configurations.",
" Red goes to blue, blue goes to yellow, yellow goes to green, green goes to orange, orange",
" goes to purple and purple goes back to red.",
" Given the configuration at any time, you know what it will be next and you know what it",
" will continue to do.",
" Of course, you may not know the law.",
" Maybe all you know is that there is a law of this type.",
" You know what I'm going to do next, what am I going to do next?",
" I'm going to draw this law as a diagram.",
" You've all seen me do this in other contexts.",
" Let's do it.",
" We have red, too hard to draw squares.",
" Red, blue, green, orange, what happened?",
" Yellow, yellow, orange, purple.",
" A law like this can be just represented by a set of lines connecting a set of arrows.",
" Red goes to blue, blue goes to green, green goes to yellow, yellow goes to orange, orange",
" goes to purple, purple goes back to red.",
" Given the assumption now that there's a discrete time interval between such events, I am not",
" assuming that the cube has any symmetry to it anymore.",
" The cube may not be symmetric at all.",
" It may have points, you know, one edge, one face, maybe tiny, another face, but if this",
" is the rule to go from one configuration to another and each step takes, let's say, a",
" microsecond, I might have no idea where I begin, but I can still tell you if I, let's",
" say it's a microsecond, a microsecond, and my job is to catch it at a particular instant",
" and ask what the color is.",
" I don't know where it started, okay?",
" But I can still tell you the probability for each one of these is one-sixth.",
" It doesn't have to do with symmetry.",
" Well, maybe it does have to do with some symmetry, but in this case, it wouldn't be the symmetry",
" of the structure of a die.",
" It would just be the fact that as it passes through these sequence of states, it spends",
" one-sixth of its time red, one-sixth of its time blue, one-sixth of a time green, and if",
" I don't know where it starts and I just take a flash, you know, a flash shot of it, my",
" probability will be one-sixth.",
" Now that one-sixth did not really depend on knowing the detailed law.",
" For example, the law could have been different.",
" Let's make up a new law.",
" Red goes to green, green goes to orange, orange goes to yellow, yellow goes to purple,",
" purple goes to blue, and blue goes back to red.",
" This shares with the previous law that there's a closed cycle of events in which you pass",
" through each color once before you cycle around.",
" You may not know which law of nature is for this system, but you can tell me again that",
" the probability will be one-sixth for each one of them.",
" So this prediction of one-sixth doesn't depend on knowing the starting point and doesn't",
" depend on knowing the law of physics.",
" It's just important to know that there is a particular kind of law.",
" Are there possible laws for the system which will not give you one-sixth?",
" Yes.",
" Let's write another law.",
" Red, blue, green, yellow, orange, purple.",
" This rule says that if you start with red, you go to blue.",
" If you start with blue, you go to green, and if you get to green, you go back to red.",
" Or if you start with purple, you go to yellow, yellow to orange, orange back to purple.",
" Notice in this case, if you're on one of these two cycles, you stay there forever.",
" If you knew you were on the upper cycle, if you knew you would start it, it doesn't matter",
" where you start, but if you knew that you started on the upper cycle somewhere, then",
" you would know that there was a one-third probability to be red, a one-third probability",
" to be blue, and a one-third probability to be green, and zero probability to be red,",
" yellow, or orange.",
" On the other hand, you could have started with the second cycle.",
" You could have started with purple.",
" Might not have known where you started, but you knew that you started in the lower triangle",
" and the lower cycle here.",
" Then you would know the probabilities of one-third for each of these and zero for each of these.",
" Now, what about a more general case?",
" The more general case might be that you know with some probability that you start on the",
" upper triangle here and with some other probability on the lower triangle.",
" In fact, let's give these triangles names.",
" Let's call this triangle the plus one triangle, and this one the minus one triangle.",
" Just giving them names, attaching to them a number, a numerical value.",
" If you hear something or other is called plus one, if you hear something or other is called",
" minus one.",
" All right, now you'll have to append, you'll have to start with something you've got to",
" get from someplace else.",
" It doesn't follow from symmetry, and it doesn't follow from cycling through the system some",
" probability that you're either on cycle plus one or cycle minus one.",
" Where might that come from?",
" Flipping somebody else's coin over here, flipping a coin over here might decide which of these",
" two.",
" It might be a biased coin, so you will have a probability to be plus one and a probability",
" to be minus one.",
" These two probabilities are not probabilities for individual colors, they're probabilities",
" for individual cycles.",
" Okay, now what's the probability for blue?",
" The probability for blue begins with the probability that you're on the first cycle, times the",
" probability that if you're on the first cycle, you get blue.",
" That's one third.",
" So the probability for blue, red, or green is one third the probability that you're on",
" the first cycle, and likewise the probability that you're at yellow will be, this is the",
" probability for red, blue, or green in this case, and this times one third will be the",
" probability for purple, yellow, or orange.",
" Okay, so in this case you need to supply another probability that you've got to get from somebody's",
" else.",
" This case here is what we call having a conservation law.",
" In this case, the conservation law would be just the conservation of this number.",
" For red, blue, and green, we've assigned the value plus one.",
" That plus one could be the energy, or it could be something else.",
" I tend to call it the zilch for some reason.",
" I call everything a zilch if there is no name for it.",
" So anyway, let's think of it as the energy to keep things familiar.",
" The energy of these three configurations might all be plus one, and the energy of these three",
" configurations might all be minus one.",
" And the point is that because the rule keeps you always on the same cycle, that quantity,",
" energy, zilch, whatever we call it, is conserved.",
" It doesn't change.",
" That's what a conservation law is.",
" A conservation law is that the configuration space, the space of possibilities, divides",
" up into cycles like this.",
" Now, the cycles don't have to have equal size.",
" Here's another case.",
" One, two, three, four.",
" You go around this way, and then the two guys over here go into each other.",
" So red goes to blue, goes to green, goes to purple.",
" That's the upper cycle here, and the lower cycle is yellow goes to orange, goes to yellow",
" goes to orange.",
" Still, we have a conservation law here.",
" It's just the number of states with one value of the conserved quantities, not the same",
" as the number of states or the other value, but still, it's a conservation law.",
" And again, somebody would have to supply for you some idea of the relative probabilities",
" of these two.",
" Where that comes from is part of the study of statistical mechanics.",
" And the other part of the study has to do with saying, if I know I'm one of these tracks,",
" how much time do I spend with each particular configuration?",
" That's what determines probabilities of statistical mechanics.",
" Some a priori probability from somewheres that tells you the probabilities for different",
" conserved quantities and cycling through the system.",
" Yeah, question?",
" No, okay.",
" So, so far, you're assuming that within any conservation arena, you will, the probabilities",
" of all the states are the same?",
" The time spent in each state is the same.",
" Right.",
" So, it's completely deterministic.",
" Laws are completely deterministic.",
" This would be classical physics.",
" Laws completely deterministic.",
" No real ambiguity of what the state is, except you're kind of lazy.",
" You didn't determine the initial condition.",
" Your timing wasn't very good.",
" Each state only lasts for a microsecond.",
" You're a lazy guy and you only have a resolution of a millisecond.",
" But nevertheless, you're able to take a very quick flash picture and pick out one of",
" the states.",
" That's the circumstance that we're talking about.",
" Yeah.",
" If we take two pictures, is it reasonable to then assume that if the first picture indicated",
" that we are in one cycle, the later one should indicate the same cycle since it couldn't",
" get out of it?",
" Yes, that's a good assumption.",
" Yes.",
" Right.",
" So, once you determine the value of some conserved quantities, then you know it.",
" And then you can reset the probabilities for it.",
" Unless, all right, so let's talk about honest energy for a minute.",
" Yes, if we have a closed system, to represent the closed system, I will just draw a box.",
" Closed now, closed means that it's not in interaction with anything else and therefore",
" can be thought of as a whole universe unto itself.",
" Okay.",
" It has an energy.",
" The energy is some function of the state of the system, whatever determines the state",
" of the system.",
" Now let's suppose we have another closed system which is built out of two identical or not",
" the identical versions of the same thing.",
" Now, if they're both closed systems, there will be two conserved quantities.",
" The energy of this system and the energy of this system and they'll both be separately",
" conserved.",
" Why?",
" Because they don't talk to each other.",
" They don't interact with each other.",
" The two energies are conserved and you could have probabilities for each of those individuals.",
" But now supposing they're connected.",
" They're connected by a little tiny tube which allows energy to flow back and forth.",
" Then there's only one conserved quantity, the total energy, and it's sort of split between",
" the two of them.",
" You can then ask, what is the probability given a total amount of energy?",
" You could ask, what's the probability that the energy of one subsystem is one thing and",
" the energy of the other subsystem is the other?",
" If the two boxes are equal, you would expect on the average they have equal energy, but",
" you can still ask, what's the probability for a given energy in this box given some",
" overall piece of information?",
" That's a circumstance where it may be that giving the probability for which cycle you're",
" on, now which cycle you're on, I'm talking about the cycle of one of these systems here,",
" may be determined by thinking about the system as part of a bigger system.",
" And we're going to do that.",
" That's important.",
" But in general, you need some other ingredient besides just cycling around through the system",
" here to tell you the relative probabilities of conserved quantities.",
" Okay.",
" So we're often flying with statistical mechanics.",
" There are bad laws.",
" By bad laws, I don't, not in the sense of DOMA or any of those kind of laws, but in",
" the sense that the rules of physics don't allow them.",
" You all know what they are.",
" The laws that violate the conservation of information.",
" The most primitive and basic rule of physics, the conservation of information.",
" Conservation of information is not a standard conservation law like this.",
" It's the rule that you keep, that you can keep track.",
" You can keep track both going forward and backward.",
" So let's just mention that again.",
" It's all work, it's all described in the classical mechanics book.",
" I'm just reviewing it now.",
" But let's take a bad law.",
" It's a possible law.",
" By bad, I mean one, two, three, four, one, two, three, four, five, six.",
" So these are the faces of a die again.",
" But the rule is wherever you are, this is red, wherever you are, you go to red.",
" Even if you are red, you go to red.",
" Okay.",
" We'll discuss in a moment what's wrong with this law.",
" But this law has one of the features that it has, is it's not reversible.",
" It's not reversible in the sense that you can go from blue to red, but you cannot go",
" from red back to blue.",
" So in that sense, it's not reversible.",
" You can predict the future wherever you are.",
" The future is very simple for this particular law, wherever you are, you'll next be at red.",
" You can make it more complicated.",
" You could make a few, you can make it more complicated.",
" But this law always winds up with red.",
" It's a bad law because it loses track of where you started.",
" Whereas these laws don't lose track.",
" If you know that you've gone through 56, 56 and a half cycles, then you know that if",
" you started at red, you'll come back and you can tell exactly where you'll be.",
" And you can also tell where you came from.",
" You can tell not only where you'll be, but exactly where you came from.",
" Well, this law, you can't say where you came from.",
" This is a law that loses information.",
" And it's exactly the kind of thing that classical physics does not allow.",
" Classical physics also doesn't allow the quantum mechanical version of it.",
" So the rule that this type of rule, that this type of law is unallowed, I give a name to.",
" It's as I said many times, there is no name for it because it's just so basically primitive",
" that everybody always forgets about it.",
" It's so basic.",
" I call it the minus first law of physics.",
" And I wish it would catch on.",
" People should start using it.",
" I mean, it is really the most basic law of physics that information is never lost, that",
" distinctions or differences between states propagate with time and you never lose track.",
" In principle, if you have the capacity to follow the system, because you may be too lazy",
" to follow the system, that's your problem.",
" But nature doesn't have that problem.",
" Nature allows, in principle, that you can reconstruct where you came from.",
" All right, so that's a bad law.",
" How do you tell the good laws from the bad laws?",
" Just by diagrammatics here, it's very simple.",
" Good laws, every state has one incoming arrow and one outgoing arrow.",
" An arrow to tell you where you came from and an arrow to tell you where you're going.",
" So those are good laws.",
" In classical mechanics, continuum classical mechanics, there is a version of this same",
" law.",
" Anybody know the name of that version?",
" Of the theorem that goes with the conservation of information?",
" It's called Leaville's theorem.",
" We studied it in classical mechanics.",
" But let me give you a counter example to Leaville's theorem.",
" Friction is an apparent contradiction.",
" Wherever you start, you come to rest.",
" It's sort of like saying wherever you start, you come to red.",
" Wherever you start, you come to rest.",
" Well, you may not know exactly where you are, but you always come to rest.",
" That seems like a violation of the laws that tell you that distinctions have to be preserved.",
" But of course, it's not really true.",
" What's really going on is that when you run the eraser through here, it's heating up the",
" surface here.",
" And if you could keep track of every molecule, you would find out that the distinctions between",
" starting points is recorded.",
" But let's imagine now that there was a fundamental law of physics.",
" By fundamental law, I mean a rock bottom fundamental law for a series of particles, for a collection",
" of particles, and the equations of motion for the particles were this.",
" d second x, that's the position of the particle, somewhere by dt squared, that's called acceleration.",
" We could put a mass in, but the mass is not doing anything.",
" There's a lot of particles, so I'll label them i.",
" Oh, we've used i to label states.",
" I should not do that.",
" Let's call it n, little n.",
" The nth particle, and what is that equal to?",
" It's equal to minus some number gamma, we've seen that number before in another context,",
" times dxn by dt.",
" Anybody remember what this formula represents?",
" Friction.",
" Viscous drag.",
" Again, it has the property that if you start with a moving particle, it will very quickly",
" come almost to rest.",
" It'll exponentially come to rest pretty quickly.",
" And so if all particles in a gas, for example, satisfied this law of physics, it's perfectly",
" deterministic.",
" It tells you what happens next, but it has the unfortunate consequence that every particle",
" just comes to rest.",
" That sounds odd.",
" It sounds like no matter what temperature you start the room, it will quickly come to",
" zero temperature.",
" That doesn't happen.",
" This is a perfectly good differential equation, but there's something wrong with it from the",
" point of view of conservation of energy.",
" There's something wrong with it from the point of view of thermodynamics.",
" If you start a closed system and you start it running, you start with a lot of kinetic",
" energy, temperature we usually call it.",
" It doesn't run to zero temperature.",
" That's not what happens.",
" In fact, this is not only a violation of energy conservation.",
" It looks like a violation of the second law of thermodynamics.",
" It says things get simpler.",
" You start with a random bunch of particles moving in random directions, and you let it",
" run and they all come to rest.",
" What you end up with is simpler and requires less information to describe than what you",
" started with.",
" That's very, very much like everything going to red.",
" Among other things, it violates the second law of thermodynamics, which generally says",
" things get worse.",
" Things get more complicated, not less complicated.",
" Okay.",
" But there's another way to say, another important way to say this rule, that every state has",
" to have one arrow in and one arrow out.",
" The thing that I called either the minus first law or the conservation of information.",
" Supposing we have a collection of states and we assign to them probabilities.",
" P of state one, P of state two, P of state three, and so forth.",
" For some subset of the states, not all of them, some subset of them.",
" All the others, we say have probability zero.",
" Okay.",
" So, for example, we can take our die and assign red, yellow, and blue probability a third,",
" and green, orange, and pink, or whatever it was, probability zero.",
" Where we got that from, doesn't matter.",
" We got it from somewhere.",
" Somebody secretly told us in our ear, it's either red, yellow, or blue, and I'm not going",
" to tell you which.",
" All right.",
" And now you follow the system.",
" You follow it as it evolves.",
" Whatever kind of law of physics, as long as it's an allowable law of physics, after a",
" while, and you're following it in detail, you're not constrained by your laziness in",
" this case.",
" You are capable of following in detail.",
" And what is the probability, what are the probabilities at a later time?",
" Well, if you don't know which the laws of physics are, you can't say, of course.",
" But you can say one thing.",
" You can say there are three states with probability one-third and three states with probability",
" zero.",
" They may get reshuffled, which ones were probable and which ones were improbable, but after",
" a certain time, there will be those same three, not the same three states, but there will",
" continue to be three states which have probability and the rest don't.",
" So in general, you could characterize these information-conserving theories by saying,",
" supposing you assign some subset of the states, let's say, N out of N states, let's say there",
" are N states altogether, that's the total number of states, and now we look at some",
" M where M is less than N, and we say for those M states, the probability for those M states",
" is one over M for these states and zero for all the others.",
" You understand why I say one over M?",
" If there are M states equally probable, then each one has probability one over M and all",
" the remaining have probability zero.",
" Then the number of states which have non-zero probability will remain constant and the probabilities",
" will remain equal to one over M. Is that clear?",
" Is that obvious?",
" That should be obvious.",
" The states may reshuffle, but the number with non-zero probability will remain fixed.",
" That's a characterization, a different characterization of the information-conserving laws.",
" For the information-non-conserving laws, everybody goes to red.",
" You may start with a probability distribution that's one over five for red, green, purple,",
" orange, and yellow, and then a little bit later, there's only one state that has a probability",
" and that's red.",
" This is another way to describe information conservation.",
" We can quantify that.",
" We can quantify that by saying let M be the number of states which all have equal, under",
" the assumption that they all have equal probability, let M be the number, let's give it a name,",
" occupied states, states which have non-zero probability with equal probability, and then",
" M, what is M characterizing?",
" M is characterizing your ignorance.",
" The bigger M is, if M is equal to N, that means equal probability for everything.",
" Maximum ignorance.",
" If M is equal to one-half N, that means you know that the system is in one out of half",
" the states.",
" You're still pretty ignorant, but you're not that ignorant.",
" You're less ignorant.",
" What's the maximum, what's the minimum amount of ignorance you can have?",
" That you know precisely what state it's in, in which case M is what?",
" M is one.",
" You know that it's in one particular state.",
" All right, so M is a measure of your ignorance.",
" Really M in relation to N is a measure of your ignorance.",
" And associated with it is the concept of entropy.",
" Now we come to the concept of entropy, notice entropy is coming before anything else.",
" Entropy is coming before temperature, it's even coming before energy.",
" Entropy is more fundamental in a certain sense than any of them, although in a certain",
" sense it's, we'll discuss entropy in a minute, but S is a logarithm of M.",
" Logarithm of the number of states that have an appreciable probability more or less all",
" equal for the specific circumstance that I talked about.",
" That entropy is conserved.",
" All that happens is the states which are occupied reshuffle, but there will always be M of them",
" with probability one over M.",
" Okay, so that's where we are.",
" And that's the conservation of entropy if we can follow the system in detail.",
" Now, of course in reality we may be again lazy, lose track of the system, and we might",
" have after a point lost track of the equations, and lost track of our timing device, and so",
" forth and so on.",
" Now we may wind up, we may have started with some, a lot of knowledge, and wound up with",
" very little knowledge.",
" That's because, again, not because the equations cause information to be lost, but because",
" we just weren't careful.",
" Perhaps we can't be careful, perhaps there are too many degrees of freedom to keep track",
" of.",
" So when that happens, the entropy increases, but it simply increases because our ignorance",
" has gone up, not because anything has really happened in the system which has, if we could",
" follow it, we would find that the entropy is conserved.",
" Okay, that's the concept of entropy in a nutshell.",
" We're going to expand on it.",
" We're going to expand on it a lot.",
" We're going to redefine it with a more careful definition.",
" But what does it measure?",
" It measures approximately the number of states that have a non-zero probability.",
" Okay.",
" The bigger it is, the less you know.",
" What's the maximum value of s?",
" Log in, log in.",
" Now of course, n could be infinite.",
" You might have an infinite number of states, and if you do, then there's no upper bound",
" to the amount of ignorance you can have.",
" But you know, in a world with only n states, your ignorance is bounded.",
" So the notion of maximum entropy is a measure of how many states there are altogether.",
" Now I said that entropy is deep and fundamental, and so it is, but there's also an aspect",
" to it which makes it in a certain sense less fundamental.",
" It's not just a property of a system.",
" It's a property of a system and your state of knowledge of the system.",
" It depends on two things.",
" It depends on characteristics of the system, and it also depends on your state of knowledge",
" of the system or the state of knowledge of the system.",
" So keep that in mind.",
" Okay.",
" Now let's talk about continuous mechanics.",
" Mechanics of particles moving around with continuous positions, continuous velocities.",
" How do we describe that?",
" How do we describe the space of states of a mechanical system, you know, a real mechanical",
" system, particles and so forth?",
" We describe it as points in phase space.",
" We learned about phase space.",
" Phase space consists of positions and momenta.",
" Momenta in simple context, momentum is mass times velocity, so roughly speaking, it's",
" the space of positions and velocities.",
" Let's draw it.",
" P is momentum.",
" It goes that way.",
" And this axis is a stand-in for all of the momentum degrees of freedom.",
" If there are 10 to the 23rd particles, there are 10 to the 23rd P's, but I can't draw more",
" than one of them.",
" Well, I could draw two of them, but then I wouldn't have any room for the Q's, for the",
" X's.",
" And horizontally, the positions of the particles, which we can call X.",
" X or P, doesn't matter.",
" All right.",
" A point here is a possible state of the system.",
" If you know a point here, you know a position and a velocity, and you can predict from that",
" through if you know the forces.",
" Okay, let's start with the analog of a probability distribution, which is zero for some set of",
" states and constant or the same for some other set, for some smaller set.",
" Well, some fraction of the states all have the same probability, and the other states",
" have zero probability.",
" We can represent that by drawing a patch in here, a subregion in the phase space, and",
" say in that subregion, there's equal probability that the system is at any point in here and",
" zero probability outside.",
" This is sort of a situation where you may know something, where you may know something",
" about the particles that they're in some subregion here.",
" For example, you know that all the particles in this room are in the room, right?",
" So that puts some boundaries on what X are.",
" You may know that all the particles have momentum which are within some range that confines",
" them to this way.",
" So a typical bit of knowledge about the room might be represented at least approximately",
" by saying that there's zero probability to be outside this region and a probability equal,",
" I won't say one, but equal probability to be in there.",
" Okay, now what happens as the system evolves?",
" As the system evolves, X and P change.",
" The equations of motion say that X and P change, if you start over here, you might go to here.",
" If you start nearby, you'll go to some nearby point and so forth.",
" And the motion of the system with time is almost like a fluid flowing in the phase space.",
" If you think of the points of the phase space as fluid points and let time go, the phase",
" space moves like a fluid.",
" In particular, this patch over here, let's call it the occupied patch, the occupied patch",
" becomes some other patch.",
" That other patch, after a certain amount of time, the system now is known to be in here.",
" After a certain amount of time, we now know that the system is in here, not in here anymore,",
" and that it has equal, in some sense, equal probability to be anywhere in there.",
" Okay?",
" There's a theorem that goes with this.",
" The theorem is called Leaville's theorem.",
" And what it says is that the volume in phase space, the amount of volume of this region",
" in the XP space, and keep in mind, the XP space may be high dimensional.",
" Not just two, if it were two dimensional, we would think of it as the area.",
" Phase space is never three dimensional.",
" It's always even dimensional.",
" It has a P for every X.",
" So the next more complicated system would be four dimensional.",
" When I speak of the volume in phase space, I mean the volume in whatever dimensionality",
" the phase space is.",
" If you follow the phase space in this manner here, Leaville's theorem, you can go back",
" to Leaville's theorem.",
" It's in the classical mechanics lecture notes.",
" It occupies, I think, a whole lecture, I think.",
" All right, you follow?",
" And it tells you whatever this evolves into, it evolves into something of the same volume.",
" In other words, roughly speaking, the same number of states.",
" It's the immediate analog of the discrete situation where if you start with M states",
" and you follow the system according to the equations of motion, you will occupy the same",
" number of states afterwards as you started with.",
" There'll be different states, but you will preserve the number of them, and the probabilities",
" will remain equal.",
" So the rule is, not the rule, the theorem says that the volume of this occupied region",
" will stay the same.",
" And a little bit better, it says that if you start with a uniform probability distribution",
" in here, it will be uniform in here.",
" So there's a very, very close analog between the discrete case and the continuous case.",
" And this is what prevents this kind of fundamental equation from this kind of equation, an equation",
" where everything comes to rest, that can't happen.",
" Okay, why not?",
" Let's see why it can't happen.",
" Let's just look on this blackboard and see why.",
" Imagine that no matter where you started, you ended up with P equals zero.",
" That would mean every point on here got mapped to the x-axis.",
" It would mean that this entire region here would get mapped to a one-dimensional region,",
" and one-dimensional region has zero area.",
" So Leeville's theorem prevents that.",
" What it says, in fact, is if the blob squeezes in one direction, it must expand in the other",
" direction.",
" The situation for the moving eraser is that if the phase space of the eraser gets shrunk,",
" it means somebody else's, some other components in the phase space, the probability distribution",
" is spread out.",
" What are the other components in this case?",
" It's the P's and X's of all the molecules that are in the table.",
" So for the case of the eraser, there's really a very high-dimensional phase space, and as",
" the eraser may come to rest, almost rest, so that the phase space squeezes this way,",
" it spreads out in the other directions, the other directions having to do with the other",
" hidden microscopic degrees of freedom.",
" Okay, so there we are with information conservation, minus first law of physics, and let's pass...",
" Let's not go to the zeroth law.",
" Let's jump the zeroth law.",
" We'll come back to the zeroth law.",
" You know what the zeroth law says?",
" Well, I'll tell you what it says.",
" We haven't defined what thermal equilibrium is, okay?",
" But it says whatever the hell thermal equilibrium is, if you have several systems, and system",
" A is in thermal equilibrium with B, and B is in thermal equilibrium with C, then A is",
" in thermal equilibrium with C.",
" We will come back to that.",
" Just put it out of your mind for the time being, because we haven't described what thermal",
" equilibrium means.",
" But we can now jump to the first law, minus one, zero, and first law.",
" And the first law is simply energy conservation.",
" It is simply energy conservation, nothing more.",
" It's really simple to write down.",
" That simplicity belies its power.",
" It is the statement that, first of all, there is a conserved quantity, and the fact that",
" we call that conserved quantity energy will play for the moment not such a big role right",
" now, but let's just say there's energy conservation.",
" What does that say?",
" That simply says DE.",
" Simply energy is DE by DT is equal to zero.",
" Now this is the law of energy conservation for a closed system.",
" If a system consists of more than one part in interaction with each other, then of course",
" any one of the parts can have a changing energy, but the sum total of all of the parts will",
" conserve energy.",
" So if a system is composed, as I drew before, of two parts with a link between them, and",
" this is called one and this is two, then this reads that DE1 by DT is equal to minus DE2",
" by DT.",
" I've really written DE1 by DT plus DE2 by DT is equal to zero, but then I transposed",
" one of them to the right hand side just to indicate, just to make graphic, that if you",
" lose energy on one side, you gain it on the other.",
" So that's the first law of thermodynamics, and that's all the first law of thermodynamics",
" says, it says energy conservation.",
" Now in this context here, there's a slightly hidden assumption.",
" We've assumed that if a system is composed of two parts, that the energy is the sum of",
" the two parts.",
" That's really not generally true.",
" If you have two systems and they interact with each other, there may be, for example,",
" forces between the two parts, so there might be a potential energy that's a function of",
" both of the coordinates.",
" For example, the energy of the solar system, being very naive, I'm thinking of the solar",
" system as two orbiting Newtonian particles.",
" The energy consists of the kinetic energy of one particle plus the kinetic energy of the",
" other particle plus a term which doesn't belong to either particle.",
" It belongs to both of them in a sense, and it's the potential energy of interaction between",
" them.",
" In that context, you really can't say that the energy is the sum of the energy of one",
" thing plus the energy of the other thing.",
" Energy conservation is still true, but you can't divide the system into two parts this",
" way.",
" On the other hand, there are many, many contexts where the interaction energies between systems",
" is negligible compared to the energy that the systems themselves have.",
" If we were to divide this table top up into blocks, let's think about it, divide the table",
" top up into blocks, how much energy is in each block?",
" Well, the amount of energy that's in each block is more or less proportional to the volume",
" of each block.",
" How much energy of interaction is there between the blocks?",
" The energy of interaction is a surface effect.",
" They interact with each other because their surface is touch, and typically, surface area",
" is small by comparison with volume.",
" So many, many, we'll come back to that.",
" We'll come back to that.",
" In many, many contexts, the energy of interaction between two systems is negligible compared",
" to the energy of either of them.",
" When that happens, you can say to a good approximation, the energy can just be represented as the",
" sum of two energies of the two parts of the system plus a teeny little thing which has",
" to do with their interactions.",
" Under those circumstances, the first law of thermodynamics, the top is always true.",
" The second has that little caveat that we're talking about systems where energy is strictly",
" additive, where you add energies.",
" Does everybody understand why I say you don't always add energies that sometimes energies",
" are not additive?",
" Yeah, actually, I was thinking that we have the same possible problem with the probabilities.",
" We assume that the outcomes were mutually exclusive.",
" Otherwise, the sum law doesn't work.",
" Yeah.",
" Yeah.",
" Okay.",
" So in all the contexts which we've talked about, the dye, if it's yellow, it can't be",
" red.",
" You say orange.",
" You say orange, orange is both yellow and red, well, we don't count that way.",
" Yeah.",
" So that's correct.",
" No, that's absolutely correct.",
" We made the assumption that what I called states, what I called states are mutually exclusive.",
" Absolutely.",
" Okay, let's come back to entropy.",
" We're not finished with entropy.",
" We've done entropy.",
" We've done energy.",
" We haven't gotten the temperature yet.",
" Just the temperature comes in behind entropy, and even energy comes in behind entropy.",
" But temperature is a highly derived quantity.",
" By highly derived, I mean it's a, despite the fact that it's the thing you feel with",
" your body, so it makes it really feel like it's something intuitive, it is a mathematically",
" derived concept, less primitive and less fundamental in either energy or entropy, but we'll come",
" to it.",
" Let's come back to entropy.",
" We define entropy, but only for certain special probability distributions.",
" Let's lay out on the horizontal axis, just to be schematic.",
" On the horizontal axis, we will put down all the different states.",
" Here's i equals one, here's i equals two, here's i equals three.",
" This axis just labels the various states, and of course, vertically, let me plot probability.",
" Okay.",
" Well, the probability, of course, is only defined on the integers here.",
" That's not very good.",
" It's some probability.",
" But let's, I don't want to have to draw such a complicated thing every time I want to draw",
" a probability distribution, let's just draw a graph.",
" Some probability distribution, or some probability for each position.",
" Now what we did was we defined entropy for a very special case, the special case being",
" where some subset have equal probabilities and the rest have zero probability.",
" For example, if our subset consists of this group over here of m, the whole group being",
" n, then all of these have the same probability, and their probabilities have to add up to",
" one, so the probability is one over n.",
" We just draw that by drawing a box like that.",
" Then we define the entropy to be the logarithm of the number of states in here.",
" Generally speaking, we don't have probability distributions like this.",
" Generally speaking, we have probability distributions which are more complicated.",
" In fact, they can be anything as long as they're positive and all add up to one.",
" So the question is how do we define entropy in a more general context where the probability",
" distribution looks like this?",
" I'm going to write down the formula, and then we're going to check that it really gives",
" us this answer when it should give us this answer.",
" For today, I'm just going to write it down and tell you this is the definition.",
" You'll get familiar with it, and you'll start to see why it's a good definition.",
" It's representing something about the probability distribution, and what it's representing is",
" in some average sense, the average number of states which are importantly contained",
" inside the probability distribution.",
" The narrower the probability distribution, the smaller the entropy will be.",
" The broader the probability distribution, the bigger the entropy will be.",
" I'll write it down for you now.",
" We'll write it down and then explore it just a little bit tonight.",
" S for a general probability distribution is, first of all, minus.",
" That's funny because this is positive, but nevertheless the formula begins with minus,",
" a sum, and it's a sum over all of the states, all of the possibilities.",
" So it's a sum over i.",
" There's a contribution for each place here.",
" The probability of i times the logarithm of the probability of i.",
" Do you remember?",
" All right, let's write something else then.",
" Remember that the average of f is equal to the summation over i, f of i times p of i.",
" This is actually the average of log p sub i.",
" It's the average of log p sub i.",
" All right, let's work this out.",
" Let's see what this gives.",
" In the special case where the probability distribution is 1 over m for m states altogether.",
" It has width m, and because it has width m, it must have height 1 over m because all the",
" probabilities have to add up to 1.",
" All right, so let's work this out.",
" Let's take the contribution for all the unoccupied states.",
" All the unoccupied states, p sub i is 0.",
" You get nothing.",
" But log of p over i is minus infinity.",
" Yeah, that's right.",
" Now what's the, all right, good.",
" So let's consider the limit of log p over p.",
" Or let's just say the limit of, no, not that's not right, the limit of p log p as p goes",
" to 0.",
" You know how to calculate that?",
" Okay, I'm going to leave it to you.",
" It's a little calculus exercise to calculate the limit as p goes to 0 of p log p.",
" It's 0.",
" The point is that p goes to 0 a lot faster than log p goes to infinity.",
" Log p as p goes to 0 goes to, is very slow.",
" p goes to 0 fast.",
" So this goes to 0.",
" You're absolutely right though.",
" That has to be, that has to be shown.",
" But p log p in the limit that p goes to 0 is 0.",
" So with that piece of knowledge, the contribution from states with very, very small probabilities,",
" probability will be very, very small.",
" And as the probability for those states goes to 0, this quantity, the contribution will",
" go to 0.",
" But what about the ones here which have significant probability?",
" They all have the same piece of body.",
" And they all have the same log piece of body.",
" The log piece of body is logarithm of 1 over m for all of them.",
" The piece of body is also 1 over m.",
" Not log 1 over m, but 1 over m.",
" So each contribution is 1 over m times log 1 over m.",
" How many contributions are there like that?",
" All right, so we multiply by m and get rid of the 1 over m there.",
" There's a minus sign here, I'll carry it along.",
" All times m because there are m such terms.",
" So the 1 over m cancels and we just left with log 1 over m.",
" What's log 1 over m?",
" Minus log m, right?",
" So that's why the minus sign was put there in the first place.",
" There's no miracle.",
" The minus sign was put there because probabilities are less than 1.",
" And so the logarithms of them are always negative.",
" So you soak up that negative with an overall negative sign and entropy is positive.",
" But this is exactly the same answer that we are, the original definition just s equals",
" log m.",
" Logarithm of the number of states.",
" But this is a definition now that makes sense even when you have a more complicated probability",
" distribution.",
" And it is a good and effective.",
" It's the average of log p.",
" For the special case where the probability distribution is constant like this, then all",
" of the probabilities in here are 1 over m and calculating the average of log 1 over m just",
" gives you this.",
" All right.",
" So this is the general definition of the entropy that's associated with a probability distribution.",
" And notice, entropy is associated with a probability distribution.",
" It's not a thing like energy which is a property of a system.",
" It's not a thing like momentum.",
" It's a thing which has to do with a specific probability distribution, probability distribution",
" on the space of possible states.",
" So that's why it's a little bit of a more obscure quantity from the point of view of,",
" you know, intuitive definition.",
" As I said, its definition has to do with both the system and your state of knowledge of",
" the system.",
" Let's do some examples.",
" Let's calculate some entropy for a couple of simple systems.",
" The first system is just going to be not a single coin but a lot of coins.",
" So we have capital N coins, N of them, and each one can be heads or tails, et cetera,",
" and so on.",
" As a matter of fact, we have no idea what the state of the system is.",
" We know nothing.",
" The probability distribution, in other words, is the same for all states.",
" Good ignorance, absolute ignorance.",
" What is the entropy associated with such a configuration?",
" There are n of these.",
" All the probabilities are equal.",
" Under the circumstance where all of the probabilities are equal, we just get to use logarithm of",
" the number of possible states.",
" The answer here is the logarithm of the number of total states.",
" How many states are there altogether?",
" Two to the N.",
" Right.",
" Two to the N.",
" Oh, let's, I'm sorry, I'm going to change definitions for a minute for a reason that",
" you'll see in a minute.",
" I'm going to call this little N.",
" Number of coins is little N.",
" Over here, big N stood for the total number of states.",
" So if I match terminologies, big N, the total number of states, is e to the 2n, no, 2 to",
" the N.",
" Sorry, I'm getting tired.",
" 2 to the N.",
" 2 to the N states altogether.",
" Two states for the first coin, two states for the second coin, and so forth.",
" 2 to the N altogether.",
" And the total number of states is capital N.",
" What's the entropy given that we know nothing?",
" Two log N.",
" Two log N.",
" N log 2.",
" S is equal to N log 2.",
" That's the logarithm of 2 to the N.",
" All right, so here we see an example of the fact that entropy is kind of additive over",
" the system.",
" It's proportional to the number of degrees of freedom in this case.",
" N times log 2.",
" And we also discover a unit of entropy.",
" The unit of entropy is called a bit.",
" That's what a bit is, an information theory.",
" It's the basic unit of entropy for a system which has only two states, up or down, heads",
" or tails or whatever.",
" The entropy is proportional to the number of bits, or in this case, the number of coins,",
" times the logarithm of 2.",
" So log 2 plays a fundamental role in information theory as the unit of entropy.",
" It does not mean that in general that entropy is an integer multiple of log 2.",
" We'll see in a second, it doesn't mean that.",
" Okay, so that's this case over here.",
" Let's try another case.",
" Let's try another state of knowledge.",
" Here our state of knowledge was zilch.",
" We knew nothing.",
" This is the maximum entropy.",
" Logarithm of capital N, or logarithm of 2 to the little n, which is n log 2.",
" One bit of entropy for each coin, if you like.",
" Okay, let's try something else.",
" Let's try a state in which what we know is, oh, something else.",
" Supposing we know the state completely.",
" In other words, that's the case where m is equal to 1.",
" That would be the case where we know that the probability is only nonzero for one state.",
" Then m is 1 and s is 0, logarithm of 0.",
" So absolute knowledge, perfect knowledge, complete knowledge corresponds to zero entropy.",
" The more you know, the smaller the entropy, excuse me.",
" Okay, let's take an interesting case.",
" Let's take our heads and tails again.",
" And here's what we know.",
" We know that all of them are heads.",
" Let's begin with that, all of them are heads, what's the entropy then?",
" Zero.",
" Except for one of them, which is tails.",
" Now, supposing we know which one is tails, what's the entropy?",
" Zero.",
" But suppose we don't know which one is tails.",
" Equal probability for all states which contain one tail and n minus one heads.",
" What's the entropy?",
" Indeed.",
" Okay, so why is it log n?",
" How many states are there with nonzero probability?",
" The answer is little n.",
" It could be this one, it could be this one, it could be this one.",
" In other words, for this particular situation, capital M, the number of states that have",
" nonzero probability is just little n.",
" All the states have the same probability.",
" So we're in exactly this situation except that M is just equal to little n.",
" N possible states, this one could be tails, this one could be tails, this one could be",
" tails.",
" They all have equal probability.",
" So capital M is n and the entropy now is, for this situation, the entropy is equal",
" to logarithm of little n.",
" Notice that that's not an integer multiple of log two in general.",
" So in general, entropy is not an integer multiple of log two.",
" Nevertheless, log two is a good unit, it's a basic unit of entropy, it's called a bit.",
" Yes in this case is equal to log n.",
" Yes?",
" The two in the log two come from the fact that you only have a probability of heads or tail,",
" right?",
" Yeah.",
" So if you had three, it'd be log three.",
" Absolutely.",
" Absolutely.",
" Now, computer scientists of course like to think in terms of two for a variety of reasons.",
" First of all, the mathematics of it is nice, but two is the smallest number which is not",
" one.",
" It's a small signature.",
" Yeah, it's a small signature, not equal to one.",
" But it's also true that the physics that goes on inside your computer is connected with",
" switches which are either on or off.",
" So counting in units of log two is very useful.",
" Excuse me.",
" Yeah?",
" It's probably not important, but what is the base of the logarithm?",
" Yeah, the standard, this is a definition, of course.",
" This is definition.",
" The definition, okay, good.",
" The, good.",
" It depends on who you are.",
" If you're an information theorist, a computer scientist often, or a disciple of Shannon,",
" then you like log to the base two.",
" In which case, this is just one, n, and the entropy here is just n, measured in units",
" of bits.",
" If you're a physicist, then you usually work in base E. Okay?",
" But the relationship is just a multiplicative, you know, log to the base E and log to the",
" base two are just related by a numerical factor that's always the same.",
" Okay.",
" So, when I write log log, I mean log to the base E, but very little would change if we",
" use some other base for the logs.",
" Okay.",
" So there we are.",
" Let me just tell you what the definition is of the entropy in phase space.",
" If we're not talking about, if we're not talking about these finite discrete systems like this,",
" we're talking about continuously infinite systems, phase space.",
" Now let's begin supposing the probability distribution is just some blob where the probabilities",
" are equal inside the blob and zero outside the blob.",
" In other words, the simple situation.",
" Then the definition of the entropy is simply, well, you could say the logarithm of the",
" number of states, but how many states are there in here?",
" Clearly a continuous infinity of them.",
" So instead, you just say it's the log of the volume of the probability distribution in",
" phase space.",
" S for a continuous system is just defined to be the logarithm of the volume in phase",
" space.",
" Now if I wrote V, you might get the sense that I'm talking about volume in space.",
" No, I'm talking about the logarithm in phase space, the volume in the phase space.",
" The phase space is high dimensional.",
" Another dimensionality of the phase space is the volume is measured in those kind of",
" in units of momentum times velocity to the power of the number of coordinates in the",
" system.",
" All right, but S is equal to the logarithm of the phase space volume.",
" Let's call it P phase, V phase space.",
" That's under the volume of the region which is occupied and has a non-zero probability",
" distribution.",
" This is the closest analog that we can think of to log M, where M represents the number",
" of equally probable states of the discrete system.",
" More generally, if we want, if we have some arbitrary probability distribution, the arbitrary",
" probability distribution, P, what would P probability?",
" What would it be a function of?",
" It would be a function of all of the coordinates and all of the momenta.",
" All of the momenta and all of the coordinates.",
" P's and Q's or P's and X's, whichever you like, all of them.",
" Probability for the system to be located at point of phase space P and X.",
" Incidentally, when you have continuous variables like that, do you write that the sum is all",
" equal to one?",
" That wouldn't make sense.",
" You can't sum over a continuously infinite set of variables.",
" It becomes integral.",
" We'll come back to this, but let's just spell it out right now.",
" That if you have a probability distribution on a phase space, the rule is that the integral",
" of it is equal one.",
" P is really a probability density on phase space.",
" It's a probability per unit cell in phase space.",
" What would you expect the entropy to be?",
" I'll give you a hint that starts out with minus.",
" Where's the formula that we had here before?",
" Let's rewrite the formula that we had before.",
" S is equal to minus summation over I of P sub I log of P sub I.",
" To go to the continuum, you simply replace sum by integral.",
" There's an integral over the phase space.",
" That's like the sum over I.",
" Then probability of P and Q times the logarithm of the probability.",
" In first approximation, I don't mean first approximation numerically.",
" I mean first conceptual approximation.",
" It's measuring the logarithm of the volume of the probability blob in phase space.",
" Okay, oh, I could know mistake.",
" In classical mechanics, X's and Q's are coordinates and P's are momenta.",
" You know that.",
" Okay, so we've now defined entropy, which, as we've seen, depends on a probability distribution.",
" I'm going to go one more step tonight and define temperature.",
" We could stop here.",
" I think that's probably enough for one night.",
" Next time we'll do temperature and then discuss the Boltzmann distribution, which is the probability",
" distribution for thermal equilibrium.",
" We haven't quite defined thermal equilibrium yet, but we will.",
" You can ask some questions.",
" I don't mind some questions now.",
" I just, I just, I have the feeling that I've probably done enough for one night.",
" Oh.",
" Well, I haven't made any such assumption.",
" I haven't made any such assumption.",
" My only assumption in calculating the various entropies was either that you know nothing",
" or I told you what you know.",
" How you got to know that and what the reasoning was and whether it had to do with knowing",
" something about the dynamics and the independence and so forth may come into your calculation",
" of what P is.",
" But in saying you know nothing, the implication was that all states are equally probable.",
" Without asking how you knew that.",
" To say that all states are equally probable is closely related to saying that there are",
" no correlations.",
" It does say that if you, all right, let's, good.",
" Let's talk about correlations for a moment.",
" To say that you know nothing means you know nothing.",
" So in particular, if you, you, you begin knowing nothing and you measure one of the coins, what",
" do you know about the other ones?",
" Nothing.",
" Nothing.",
" You started knowing nothing about them.",
" You measured one of them.",
" You still know nothing about the other ones.",
" Of course, you know about the one that you've measured.",
" Now let's take the other, the other case that we studied.",
" Anything.",
" We know that all coins are heads except for one which is tails.",
" And we now measure one of them and we find that it's tails.",
" What can we say about some other one?",
" It's surely heads.",
" What if we measure that that one is heads?",
" What do we know about the other ones?",
" It changes the probability.",
" It changes the probability.",
" Just a new probability that one of the other ones is tails.",
" Right.",
" It's one over n, one over n minus one instead of one over, sorry, one over, yeah, one over",
" n minus one.",
" So that's correlation.",
" That's correlation where when you measure something, you learn something new or the",
" probability distribution for the other things is modified by measuring something.",
" That's called correlation.",
" For the complete ignorance, there is no correlation.",
" For any other kind of configuration in general, there's very likely to be some, not necessarily,",
" but there's very likely to be some correlation.",
" Correlation as I said means you learn something about the probability distribution of other",
" things by measuring the first one.",
" Or you modify the probability distribution.",
" Good.",
" Okay.",
" I don't know if that's what you asked about or not, but yeah.",
" Okay.",
" All right.",
" Well, is that original system there?",
" We had two parts.",
" Once we measured one thing, we had to conserve quantity.",
" What's that again?",
" We had to conserve quantity there that once we did one measurement, we knew which side",
" we were.",
" Yeah.",
" Yeah.",
" Incidentally, entropy is additive.",
" It's additive.",
" It's the sum of the entropies of all the individuals.",
" It's proportional to a number of things.",
" It's additive whenever there is no correlation.",
" When there's no correlation, it's additive.",
" Now, so uncorrelated systems have additive entropies.",
" We'll come back.",
" That's a theme that we'll come back to.",
" Is an interesting question.",
" Here's what you know, you know that if you measure a coin up, then with three-quarter",
" probability, it's too, they're laid out in a row.",
" If you measure one of them and it's up, then the probability for its neighbors is three-quarters",
" to be down.",
" That's what you're given.",
" That's all you know.",
" All you know is that if one of them is up, if any one of them is up, its neighbors are",
" three-quarters likely to be down.",
" It's an interesting thing to try to calculate the entropy of such a distribution.",
" That is correlated, of course.",
" That is correlated because when you measure one, you immediately know something about",
" its neighbor.",
" Well, make up your own example like that.",
" Make up your own example like that and compute the entropy.",
" You can learn something from it.",
" Any other questions before we go home?",
" Not particularly.",
" The formula here is due to Boltzmann.",
" I think it's just the one that's on Boltzmann's tomb, I'm not sure.",
" Now what's on Boltzmann's tomb?",
" He meant this.",
" Well he did write this.",
" This is Boltzmann's final formula for entropy.",
" The only difference between the Shannon entropy and the Boltzmann entropy is that Shannon",
" used log two.",
" Now, of course, Shannon discovered this entirely by himself.",
" He didn't know Boltzmann's work from an entirely different direction from information theory",
" rather than thermodynamics, but none of it would have surprised Boltzmann.",
" Nor do I think Boltzmann's definition would have surprised Shannon.",
" So they're really the same thing.",
" There's no real point in comparing them because they're comparing them, they are the same.",
" There's no real difference.",
" Shannon may have, I don't know whether he did or not put the minus sign in here.",
" If you don't put the minus sign in there, it's called information.",
" If you put the minus sign, it's called lack of information or entropy.",
" So I don't know which Shannon wrote down.",
" Anyway.",
" Shannon wrote down entropy.",
" He did.",
" Yeah.",
" Is there a simple way to relate to Heisenberg's uncertainty principle?",
" No, no, this is a separate issue.",
" This doesn't have to do with quantum mechanical uncertainty.",
" It has to do with the uncertainty implicit in mixed states, not the uncertainty implicit",
" in pure states.",
" Okay.",
" Oh, oh boy.",
" Yeah, there's a conversion factor.",
" You know, C equals H bar equals G equals Boltzmann's constant equals one.",
" Very natural.",
" Boltzmann's constant was a conversion factor from temperature to energy.",
" The natural unit for temperature is really energy.",
" But the energy of a molecule, for example, is approximately equal to its temperature",
" in certain units.",
" Those units contain a conversion factor K Boltzmann.",
" I'll remind me to talk about K Boltzmann.",
" For more, please visit us at stanford.edu."
],
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10.5446/12996 (DOI) | Putting Together the Pieces of the Universe | https://av.tib.eu/media/12996 | https://tib.flowcenter.de/mfc/medialink/3/deaeccddd1f414201a931e7a5a0d1ba60969f81a099c9f943db07cf9dd5f5abde9/Putting_Together_the_Pieces_of_the_Universe_flash9.mp4 | CC Attribution - ShareAlike 3.0 Unported:
You are free to use, adapt and copy, distribute and transmit the work or content in adapted or unchanged form for any legal and non-commercial purpose as long as the work is attributed to the author in the manner specified by the author or licensor and the work or content is shared also in adapted form only under the conditions of this | Physics | Lecture | 2009 | Bullock, James | null | A lecture delivered by UCI Professor James Bullock on February 11, 2009. James Bullock, Associate Professor of Physics and Astronomy at UC Irvine, is part of a team of scientists who believe they have discovered the minimum mass for galaxies in the universe -- 10 million times the mass of the sun. This mass could be the smallest known "building block" of the mysterious, invisible substance called dark matter. Stars that form within these building blocks clump together and turn into galaxies. Dark matter governs the growth of structure in the universe. Without it, galaxies like our own Milky Way would not exist. Dark matter's gravity attracts normal matter and causes galaxies to form small galaxies and to merge to create larger galaxies. | James Bullock and we're very proud to have him. He's a man who is asking those big questions. Has the universe always existed? Did time have a beginning? What makes us think that the earth goes around the sun? What is a black hole? How fast is light? So we're about to hear from someone who asks the big, big questions. Would you please give a warm, inside-edge welcome to James Bullock. Thanks very much. It's a pleasure to be here. I thought, so what I'm going to do today is talk about some of the things we're trying to understand in the field of cosmology. I actually work in that building right over there. That's the physics and astronomy department. We have a center for cosmology there that was started about four years ago. And the center is all about trying to answer the kinds of big questions that we were just hearing about. And that's what we're doing. And I thought what I would do today is just sort of give you a little bit of an overview of what we're trying to do, some of the big questions that we have. And I'm going to do my best to leave some time at the end for questions. Normally when I give these kinds of talks, I like to keep it very free-form and open. So if anyone has any comments or questions at any time, I'm happy to address them. Now, I understand that since we're recording this, that might make it a little bit logistically difficult to have people running around with microphones. So what I'm going to do then is do my best to make sure I don't talk too much and leave some time at the end just for general questions. If you have an urgent question at any time and you just need to ask it, we can do that. So let's just keep it open there. So I just thought I would start with this. And one of the things I like to do when I talk about cosmology is to show this picture. This is a picture that was taken by a photographer named Art Roche. And I thought it was really beautiful. But one of the things that I find very attractive about it is he was telling me when he took this picture, he was in this canyon in the southwest. And he noticed that there were petroglyphs, sort of ancient petroglyphs left on the canyon walls. And this reminds me, I think, of really what we're trying to do in cosmology, which is it's a scientific exploration of the kinds of questions that you ask when you look up at the night sky. So you can imagine the people who made these petroglyphs, these ancient peoples, who were drawing into the canyon walls. And when they looked up at the night sky, they probably had similar questions to the kind of questions that we have when we look up at the night sky. So how old is the universe? Was it always there? What's the universe made of? How did structure like us come to be? So these are the kind of questions that a lot of us has asked ourselves. And in many ways, cosmology is the oldest science, because you could imagine people have always asked themselves these kinds of questions. What we're trying to do is to answer these questions in a scientific context. So we're trying to make testable predictions and see if they come true, and use that to build up a theory for how the universe began and emerged. And for the first time, trying to build one that's in a scientifically tested context. And that's what we're trying to do. So what I'd like to do is tell you a little bit about what we believe about the universe, so how we think the universe came to be and some basic facts. And then I'm going to go on and tell a story of how we began to shape these ideas. So this is kind of a cartoon picture of the overview of modern scientific cosmology, and our present understanding of what the universe is and how it began. So the first order picture here is that time and space, time and space itself, began in something that we call the Big Bang, which is just a word, about 14 billion years ago. The universe was very, very hot at early times. The universe has been expanding. So today the universe is quite big, and in early times, the universe was smaller and smaller. It's expanding. So if you go back in time like a movie, it's hot at very early times. And in fact, it was so hot at early times that, you know, as you crank up the temperature, things start to melt. And if you keep cranking up the temperature, even molecules can't hang together anymore. If you keep cranking up the temperature, even atoms, it becomes so hot that even atoms can't hang together anymore. Electrons will fly off of protons. In a very, very early universe, it was so hot that even elementary particles couldn't exist, and we were down to the most fundamental constituents of nature. So in that sense, it was a very smooth and, in some sense, simple beginning. And from this elegant beginning, this very, very simple beginning, emerged as the universe cools and cools over time, more complicated structures can begin to form. And then you can start making planets, et cetera. And we have a situation where we have really the real primordial soup, okay? The early universe's primordial soup. And then from this, structure grew. And eventually, we end up with galaxies like ours, the galaxy that we live in, the Milky Way. And in these galaxies, there are stars. In our galaxy, there are billions of stars. And around one of these stars orbits a planet that's very low mass, that's mostly rock, and that's the Earth. So what we'd like to understand is how this picture emerges. Another thing, oh, one thing I did want to mention is this, you know, numbers like 14 billion don't seem that big anymore. We're hearing about, you know, the national debt and the... But 14 billion is a big word, is still a big number. So 14 billion is a very long time. If you took the entire age of the universe and scrunched it down into one year, the scale of reference for that is that Shakespeare, okay, living in the 17th century, wrote his plays about a second ago. So you take the whole expanse of time and scrunch it into one year, Shakespeare was walking around about a second ago. So that's, these are the time frames we're talking about. So true cosmological time frames. Now, another thing we would like to understand is in addition to sort of the size of the universe, the age of the universe, we'd like to understand what the fundamental constituents of the universe are, okay? And this represents a pie chart of our current understanding of the composition of the cosmos. And I'll talk a little bit about these different pieces here, it's just words, but I thought I would just give you an overview now. The thing you notice here is this area right here, this little yellow sliver in this pie chart, is supposed to be representative of heavy elements. And that heavy elements just means any kind of atom that's heavier than hydrogen or helium. So basically us and the Earth. And in terms of a global sort of composition of the universe, that represents only about 0.03% of the composition. So a tiny, tiny sliver of what we believe is out there. About, only about 5% of the universe is made out of things that we have a really good understanding of what they are. Okay, so the entire periodic table of elements, okay? Everything you learn in chemistry class. In fact, everything that they study in every department on this campus, except for ours, is in this piece of the pie right here, okay? I put neutrinos there purposefully, this is a kind of elementary particle that was discovered by Fred Reines, who's our Nobel Laureate over there. So I have to put that on the chart, even though it's 0.3%. But the rest of this chart is all stuff that we really don't understand. And I'll talk a bit about this. About 25% of the universe consists of something we call dark matter. Dark matter is some weird stuff that I'll talk about later, and there's some even weirder stuff called dark energy, and that makes up 70% of the universe. So this is really a statement of our ignorance, this pie chart. We have a pretty good idea of what we don't understand, but as we all know, that's how you start. You have to understand what you don't understand first. Okay, so this is our ignorance chart here. These are the big questions we're trying to answer. Now, sorry. So as I mentioned, cosmology is in many ways the oldest science. And as far as I know, every culture on Earth has had their own story of cosmology. So how does the Earth begin? How old is the Earth? How did we emerge? One of the oldest pictures, one of the oldest cosmological models was actually that of Aristotle, and then later on refined by Ptolemy. And in Aristotle's model, the Earth sits at the center of the universe. And all the planets and the sun orbit the Earth, and the stars extend out in a celestial sphere. And this was the picture they had. And, you know, Aristotle was not a dumb guy, right? When you look up out the sky, it looks like the sun is going around the Earth, okay? So, and in fact, this is the longest lasting scientific cosmological model in history. It lasted 1400 years. But eventually, this idea was broken by Copernicus and Galileo and Newton and people like that. What Copernicus said is he said, well, you know, if you actually put the sun at the center and let the planets go around the sun, I can also explain all the observations. And to me, that seems just prettier. It just seems a little bit more elegant. But that was kind of the end of it, okay? And there was sort of this, it wasn't clear whether that was really true or what. The thing that really changed the way people saw the universe was when Galileo turned a telescope to the heavens and he actually started testing some of these ideas. And he showed that there were moons going around Jupiter and that the moon itself was corrupt. That is, it had mountains and things and it wasn't this perfect celestial sphere kind of thing that Aristotle thought was going on. And from this piece of technology and the application of this technology and direct observation, it sort of shattered this idea that people had for a very long time. A lot of very smart people had for a very long time. So it was really the tools that Galileo had that allowed us to sort of extend this idea. So this began, this began sort of the sort of modern theory of cosmology where, you know, at that time cosmology was the solar system. This is everything. Now, one of the things that's remarkable about this is not only does this move the position of the Earth relative to the sun, but it transforms sort of how it sort of, it's representative of how bizarre the universe had to be if this was true. It wasn't that no one had ever conceived of the idea that the Earth might be going around the sun before. There were ancient Greeks who proposed that idea. But people decided that was crazy and the reason why they decided that was crazy is if you have something that's going around the sun, let's say us, we're going around the sun, okay? We're moving a big distance from one time of the year to the next. So if that's true and you're moving a lot and you look at something and you're moving like this, the position of that thing on the horizon will shift a bit, right? When you're in a train, you see stuff go by and you have to turn your head. The only way you don't have to turn your head on a train is when you're looking at, say, a mountain peak that's really, really far away. So what this meant is if this is true, this meant the universe, the stars that were really far away, which we never see shift from one time of the year to the next, must be really, really far away. And so by putting the sun at the center and having the Earth go around the sun, it was actually bizarre in many ways because it meant the universe was much, much bigger than anyone had ever imagined before. So one of the things that's kind of interesting about scientific cosmology is I think almost every step of the way the data tell us something that's much crazier than anyone had ever thought of before. And I think it's true every step of the way. If you look how cosmology proceeds, it's a crazier universe than anyone had ever imagined. Even very creative people. So today, I mean, cosmology at the end of the 20th century, it progressed significantly. And people had finally figured out how to measure the tiny wiggles and distances to stars on the horizon, and that allowed them to figure out how far away the stars were. And it was realized that the sun, our sun, was just one among billions of stars in this galaxy. And the galaxy is huge. Okay, the galaxy, it takes light 100,000 years to cross our galaxy. So just to explain what I'm trying to say here is that if I have a flashlight and I turn the flashlight on, light travels from my flashlight at 186,000 miles a second. So if I turn a flashlight on, a light beam could go around the Earth 10 times in one second. It moves really fast, but it's moving at finite speed. Light that leaves the sun takes eight minutes to get here. So, which isn't that long a time, but it's still, you know, it's a delay. Light takes 100,000 years to cross the galaxy to give you an idea of size. Okay, it's big, very big. And in fact, if you take the sun, okay, the sun is big, we really don't have a concept of how big it is, but the Earth, we have some rough idea of how big it is, even though it's hard to conceptualize. You can place 100 Earths across the face of the sun. The sun is big. If you took the sun and shrank it down to the size of a grain of sand, okay, our galaxy would be the size of the Earth. So the size of our sun compared to the size of the galaxy is like the size of a grain of sand to the Earth. So these are the distances that we're trying to deal with, okay. Now the thing is, at the end of the 20th century, pretty soon, Edwin Hubble would realize, Edwin Hubble looking through his telescope, he would soon prove that actually there are blobs of stuff in the sky that we call nebulae, that at the time people didn't know what they were, he would eventually figure out that these were actually distant galaxies of no own. So that the universe is actually filled with billions of galaxies like ours. So yes, billions and billions, just like Carl Sagan, he was right. Okay. So this is our current picture of the Milky Way. The Milky Way, our galaxy, is a disk of stars. It's got a bright thing in the middle, we call the bulge. It's about 100,000 light-years across, and we live kind of at the edge here, sort of in the middle, kind of out, sort of outer outskirts. Here's the sort of blow-up of the sun and its circular planets going around it. And like I said, it takes eight minutes from light to get to the sun to the earth, but it takes 100,000 years for light to go across the galaxy. So it's a pretty big place. But this is nothing. 100,000 years of travel time for light is small potatoes compared to the size of the universe. This is the nearest big galaxy to us, called the Andromeda galaxy. You can see this with your eye sometimes when it's dark. It takes light two and a half million years to reach us from this galaxy. And so just think about this. This picture is what Andromeda looked like two and a half million years ago. So before there were homo sapiens to look up at the sky, that's when this light left. Okay, so we are looking back in time. And the farther things are away, the further back in time you look. And that's one of the techniques that we try to use when we try to understand how the universe is assembled over time. Because as we look at stuff that's further away, we're looking at what the universe was like long ago. And from this, you can imagine trying to build a movie, sort of how stuff is assembled over time. So you use this fact that light can't travel infinitely fast to your advantage. And you just have to actually look at what the universe looked like in the past. So we can make maps of what our local environment looks like. And this is a lot of astronomers like to do this. Here is our galaxy, the Milky Way. And here is Andromeda, the one I was just showing you, two and a half million light years away. And there's lots of other little galaxies all around. We are the two big boys on the block, but there are these little things that float around our own things and orbit us. They're satellites. They're galaxies that are satellites of our galaxy, just like there are planets that are satellites of the sun orbiting around us. And gravity is doing all this attraction. And you can zoom out again. So this is three million light years, this arrow. And if we zoom out again, this is 30 million light years. And we can make maps there, too. And they're still naming things, but eventually you stop naming things. There's so many things. But all these guys have names, you know. They're named after the constellations. A lot of them are named after the constellations. You have to look through to see them, right? You have to look through the fornax constellation to see the fornax galaxy or whatever. And that's when the naming convention comes from. So this is 30 million light years. And we can stop on this point. Every dot on this picture is a galaxy. And we can keep zooming out. So this is 300 million light years. This is where we kind of stop naming stuff. It gets kind of ridiculous. It keeps going, okay? It's sort of arbitrarily cut off in a circle. But it keeps going. So galaxies like to live around other galaxies. They cluster together in clusters of galaxies and what we call superclusters. And there's giant structures in the sky. And one of the things we like to understand is why, how many, these kinds of things. And in fact, so all of these things I had just shown you were cartoon pictures. But this is actually a picture of real data. This is the largest continuous map of the universe that's ever been made by a survey called the Sloan Digital Sky Survey. You've heard of the Alfred P. Sloan Foundation. They funded this sky survey to take a huge section of the sky and just look very deeply and try to map continuously all the galaxies. This is a beautiful survey. The total amount of data is 15 terabytes, which is equivalent to about the Library of Congress. So that's the kind of data that they've taken for this one, this one picture. So these are the kind of maps that we're trying to make. We want to know what the universe looks like. Now there's another technique. Rather than looking at sort of broadly at everything, we could point at one point on the sky and go very, very deep. And the way you do that is the same way you'd open up a shutter on a camera and you could take a really faint image. You do that with a telescope. Let me back up. That's what I'm going to show you. I just want to explain it first. So you know when you overexpose your camera, it's not good, but you can take pictures of things that are very, very faint by keeping your shutter open for a long time. That's what we can do with the telescope. So with the telescope, the lens is sort of eight meters across. These are the biggest telescopes in the world, and you can point them at one point on the sky and leave the shutter open. So you can see stuff that's really far away. The Hubble Space Telescope is a big telescope like this. It's not quite that big, but you've heard of the Hubble Space Telescope. So one thing that the Hubble folks did is they picked a place on the sky that was completely dark, nothing there. And they pointed the Hubble Space Telescope there for 12 days straight. And so the point they pointed is somewhere in here. Here's a constellation you might be familiar with. And what I'm going to show you now is a movie that zooms in to this point. Notice that it's complete blackness that they're going to zoom into. And what they're doing is they're showing you progressively deeper and deeper images of the same part of the sky. And what we're looking at now is we're seeing nothing but galaxies. We're way past the stars. They're going deeper and deeper into sort of a deep pencil beam into the universe. And this is it. This is called the Hubble Ultra Deep Field. It's a deep region of space that Hubble just sat on for 12 days. And the light from these galaxies left these galaxies about 12 billion years ago. So this is actually a movie, a snapshot of what the universe looked like, something like 12 billion years ago. And notice that the galaxies look kind of messed up. They look like they're not fully formed yet. So these are the kind of movies that you try to make and then interpret. We want to understand how from this kind of stuff, emerge the galaxies like we see. So we like to ask, where did all this stuff come from? How did we get here? So one of the things you do when you have a question, what things sometimes people do is you ask the smartest person you know. And so the smartest person I know is Albert Einstein. So how do we begin to sort of take measure of these kinds of observations? It turns out, especially in the context of a theory that seems to be, we seem to have a universe that's expanding. You want to understand the nature of space and time. It's sort of all grounds there. And really the beginning of modern cosmology was really what this guy here, when he had a short haircut. In 1905, he proposed this theory that's called the special theory of relativity. And as part of this theory, you know, he had this famous equation, E equals MC squared. But lying at the backbone of this derivation of this equation, was that there was some kind of interrelationship between space and time. What Einstein realized is, the actual way in which time ticks along, is linked in some way with space and how fast people move through it. This was an earth-shattering conclusion that he made. And what it meant was, our previous concepts of space and time were wrong. And that, in fact, the ideas that Newton had about how gravity operates on Earth, and why apples fall from trees, had to be revised. Now he realized this in 1905, and then for the next 10 years, he said, oh, I need to fix gravity. I need to figure out how gravity really works, because E equals doesn't equal MC squared, according to Newton's gravity, the first order. So now we're talking about 10 years of Einstein time. Okay? I don't know how many years of my time that would be. So it's like 600 years of my time. Then in 1915, he comes upon what's called the general theory of relativity. And in this theory, by the way, it's largely, is regarded by most professional physicists as the single greatest achievement ever by anyone, period. I mean, this is a very impressive thing. What he realized is that you can understand gravity is actually a warping of space. So that if you took, I mean, a way of sort of characterizing it, is if you took a bowling ball and set it on a mattress, and there's like a dip in the mattress, you can imagine taking a marble and flicking it on the mattress, and the marble would roll and bend as it approached that dip in the mattress. It's in a similar way that Einstein understood how planets go around the sun, and that there's a dip in space time around the sun, and the planets go around on circles, because that seems to them to be a straight line in that curved space. He described all gravity that way. Now the thing, the sort of backup ramification of this, is that space is not something that's fixed that you just travel through. It can warp and evolve and twist. So if you think about this a little bit, it's sort of, it's sort of, you're now kind of in this weird position where you can have space itself, not being static, not always being the same. So soon after this paper came out, a guy named Friedman read it, and he realized that, hey, you know, if I look at your equations, I realize that the universe should be expanding according to your equations of gravity. Now Einstein said, uh-oh, that sounds crazy. Even to me, that's crazy. Okay, even this great creative genius that Einstein was couldn't accept that, even though it was his own theory. And so he changed his equations. He changed his theory to account for it. And he added basically a fudge factor that's called, that we call lambda, which is now called the cosmological constant. It's sort of like put a wedge in the door and kept the door from closing. He just stuck it in. He could do it, it was his theory, he did it. But of course, this is widely regarded now, and it's claimed that he said it was his greatest blunder. You know, he could have predicted that the universe is expanding, but he didn't. But then in 1929, the same guy, Edwin Hubble, using the 100-inch Hooker telescope, which is not too far from here, saw that as he looked at galaxies that were more distant to us, they were moving away from us faster. And this implies that the universe is expanding. So in some level, it fit in with the theory that Einstein had proposed, but he didn't quite have the guts to stick with the prediction. Now, what this gives rise to an idea is that we're leaving a... And then a lot of data came after that. People studied this for a long time. And now we're up to this idea where, because the universe is expanding, if you just run the movie backwards, it was smaller in the past. And if you take something and you scrunch it down, smaller and smaller and smaller, the atoms in it or the gas in it will get hotter. You might notice this sometime if you're holding a bicycle pump and pumping it, it'll get hot. If you scrunch air together, it tends to get hotter as it's compressed. Same kind of thing with the universe. The universe is scrunched, it's smaller, it's hotter. Earlier than that, it was smaller and even hotter. So it's now about 14 billion years since the Big Bang, and now it's very cool. It's about three degrees above absolute zero in empty space. But the universe at early times was very, very hot. And it turns out it was so hot in the early universe, there's radiation that's left over from that. And it's realized that that should be observed. And in fact, in 1965, Tensien and Wilson discovered the glow from the Big Bang, which is called the cosmic microwave background radiation. And from that, they were given a Nobel Prize. And it sort of solidified this idea that in the past, the universe was hot. So what we have now is a picture of the universe where not only doesn't have this map, but it's moving away. So I thought I would mention just a couple more words, and then I'll stop and open it up for questions. But when you look at one of these galaxies in particular, it turns out now we don't have to just, we can't just, we don't just sort of take pictures of them and see where they are. We can actually study them in detail and understand things like, how fast are they spinning around on edge? Because it's a disk. It turns out that us, in our own galaxy, we're spinning around, or orbiting around the center, spinning like a disk. And in fact, if you look at the stars in the galaxies, you expect them to spin around, but as you go out to the edge of the galaxy, you expect it to go around more slowly because there's less gravity out there. And in fact, Pluto, if you want to call it a planet, is going around more slowly around the sun than the earth. Because it's further away, there's less gravity from the sun. And this is expected, this is well understood physics. And so when people looked at these galaxies, what they expect to see is that they speed, they rotate really fast in the center, and then the speed rolls off quickly at large distances. When the 70s, there's a woman named Vera Rubin who went out to actually measure how fast they were spinning around. And what Vera discovered is, that's not what's happening. In fact, they're still going really fast way out here. And the way we eventually came to understand this is that there are big extended distributions of matter all around these galaxies. And this matter is called dark matter, because we can't see it. That's why it's dark. Okay? Vera Rubin is credited with this observation, although there are a lot of other people who found evidence for it, even before Vera Rubin. Just an interesting tidbit. Her son is a professor of mathematics in the math department here. There's a lot of more evidence for dark matter by studying galaxy clusters. Clusters of these galaxies, we can measure how fast stuff's moving in those galaxy clusters. And there's a lot of evidence that it's there. It's not just this rotation group. There's tremendous amounts of evidence. Another question that people had, and they were specifically going after this issue in the 80s, was, is the universe ever going to stop expanding and re-collapse and have a big crunch? In order for us to know what's going to happen, it kind of goes something like this. Imagine that I took this pointer and I threw it up in the air. Eventually it's going to come back down. It's going to come back down because gravity's tugging on it. But if I lived on a planet that was really, really light, that didn't have much gravity, the force of gravity would be less, be like on the moon. And if I, you can imagine me throwing this up and this escaping the moon and never coming back down. So the question with the universe is, it's going up now, will it come back down? And the crucial question is, how much mass is there? If there's a lot of mass, it will slow down but never come back down. Sorry, if there's not much mass, it might slow down but not come back down. If there's a lot of mass, it will slow down and then re-collapse. So people in the 80s were looking to see if this was what's going to happen. So imagine this is distance and this is time and you throw a ball up, you might expect it to come back down, or it might keep going up, depending on how much mass the planet has or how much mass the universe has. So a very heavy universe, we expect this. In a very light universe, we just expect that ball to just keep going up. The universe is expanding. So a bunch of astronomers, one part of the team was led by a team at Berkeley, was looking at this in the late, actually turned into late 90s when they actually got the result. So this is a big crunch picture and this is a not big crunch picture. The universe keeps expanding and this is what they wanted to know. So they did the observation and this is what they found. So what they found is, you throw the ball up and rather than it coming back down or slowing down, it speeds up as it goes up. That's weird. It's that weird. It's exactly that weird. That's called dark energy. So there's some force in the universe that's making the universe expand at accelerated rate. We do not understand what this is. The thing that's kind of fun about it is, our ideas for this, well, let me tell you a little bit about our ideas. So this is the dark energy piece. This is the thing that makes the universe accelerate. This is the dark matter piece. This is the thing that makes galaxies spin around fast. And this is the piece that we understand. Chemistry, that's easy, right? No. It's actually much harder than this. Our questions are so simple that it's actually much easier than chemistry. So here's the big questions. What's the dark matter? What's the dark energy? How did galaxies like ours emerge from this early universe? What was going on in the early universe anyway? So these are the kind of questions we're trying to answer. What is the dark matter? I'll tell you what we know. It's dark. It doesn't shine. What that really means is, it doesn't interact with light. It doesn't interact with things electromagnetically. So if I had a ball of dark matter, and it was traveling through the wall, it would go right through the wall. Because the only reason balls bounce off walls is because the electrons hit each other and make it bounce. It's electrical repulsion. There is no electrical repulsion with dark matter. It doesn't interact with electricity. It doesn't interact with light. But it does attract to other matter via gravity. And most cosmologists believe that the dark matter is a fundamental particle of nature that we just haven't discovered yet. And there are big programs to look for it. There's a big particle accelerator in Switzerland called the Large Hadron Collider. Where they're taking particles and zooming them around a 17-mile loop. They're going to slam them together, recreate the hot temperatures of the universe, and try to make dark matter. And try to observe it. That's going on now. There are also satellite projects. They're looking for glows of high-energy gamma rays, which might be evidence of this dark matter. And also, we're looking for it directly. We think there are dark matter particles streaming through the earth all the time. We don't feel them. But we're trying to build detectors that will feel them. And by detecting them, it would be really interesting. So this is the ways we're looking for it. The dark energy is much harder. It's dark. It doesn't shine. It's energy. It causes the universe to accelerate. But it does not attract stuff together like normal matter. In fact, it's a lot like the cosmological constant that Einstein proposed. In fact, it's possible that the dark energy is this cosmological constant, that Einstein thought was his greatest mistake. It might have been his greatest prediction. Guy's always right. Quintessence. It's another idea. So there are other ideas, and these are just words. Another possibility is that Einstein's theory of gravity is wrong. And our whole interpretation of why the universe is slowing down or speeding up is just wrong because it's based on Einstein gravity. So all the young physicists hope that this is true because they want to be the one to know what the right number, the right answer. Let me just flash on the future, and then I'll take a few questions. So how are we going to try to answer some of these questions? UC Irvine is involved in a survey called the Large Synoptic Survey Telescope. This is that little Sloan diagram, which is the largest continuous map of the universe that's ever been made, shown on the scale of what we want to do. So we want to make, we want to map about a quarter of the visible universe with this project. This is going to be a telescope in Chile. We're also involved in something called the 30-meter telescope. The 30-meter telescope will be the largest optical telescope that's ever been built. And there's something else that's coming online called the Hubble Space Telescope, sorry, called the James Webb Space Telescope, which is the successor of the Hubble Space Telescope. It's 10 times more powerful than Hubble. So these things are coming online. Let me show you, just to give you some perspective about what we're getting ready to do. The Hooker Telescope was 100 inches across, which is big. This is the telescope that Edwin Hubble used to discover that the universe was expanding. At the time, he didn't know he was going to discover that. And the Palomar 200-inch telescope, this was the biggest telescope in the world in 1949. It was this telescope, of course, when it was built, no one had any idea that eventually they would be able to measure galaxies spinning around in discovered arc matter. It was with the Keck Telescope, which was built in 1993, which is currently the largest telescope in the world, that we used to discover that the universe is accelerating its expansion. But when this telescope was built, if you said that to somebody, they would have thought you were crazy. Okay? So now, this is the 30-meter telescope. So who knows what we're going to find? Okay. Now, I will close then with saying, this is the real future. Okay? We're over there across the building over there, and the people who really do the work are the graduate students and all these young people. And so, I, you know, I was asked if I had anything to sell. I don't have anything to sell. But what I do kind of have to sell is this center in really these people. There are people over there who are doing really great work. And if you're interested in that work, you can contact me. If you're interested in supporting that work, you can contact me. Specifically, supporting maybe one of these people to go to one of these telescopes and do some observations. That would really help us. But anyway, so let's leave, let's leave with this picture. That's it. I can take a couple questions. I'm sorry I went kind of long. This hand was up first. Yeah. Thank you very much. Is this on? Yeah. Okay. If I can, if we can look into the far reaches in order to see back in time and see a very primitive part of the, of the universe. Yes. If I were able to stand in that primitive part at that far away time, Yes. Would I eventually see back to where we are now and did that exist? Okay. So if you were in that far away place of the universe and you looked towards us, Yeah. You would see us as we were 10 billion years ago. So there would be no sun. Was there a Milky Way? The Milky Way was, yes. So that's a good question. The Milky Way we think basically came into existence about 11 to 12 billion years ago. And the sun was born in a sort of second generation of stars about five billion years ago. Yes. Sure. As you know, the, in the subatomic world, gravity has no effect. Yes. And so that's why I actually respect Niels Bohr and those guys a lot more than Einstein. But that's me. Einstein contributed to quantum mechanics too. The point I'm making is have you, have you guys gleaned into the subatomic world to try to figure out what happened? Yes. That's a great question. So one thing I skipped over really fast is in fact that's a lot of what particle physics tries to do. So in the very early universe, it was very, very hot. So that there really was nothing but some of atomic particles. There was nothing on the big scale. The universe was tiny and very, very hot at this time. And what particle accelerators do, what particle physicists do is they take particles and slam them together at very high energies to create very high temperatures and see what flies out to figure out what the subatomic world is made of. Okay, Richard Feynman said it's like taking a switch watch, throwing it against the wall and watching what flies out to figure out what it's made of. Okay. That's what, that's what's done. And that has deep ties to cosmology because cosmology unites these fields. Subatomic physics governs what's going on in the very early universe, which sets the stage for all that emerges. And in fact, all the structure in the universe, we believe, originated in tiny fluctuations that were originally quantum mechanical processes that were developed some 14 billion years ago in a process known as cosmic inflation that eventually grew to be the galaxies we are today. So without quantum mechanics, there would be nothing, according to that. Earlier we were talking about Aspen and the wonderful research institute there. Last summer I heard a lecture again on the universe and they were talking about, what I want to say is energy polarization when you have a galaxy that is flat and there are these energy spears, if you will, one going up and one going down, if you will. Yes, yes. And how that creates the energy kind of like the ying and yang. And I was wondering if you could fold that into what you're talking about. A little bit. So the jets you're talking about in galaxies, a lot of galaxies are observed to have polar jets of material streaming out of them like this. What that actually is fueled by is a black hole at their center, a very massive black hole that creates a very strong gravitational field and that heats up matter, makes it very hot and at the same time it creates a magnetic field that twists and that's what drives those columns out. And so that's a very interesting subject and that's what people are using that to study black holes. And black holes are really fundamental objects because they sort of sit at the cusp, the threshold of us understanding gravity in the extreme. So another way we're trying to test our theories of gravity is by studying these black holes that create these kind of jets. I have a rather pedestrian question. I was noticing that when you're showing a galaxy that it all seems to be rotating on one plane rather than all these others. Now within that plane, does the planets around the sun rotate on that same plane? That's a great question. And how about the moon around the earth? Yes. So comment on that. What they do and why. Yes. So the planets do not orbit the sun in the same plane that the galaxy is going around. It's tilted by about 60 degrees and it turns out that the stars are so far away from each other that they don't really, that kind of effect is small. If our sun was the size of a basketball, the nearest star would be in Hawaii. So the size scales are very big. They don't really interact with each other very strongly. And so there's really not a strong correlation. It has to do with smaller scale processes, almost like weather we believe in the early galaxy. Told the planet what plane to align on basically. And the moon similarly. The moon is sort of orbiting at a slightly different plane as well. That's why we have phases of the moon because it's not exactly aligned up with the earth and the sun. One more. Thank you. Could you tell us about string theory and how does that relate to any of these, please? I know it's the last question you have. I have 30 seconds to explain to you string theory. Okay. So the basic idea of string theory is that there's this idea in particle physics that all the elementary things are single points. Okay. An electron is a single point and you model it that way. String theory says no, no, maybe they're strings. And it turns out if they're strings, a lot of ugliness goes away. There are a lot of divergences that are a lot of infinities that pop up everywhere. Remember when you're doing basic math and there are infinities everywhere and that's bad, does not exist. It turns out particle physicists get that all the time. But if you do string theory, a lot of that stuff goes away. String theory is an idea that the subatomic world is really based on these one-dimensional loops. And that in fact, on very, very small scales, the universe is multi-dimensional, as many more dimensions than just four. But the fundamental reason for that is if you don't do that, theories break. Trying to unify gravity and quantum mechanics, this issue that was brought up here, cannot happen. So you can't understand gravity in a modern sense unless you do something like string theory. We really don't know where it's going. There are no testable predictions of string theory right now, but a lot of extremely smart people work on the problem. So, we'll see. Okay. Thanks. | {
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"text": [
" James Bullock and we're very proud to have him. He's a man who is asking those big questions.",
" Has the universe always existed? Did time have a beginning?",
" What makes us think that the earth goes around the sun?",
" What is a black hole? How fast is light?",
" So we're about to hear from someone who asks the big, big questions.",
" Would you please give a warm, inside-edge welcome to James Bullock.",
" Thanks very much. It's a pleasure to be here.",
" I thought, so what I'm going to do today is talk about some of the things we're trying to understand in the field of cosmology.",
" I actually work in that building right over there. That's the physics and astronomy department.",
" We have a center for cosmology there that was started about four years ago.",
" And the center is all about trying to answer the kinds of big questions that we were just hearing about.",
" And that's what we're doing. And I thought what I would do today is just sort of give you a little bit of an overview",
" of what we're trying to do, some of the big questions that we have.",
" And I'm going to do my best to leave some time at the end for questions.",
" Normally when I give these kinds of talks, I like to keep it very free-form and open.",
" So if anyone has any comments or questions at any time, I'm happy to address them.",
" Now, I understand that since we're recording this, that might make it a little bit logistically difficult to have people running around with microphones.",
" So what I'm going to do then is do my best to make sure I don't talk too much and leave some time at the end just for general questions.",
" If you have an urgent question at any time and you just need to ask it, we can do that.",
" So let's just keep it open there.",
" So I just thought I would start with this.",
" And one of the things I like to do when I talk about cosmology is to show this picture.",
" This is a picture that was taken by a photographer named Art Roche. And I thought it was really beautiful.",
" But one of the things that I find very attractive about it is he was telling me when he took this picture,",
" he was in this canyon in the southwest.",
" And he noticed that there were petroglyphs, sort of ancient petroglyphs left on the canyon walls.",
" And this reminds me, I think, of really what we're trying to do in cosmology,",
" which is it's a scientific exploration of the kinds of questions that you ask when you look up at the night sky.",
" So you can imagine the people who made these petroglyphs, these ancient peoples,",
" who were drawing into the canyon walls.",
" And when they looked up at the night sky, they probably had similar questions to the kind of questions that we have when we look up at the night sky.",
" So how old is the universe? Was it always there?",
" What's the universe made of?",
" How did structure like us come to be?",
" So these are the kind of questions that a lot of us has asked ourselves.",
" And in many ways, cosmology is the oldest science,",
" because you could imagine people have always asked themselves these kinds of questions.",
" What we're trying to do is to answer these questions in a scientific context.",
" So we're trying to make testable predictions and see if they come true,",
" and use that to build up a theory for how the universe began and emerged.",
" And for the first time, trying to build one that's in a scientifically tested context.",
" And that's what we're trying to do.",
" So what I'd like to do is tell you a little bit about what we believe about the universe,",
" so how we think the universe came to be and some basic facts.",
" And then I'm going to go on and tell a story of how we began to shape these ideas.",
" So this is kind of a cartoon picture of the overview of modern scientific cosmology,",
" and our present understanding of what the universe is and how it began.",
" So the first order picture here is that time and space, time and space itself,",
" began in something that we call the Big Bang, which is just a word, about 14 billion years ago.",
" The universe was very, very hot at early times.",
" The universe has been expanding.",
" So today the universe is quite big, and in early times, the universe was smaller and smaller.",
" It's expanding.",
" So if you go back in time like a movie, it's hot at very early times.",
" And in fact, it was so hot at early times that, you know, as you crank up the temperature,",
" things start to melt.",
" And if you keep cranking up the temperature, even molecules can't hang together anymore.",
" If you keep cranking up the temperature, even atoms, it becomes so hot that even atoms can't hang together anymore.",
" Electrons will fly off of protons.",
" In a very, very early universe, it was so hot that even elementary particles couldn't exist,",
" and we were down to the most fundamental constituents of nature.",
" So in that sense, it was a very smooth and, in some sense, simple beginning.",
" And from this elegant beginning, this very, very simple beginning,",
" emerged as the universe cools and cools over time, more complicated structures can begin to form.",
" And then you can start making planets, et cetera.",
" And we have a situation where we have really the real primordial soup, okay?",
" The early universe's primordial soup.",
" And then from this, structure grew.",
" And eventually, we end up with galaxies like ours, the galaxy that we live in, the Milky Way.",
" And in these galaxies, there are stars.",
" In our galaxy, there are billions of stars.",
" And around one of these stars orbits a planet that's very low mass, that's mostly rock, and that's the Earth.",
" So what we'd like to understand is how this picture emerges.",
" Another thing, oh, one thing I did want to mention is this, you know,",
" numbers like 14 billion don't seem that big anymore.",
" We're hearing about, you know, the national debt and the...",
" But 14 billion is a big word, is still a big number.",
" So 14 billion is a very long time.",
" If you took the entire age of the universe and scrunched it down into one year,",
" the scale of reference for that is that Shakespeare, okay, living in the 17th century,",
" wrote his plays about a second ago.",
" So you take the whole expanse of time and scrunch it into one year,",
" Shakespeare was walking around about a second ago.",
" So that's, these are the time frames we're talking about.",
" So true cosmological time frames.",
" Now, another thing we would like to understand is in addition to sort of the size of the universe,",
" the age of the universe, we'd like to understand what the fundamental constituents of the universe are, okay?",
" And this represents a pie chart of our current understanding of the composition of the cosmos.",
" And I'll talk a little bit about these different pieces here, it's just words,",
" but I thought I would just give you an overview now.",
" The thing you notice here is this area right here, this little yellow sliver in this pie chart,",
" is supposed to be representative of heavy elements.",
" And that heavy elements just means any kind of atom that's heavier than hydrogen or helium.",
" So basically us and the Earth.",
" And in terms of a global sort of composition of the universe,",
" that represents only about 0.03% of the composition.",
" So a tiny, tiny sliver of what we believe is out there.",
" About, only about 5% of the universe is made out of things that we have a really good understanding of what they are.",
" Okay, so the entire periodic table of elements, okay?",
" Everything you learn in chemistry class.",
" In fact, everything that they study in every department on this campus, except for ours,",
" is in this piece of the pie right here, okay?",
" I put neutrinos there purposefully, this is a kind of elementary particle that was discovered by Fred Reines,",
" who's our Nobel Laureate over there.",
" So I have to put that on the chart, even though it's 0.3%.",
" But the rest of this chart is all stuff that we really don't understand.",
" And I'll talk a bit about this.",
" About 25% of the universe consists of something we call dark matter.",
" Dark matter is some weird stuff that I'll talk about later,",
" and there's some even weirder stuff called dark energy, and that makes up 70% of the universe.",
" So this is really a statement of our ignorance, this pie chart.",
" We have a pretty good idea of what we don't understand, but as we all know, that's how you start.",
" You have to understand what you don't understand first.",
" Okay, so this is our ignorance chart here.",
" These are the big questions we're trying to answer.",
" Now, sorry.",
" So as I mentioned, cosmology is in many ways the oldest science.",
" And as far as I know, every culture on Earth has had their own story of cosmology.",
" So how does the Earth begin? How old is the Earth? How did we emerge?",
" One of the oldest pictures, one of the oldest cosmological models was actually that of Aristotle,",
" and then later on refined by Ptolemy.",
" And in Aristotle's model, the Earth sits at the center of the universe.",
" And all the planets and the sun orbit the Earth, and the stars extend out in a celestial sphere.",
" And this was the picture they had.",
" And, you know, Aristotle was not a dumb guy, right?",
" When you look up out the sky, it looks like the sun is going around the Earth, okay?",
" So, and in fact, this is the longest lasting scientific cosmological model in history.",
" It lasted 1400 years.",
" But eventually, this idea was broken by Copernicus and Galileo and Newton and people like that.",
" What Copernicus said is he said, well, you know, if you actually put the sun at the center",
" and let the planets go around the sun, I can also explain all the observations.",
" And to me, that seems just prettier.",
" It just seems a little bit more elegant.",
" But that was kind of the end of it, okay?",
" And there was sort of this, it wasn't clear whether that was really true or what.",
" The thing that really changed the way people saw the universe was when Galileo turned a telescope to the heavens",
" and he actually started testing some of these ideas.",
" And he showed that there were moons going around Jupiter and that the moon itself was corrupt.",
" That is, it had mountains and things and it wasn't this perfect celestial sphere kind of thing that Aristotle thought was going on.",
" And from this piece of technology and the application of this technology and direct observation,",
" it sort of shattered this idea that people had for a very long time.",
" A lot of very smart people had for a very long time.",
" So it was really the tools that Galileo had that allowed us to sort of extend this idea.",
" So this began, this began sort of the sort of modern theory of cosmology",
" where, you know, at that time cosmology was the solar system.",
" This is everything.",
" Now, one of the things that's remarkable about this is not only does this move the position of the Earth relative to the sun,",
" but it transforms sort of how it sort of, it's representative of how bizarre the universe had to be if this was true.",
" It wasn't that no one had ever conceived of the idea that the Earth might be going around the sun before.",
" There were ancient Greeks who proposed that idea.",
" But people decided that was crazy and the reason why they decided that was crazy is",
" if you have something that's going around the sun, let's say us, we're going around the sun, okay?",
" We're moving a big distance from one time of the year to the next.",
" So if that's true and you're moving a lot and you look at something and you're moving like this,",
" the position of that thing on the horizon will shift a bit, right?",
" When you're in a train, you see stuff go by and you have to turn your head.",
" The only way you don't have to turn your head on a train is when you're looking at, say, a mountain peak that's really, really far away.",
" So what this meant is if this is true, this meant the universe, the stars that were really far away,",
" which we never see shift from one time of the year to the next, must be really, really far away.",
" And so by putting the sun at the center and having the Earth go around the sun,",
" it was actually bizarre in many ways because it meant the universe was much, much bigger than anyone had ever imagined before.",
" So one of the things that's kind of interesting about scientific cosmology is I think almost every step of the way",
" the data tell us something that's much crazier than anyone had ever thought of before.",
" And I think it's true every step of the way.",
" If you look how cosmology proceeds, it's a crazier universe than anyone had ever imagined.",
" Even very creative people.",
" So today, I mean, cosmology at the end of the 20th century, it progressed significantly.",
" And people had finally figured out how to measure the tiny wiggles and distances to stars on the horizon,",
" and that allowed them to figure out how far away the stars were.",
" And it was realized that the sun, our sun, was just one among billions of stars in this galaxy.",
" And the galaxy is huge.",
" Okay, the galaxy, it takes light 100,000 years to cross our galaxy.",
" So just to explain what I'm trying to say here is that if I have a flashlight and I turn the flashlight on,",
" light travels from my flashlight at 186,000 miles a second.",
" So if I turn a flashlight on, a light beam could go around the Earth 10 times in one second.",
" It moves really fast, but it's moving at finite speed.",
" Light that leaves the sun takes eight minutes to get here.",
" So, which isn't that long a time, but it's still, you know, it's a delay.",
" Light takes 100,000 years to cross the galaxy to give you an idea of size.",
" Okay, it's big, very big.",
" And in fact, if you take the sun, okay, the sun is big, we really don't have a concept of how big it is,",
" but the Earth, we have some rough idea of how big it is, even though it's hard to conceptualize.",
" You can place 100 Earths across the face of the sun.",
" The sun is big.",
" If you took the sun and shrank it down to the size of a grain of sand, okay,",
" our galaxy would be the size of the Earth.",
" So the size of our sun compared to the size of the galaxy is like the size of a grain of sand to the Earth.",
" So these are the distances that we're trying to deal with, okay.",
" Now the thing is, at the end of the 20th century, pretty soon, Edwin Hubble would realize,",
" Edwin Hubble looking through his telescope, he would soon prove that actually there are blobs of stuff in the sky that we call nebulae,",
" that at the time people didn't know what they were, he would eventually figure out that these were actually distant galaxies of no own.",
" So that the universe is actually filled with billions of galaxies like ours.",
" So yes, billions and billions, just like Carl Sagan, he was right.",
" Okay.",
" So this is our current picture of the Milky Way.",
" The Milky Way, our galaxy, is a disk of stars.",
" It's got a bright thing in the middle, we call the bulge.",
" It's about 100,000 light-years across, and we live kind of at the edge here,",
" sort of in the middle, kind of out, sort of outer outskirts.",
" Here's the sort of blow-up of the sun and its circular planets going around it.",
" And like I said, it takes eight minutes from light to get to the sun to the earth,",
" but it takes 100,000 years for light to go across the galaxy.",
" So it's a pretty big place.",
" But this is nothing.",
" 100,000 years of travel time for light is small potatoes compared to the size of the universe.",
" This is the nearest big galaxy to us, called the Andromeda galaxy.",
" You can see this with your eye sometimes when it's dark.",
" It takes light two and a half million years to reach us from this galaxy.",
" And so just think about this.",
" This picture is what Andromeda looked like two and a half million years ago.",
" So before there were homo sapiens to look up at the sky, that's when this light left.",
" Okay, so we are looking back in time.",
" And the farther things are away, the further back in time you look.",
" And that's one of the techniques that we try to use",
" when we try to understand how the universe is assembled over time.",
" Because as we look at stuff that's further away,",
" we're looking at what the universe was like long ago.",
" And from this, you can imagine trying to build a movie,",
" sort of how stuff is assembled over time.",
" So you use this fact that light can't travel infinitely fast to your advantage.",
" And you just have to actually look at what the universe looked like in the past.",
" So we can make maps of what our local environment looks like.",
" And this is a lot of astronomers like to do this.",
" Here is our galaxy, the Milky Way.",
" And here is Andromeda, the one I was just showing you, two and a half million light years away.",
" And there's lots of other little galaxies all around.",
" We are the two big boys on the block,",
" but there are these little things that float around our own things and orbit us.",
" They're satellites.",
" They're galaxies that are satellites of our galaxy,",
" just like there are planets that are satellites of the sun orbiting around us.",
" And gravity is doing all this attraction.",
" And you can zoom out again.",
" So this is three million light years, this arrow.",
" And if we zoom out again, this is 30 million light years.",
" And we can make maps there, too.",
" And they're still naming things, but eventually you stop naming things.",
" There's so many things.",
" But all these guys have names, you know.",
" They're named after the constellations.",
" A lot of them are named after the constellations.",
" You have to look through to see them, right?",
" You have to look through the fornax constellation to see the fornax galaxy or whatever.",
" And that's when the naming convention comes from.",
" So this is 30 million light years.",
" And we can stop on this point. Every dot on this picture is a galaxy.",
" And we can keep zooming out.",
" So this is 300 million light years.",
" This is where we kind of stop naming stuff.",
" It gets kind of ridiculous.",
" It keeps going, okay?",
" It's sort of arbitrarily cut off in a circle.",
" But it keeps going.",
" So galaxies like to live around other galaxies.",
" They cluster together in clusters of galaxies and what we call superclusters.",
" And there's giant structures in the sky.",
" And one of the things we like to understand is why, how many, these kinds of things.",
" And in fact, so all of these things I had just shown you were cartoon pictures.",
" But this is actually a picture of real data.",
" This is the largest continuous map of the universe that's ever been made",
" by a survey called the Sloan Digital Sky Survey.",
" You've heard of the Alfred P. Sloan Foundation.",
" They funded this sky survey to take a huge section of the sky",
" and just look very deeply and try to map continuously all the galaxies.",
" This is a beautiful survey.",
" The total amount of data is 15 terabytes,",
" which is equivalent to about the Library of Congress.",
" So that's the kind of data that they've taken for this one, this one picture.",
" So these are the kind of maps that we're trying to make.",
" We want to know what the universe looks like.",
" Now there's another technique.",
" Rather than looking at sort of broadly at everything,",
" we could point at one point on the sky and go very, very deep.",
" And the way you do that is the same way you'd open up a shutter on a camera",
" and you could take a really faint image.",
" You do that with a telescope.",
" Let me back up.",
" That's what I'm going to show you.",
" I just want to explain it first.",
" So you know when you overexpose your camera, it's not good,",
" but you can take pictures of things that are very, very faint",
" by keeping your shutter open for a long time.",
" That's what we can do with the telescope.",
" So with the telescope, the lens is sort of eight meters across.",
" These are the biggest telescopes in the world,",
" and you can point them at one point on the sky and leave the shutter open.",
" So you can see stuff that's really far away.",
" The Hubble Space Telescope is a big telescope like this.",
" It's not quite that big, but you've heard of the Hubble Space Telescope.",
" So one thing that the Hubble folks did is they picked a place on the sky",
" that was completely dark, nothing there.",
" And they pointed the Hubble Space Telescope there for 12 days straight.",
" And so the point they pointed is somewhere in here.",
" Here's a constellation you might be familiar with.",
" And what I'm going to show you now is a movie that zooms in to this point.",
" Notice that it's complete blackness that they're going to zoom into.",
" And what they're doing is they're showing you progressively deeper and deeper images",
" of the same part of the sky.",
" And what we're looking at now is we're seeing nothing but galaxies.",
" We're way past the stars.",
" They're going deeper and deeper into sort of a deep pencil beam into the universe.",
" And this is it.",
" This is called the Hubble Ultra Deep Field.",
" It's a deep region of space that Hubble just sat on for 12 days.",
" And the light from these galaxies left these galaxies about 12 billion years ago.",
" So this is actually a movie, a snapshot of what the universe looked like,",
" something like 12 billion years ago.",
" And notice that the galaxies look kind of messed up.",
" They look like they're not fully formed yet.",
" So these are the kind of movies that you try to make and then interpret.",
" We want to understand how from this kind of stuff,",
" emerge the galaxies like we see.",
" So we like to ask, where did all this stuff come from?",
" How did we get here?",
" So one of the things you do when you have a question,",
" what things sometimes people do is you ask the smartest person you know.",
" And so the smartest person I know is Albert Einstein.",
" So how do we begin to sort of take measure of these kinds of observations?",
" It turns out, especially in the context of a theory that seems to be,",
" we seem to have a universe that's expanding.",
" You want to understand the nature of space and time.",
" It's sort of all grounds there.",
" And really the beginning of modern cosmology was really what this guy here,",
" when he had a short haircut.",
" In 1905, he proposed this theory that's called the special theory of relativity.",
" And as part of this theory, you know, he had this famous equation,",
" E equals MC squared.",
" But lying at the backbone of this derivation of this equation,",
" was that there was some kind of interrelationship between space and time.",
" What Einstein realized is, the actual way in which time ticks along,",
" is linked in some way with space and how fast people move through it.",
" This was an earth-shattering conclusion that he made.",
" And what it meant was, our previous concepts of space and time were wrong.",
" And that, in fact, the ideas that Newton had about how gravity operates on Earth,",
" and why apples fall from trees, had to be revised.",
" Now he realized this in 1905, and then for the next 10 years,",
" he said, oh, I need to fix gravity.",
" I need to figure out how gravity really works,",
" because E equals doesn't equal MC squared, according to Newton's gravity,",
" the first order.",
" So now we're talking about 10 years of Einstein time.",
" Okay?",
" I don't know how many years of my time that would be.",
" So it's like 600 years of my time.",
" Then in 1915, he comes upon what's called the general theory of relativity.",
" And in this theory, by the way, it's largely, is regarded by most professional physicists",
" as the single greatest achievement ever by anyone, period.",
" I mean, this is a very impressive thing.",
" What he realized is that you can understand gravity is actually a warping of space.",
" So that if you took, I mean, a way of sort of characterizing it,",
" is if you took a bowling ball and set it on a mattress,",
" and there's like a dip in the mattress,",
" you can imagine taking a marble and flicking it on the mattress, and the marble would roll",
" and bend as it approached that dip in the mattress.",
" It's in a similar way that Einstein understood how planets go around the sun,",
" and that there's a dip in space time around the sun,",
" and the planets go around on circles,",
" because that seems to them to be a straight line in that curved space.",
" He described all gravity that way.",
" Now the thing, the sort of backup ramification of this,",
" is that space is not something that's fixed that you just travel through.",
" It can warp and evolve and twist.",
" So if you think about this a little bit,",
" it's sort of, it's sort of, you're now kind of in this weird position",
" where you can have space itself, not being static, not always being the same.",
" So soon after this paper came out, a guy named Friedman read it,",
" and he realized that, hey, you know, if I look at your equations,",
" I realize that the universe should be expanding according to your equations of gravity.",
" Now Einstein said, uh-oh, that sounds crazy.",
" Even to me, that's crazy.",
" Okay, even this great creative genius that Einstein was couldn't accept that,",
" even though it was his own theory.",
" And so he changed his equations.",
" He changed his theory to account for it.",
" And he added basically a fudge factor that's called, that we call lambda,",
" which is now called the cosmological constant.",
" It's sort of like put a wedge in the door and kept the door from closing.",
" He just stuck it in.",
" He could do it, it was his theory, he did it.",
" But of course, this is widely regarded now,",
" and it's claimed that he said it was his greatest blunder.",
" You know, he could have predicted that the universe is expanding,",
" but he didn't.",
" But then in 1929, the same guy, Edwin Hubble, using the 100-inch Hooker telescope,",
" which is not too far from here,",
" saw that as he looked at galaxies that were more distant to us,",
" they were moving away from us faster.",
" And this implies that the universe is expanding.",
" So in some level, it fit in with the theory that Einstein had proposed,",
" but he didn't quite have the guts to stick with the prediction.",
" Now, what this gives rise to an idea is that we're leaving a...",
" And then a lot of data came after that.",
" People studied this for a long time.",
" And now we're up to this idea where, because the universe is expanding,",
" if you just run the movie backwards, it was smaller in the past.",
" And if you take something and you scrunch it down,",
" smaller and smaller and smaller,",
" the atoms in it or the gas in it will get hotter.",
" You might notice this sometime if you're holding a bicycle pump and pumping it,",
" it'll get hot.",
" If you scrunch air together, it tends to get hotter as it's compressed.",
" Same kind of thing with the universe.",
" The universe is scrunched, it's smaller, it's hotter.",
" Earlier than that, it was smaller and even hotter.",
" So it's now about 14 billion years since the Big Bang,",
" and now it's very cool.",
" It's about three degrees above absolute zero in empty space.",
" But the universe at early times was very, very hot.",
" And it turns out it was so hot in the early universe,",
" there's radiation that's left over from that.",
" And it's realized that that should be observed.",
" And in fact, in 1965,",
" Tensien and Wilson discovered the glow from the Big Bang,",
" which is called the cosmic microwave background radiation.",
" And from that, they were given a Nobel Prize.",
" And it sort of solidified this idea that in the past, the universe was hot.",
" So what we have now is a picture of the universe where not only doesn't have this map,",
" but it's moving away.",
" So I thought I would mention just a couple more words, and then I'll stop and open it up for questions.",
" But when you look at one of these galaxies in particular,",
" it turns out now we don't have to just, we can't just,",
" we don't just sort of take pictures of them and see where they are.",
" We can actually study them in detail and understand things like,",
" how fast are they spinning around on edge?",
" Because it's a disk.",
" It turns out that us, in our own galaxy, we're spinning around,",
" or orbiting around the center, spinning like a disk.",
" And in fact, if you look at the stars in the galaxies,",
" you expect them to spin around, but as you go out to the edge of the galaxy,",
" you expect it to go around more slowly because there's less gravity out there.",
" And in fact, Pluto, if you want to call it a planet,",
" is going around more slowly around the sun than the earth.",
" Because it's further away, there's less gravity from the sun.",
" And this is expected, this is well understood physics.",
" And so when people looked at these galaxies, what they expect to see",
" is that they speed, they rotate really fast in the center,",
" and then the speed rolls off quickly at large distances.",
" When the 70s, there's a woman named Vera Rubin who went out",
" to actually measure how fast they were spinning around.",
" And what Vera discovered is, that's not what's happening.",
" In fact, they're still going really fast way out here.",
" And the way we eventually came to understand this is",
" that there are big extended distributions of matter all around these galaxies.",
" And this matter is called dark matter, because we can't see it.",
" That's why it's dark.",
" Okay? Vera Rubin is credited with this observation,",
" although there are a lot of other people who found evidence for it,",
" even before Vera Rubin.",
" Just an interesting tidbit.",
" Her son is a professor of mathematics in the math department here.",
" There's a lot of more evidence for dark matter",
" by studying galaxy clusters.",
" Clusters of these galaxies, we can measure how fast stuff's moving",
" in those galaxy clusters. And there's a lot of evidence that it's there.",
" It's not just this rotation group. There's tremendous amounts of evidence.",
" Another question that people had, and they were specifically going after this issue in the 80s,",
" was, is the universe ever going to stop expanding and re-collapse",
" and have a big crunch?",
" In order for us to know what's going to happen, it kind of goes something like this.",
" Imagine that I took this pointer and I threw it up in the air.",
" Eventually it's going to come back down. It's going to come back down because gravity's tugging on it.",
" But if I lived on a planet that was really, really light, that didn't have much gravity,",
" the force of gravity would be less, be like on the moon.",
" And if I, you can imagine me throwing this up and this escaping the moon",
" and never coming back down.",
" So the question with the universe is, it's going up now, will it come back down?",
" And the crucial question is, how much mass is there?",
" If there's a lot of mass, it will slow down but never come back down.",
" Sorry, if there's not much mass, it might slow down but not come back down.",
" If there's a lot of mass, it will slow down and then re-collapse.",
" So people in the 80s were looking to see if this was what's going to happen.",
" So imagine this is distance and this is time and you throw a ball up,",
" you might expect it to come back down, or it might keep going up,",
" depending on how much mass the planet has or how much mass the universe has.",
" So a very heavy universe, we expect this.",
" In a very light universe, we just expect that ball to just keep going up.",
" The universe is expanding.",
" So a bunch of astronomers, one part of the team was led by a team at Berkeley,",
" was looking at this in the late, actually turned into late 90s when they actually got the result.",
" So this is a big crunch picture and this is a not big crunch picture.",
" The universe keeps expanding and this is what they wanted to know.",
" So they did the observation and this is what they found.",
" So what they found is, you throw the ball up and rather than it coming back down or slowing down,",
" it speeds up as it goes up.",
" That's weird. It's that weird. It's exactly that weird.",
" That's called dark energy.",
" So there's some force in the universe that's making the universe expand at accelerated rate.",
" We do not understand what this is.",
" The thing that's kind of fun about it is, our ideas for this,",
" well, let me tell you a little bit about our ideas.",
" So this is the dark energy piece.",
" This is the thing that makes the universe accelerate.",
" This is the dark matter piece.",
" This is the thing that makes galaxies spin around fast.",
" And this is the piece that we understand.",
" Chemistry, that's easy, right?",
" No. It's actually much harder than this.",
" Our questions are so simple that it's actually much easier than chemistry.",
" So here's the big questions.",
" What's the dark matter? What's the dark energy?",
" How did galaxies like ours emerge from this early universe?",
" What was going on in the early universe anyway?",
" So these are the kind of questions we're trying to answer.",
" What is the dark matter?",
" I'll tell you what we know.",
" It's dark. It doesn't shine.",
" What that really means is, it doesn't interact with light.",
" It doesn't interact with things electromagnetically.",
" So if I had a ball of dark matter, and it was traveling through the wall,",
" it would go right through the wall.",
" Because the only reason balls bounce off walls is because the electrons hit each other",
" and make it bounce.",
" It's electrical repulsion.",
" There is no electrical repulsion with dark matter.",
" It doesn't interact with electricity.",
" It doesn't interact with light.",
" But it does attract to other matter via gravity.",
" And most cosmologists believe that the dark matter is a fundamental particle of nature",
" that we just haven't discovered yet.",
" And there are big programs to look for it.",
" There's a big particle accelerator in Switzerland called the Large Hadron Collider.",
" Where they're taking particles and zooming them around a 17-mile loop.",
" They're going to slam them together, recreate the hot temperatures of the universe,",
" and try to make dark matter.",
" And try to observe it.",
" That's going on now.",
" There are also satellite projects.",
" They're looking for glows of high-energy gamma rays,",
" which might be evidence of this dark matter.",
" And also, we're looking for it directly.",
" We think there are dark matter particles streaming through the earth all the time.",
" We don't feel them.",
" But we're trying to build detectors that will feel them.",
" And by detecting them, it would be really interesting.",
" So this is the ways we're looking for it.",
" The dark energy is much harder.",
" It's dark. It doesn't shine.",
" It's energy. It causes the universe to accelerate.",
" But it does not attract stuff together like normal matter.",
" In fact, it's a lot like the cosmological constant that Einstein proposed.",
" In fact, it's possible that the dark energy is this cosmological constant,",
" that Einstein thought was his greatest mistake.",
" It might have been his greatest prediction.",
" Guy's always right.",
" Quintessence.",
" It's another idea.",
" So there are other ideas, and these are just words.",
" Another possibility is that Einstein's theory of gravity is wrong.",
" And our whole interpretation of why the universe is slowing down or speeding up",
" is just wrong because it's based on Einstein gravity.",
" So all the young physicists hope that this is true because they want to be the one to",
" know what the right number, the right answer.",
" Let me just flash on the future, and then I'll take a few questions.",
" So how are we going to try to answer some of these questions?",
" UC Irvine is involved in a survey called the Large Synoptic Survey Telescope.",
" This is that little Sloan diagram, which is the largest continuous map of the universe",
" that's ever been made, shown on the scale of what we want to do.",
" So we want to make, we want to map about a quarter of the visible universe",
" with this project.",
" This is going to be a telescope in Chile.",
" We're also involved in something called the 30-meter telescope.",
" The 30-meter telescope will be the largest optical telescope that's ever been built.",
" And there's something else that's coming online called the Hubble Space Telescope,",
" sorry, called the James Webb Space Telescope, which is the successor of the Hubble Space Telescope.",
" It's 10 times more powerful than Hubble.",
" So these things are coming online.",
" Let me show you, just to give you some perspective about what we're getting ready to do.",
" The Hooker Telescope was 100 inches across, which is big.",
" This is the telescope that Edwin Hubble used to discover that the universe was expanding.",
" At the time, he didn't know he was going to discover that.",
" And the Palomar 200-inch telescope, this was the biggest telescope in the world in 1949.",
" It was this telescope, of course, when it was built, no one had any idea",
" that eventually they would be able to measure galaxies spinning around in discovered arc matter.",
" It was with the Keck Telescope, which was built in 1993,",
" which is currently the largest telescope in the world,",
" that we used to discover that the universe is accelerating its expansion.",
" But when this telescope was built, if you said that to somebody, they would have thought you were crazy.",
" Okay? So now, this is the 30-meter telescope.",
" So who knows what we're going to find?",
" Okay. Now, I will close then with saying, this is the real future.",
" Okay? We're over there across the building over there,",
" and the people who really do the work are the graduate students and all these young people.",
" And so, I, you know, I was asked if I had anything to sell.",
" I don't have anything to sell.",
" But what I do kind of have to sell is this center in really these people.",
" There are people over there who are doing really great work.",
" And if you're interested in that work, you can contact me.",
" If you're interested in supporting that work, you can contact me.",
" Specifically, supporting maybe one of these people to go to one of these telescopes",
" and do some observations. That would really help us.",
" But anyway, so let's leave, let's leave with this picture.",
" That's it. I can take a couple questions. I'm sorry I went kind of long.",
" This hand was up first.",
" Yeah.",
" Thank you very much. Is this on?",
" Yeah.",
" Okay. If I can, if we can look into the far reaches in order to see back in time",
" and see a very primitive part of the, of the universe.",
" Yes.",
" If I were able to stand in that primitive part at that far away time,",
" Yes.",
" Would I eventually see back to where we are now and did that exist?",
" Okay. So if you were in that far away place of the universe and you looked towards us,",
" Yeah.",
" You would see us as we were 10 billion years ago.",
" So there would be no sun.",
" Was there a Milky Way?",
" The Milky Way was, yes. So that's a good question.",
" The Milky Way we think basically came into existence about 11 to 12 billion years ago.",
" And the sun was born in a sort of second generation of stars about five billion years ago.",
" Yes. Sure.",
" As you know, the, in the subatomic world, gravity has no effect.",
" Yes.",
" And so that's why I actually respect Niels Bohr and those guys a lot more than Einstein.",
" But that's me.",
" Einstein contributed to quantum mechanics too.",
" The point I'm making is have you, have you guys gleaned into the subatomic world to try to figure out what happened?",
" Yes. That's a great question.",
" So one thing I skipped over really fast is in fact that's a lot of what particle physics tries to do.",
" So in the very early universe, it was very, very hot.",
" So that there really was nothing but some of atomic particles.",
" There was nothing on the big scale.",
" The universe was tiny and very, very hot at this time.",
" And what particle accelerators do, what particle physicists do is they take particles and slam them together at very high energies to create very high temperatures and see what flies out to figure out what the subatomic world is made of.",
" Okay, Richard Feynman said it's like taking a switch watch, throwing it against the wall and watching what flies out to figure out what it's made of.",
" Okay. That's what, that's what's done.",
" And that has deep ties to cosmology because cosmology unites these fields.",
" Subatomic physics governs what's going on in the very early universe, which sets the stage for all that emerges.",
" And in fact, all the structure in the universe, we believe, originated in tiny fluctuations that were originally quantum mechanical processes that were developed some 14 billion years ago in a process known as cosmic inflation that eventually grew to be the galaxies we are today.",
" So without quantum mechanics, there would be nothing, according to that.",
" Earlier we were talking about Aspen and the wonderful research institute there.",
" Last summer I heard a lecture again on the universe and they were talking about, what I want to say is energy polarization when you have a galaxy that is flat and there are these energy spears, if you will, one going up and one going down, if you will.",
" Yes, yes. And how that creates the energy kind of like the ying and yang.",
" And I was wondering if you could fold that into what you're talking about.",
" A little bit. So the jets you're talking about in galaxies, a lot of galaxies are observed to have polar jets of material streaming out of them like this.",
" What that actually is fueled by is a black hole at their center, a very massive black hole that creates a very strong gravitational field and that heats up matter, makes it very hot and at the same time it creates a magnetic field that twists and that's what drives those columns out.",
" And so that's a very interesting subject and that's what people are using that to study black holes.",
" And black holes are really fundamental objects because they sort of sit at the cusp, the threshold of us understanding gravity in the extreme.",
" So another way we're trying to test our theories of gravity is by studying these black holes that create these kind of jets.",
" I have a rather pedestrian question. I was noticing that when you're showing a galaxy that it all seems to be rotating on one plane rather than all these others.",
" Now within that plane, does the planets around the sun rotate on that same plane?",
" That's a great question.",
" And how about the moon around the earth?",
" Yes.",
" So comment on that.",
" What they do and why.",
" Yes.",
" So the planets do not orbit the sun in the same plane that the galaxy is going around.",
" It's tilted by about 60 degrees and it turns out that the stars are so far away from each other that they don't really, that kind of effect is small.",
" If our sun was the size of a basketball, the nearest star would be in Hawaii.",
" So the size scales are very big.",
" They don't really interact with each other very strongly.",
" And so there's really not a strong correlation. It has to do with smaller scale processes, almost like weather we believe in the early galaxy.",
" Told the planet what plane to align on basically.",
" And the moon similarly.",
" The moon is sort of orbiting at a slightly different plane as well.",
" That's why we have phases of the moon because it's not exactly aligned up with the earth and the sun.",
" One more. Thank you.",
" Could you tell us about string theory and how does that relate to any of these, please?",
" I know it's the last question you have.",
" I have 30 seconds to explain to you string theory.",
" Okay.",
" So the basic idea of string theory is that there's this idea in particle physics that all the elementary things are single points.",
" Okay.",
" An electron is a single point and you model it that way.",
" String theory says no, no, maybe they're strings.",
" And it turns out if they're strings, a lot of ugliness goes away.",
" There are a lot of divergences that are a lot of infinities that pop up everywhere.",
" Remember when you're doing basic math and there are infinities everywhere and that's bad, does not exist.",
" It turns out particle physicists get that all the time.",
" But if you do string theory, a lot of that stuff goes away.",
" String theory is an idea that the subatomic world is really based on these one-dimensional loops.",
" And that in fact, on very, very small scales, the universe is multi-dimensional, as many more dimensions than just four.",
" But the fundamental reason for that is if you don't do that, theories break.",
" Trying to unify gravity and quantum mechanics, this issue that was brought up here, cannot happen.",
" So you can't understand gravity in a modern sense unless you do something like string theory.",
" We really don't know where it's going.",
" There are no testable predictions of string theory right now, but a lot of extremely smart people work on the problem.",
" So, we'll see.",
" Okay.",
" Thanks."
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} | en |