Talk:H-theorem

message: how did boltzmann come up with his definition for H?
(Yosofun 06:42, 12 Apr 2005 (UTC)
thread_title:  Origin of the H formula ? 

message: I'm cutting the following line:
:The final resolution of the time reversibility paradox is found in the Fluctuation Theorem.
There is stuff which could be said about the Fluctuation Theorem as a modern generalisation of work on the prediction of entropy increase.  
But the FT has nothing to say about the Arrow of time, because, like Boltzmann's H-theorem, the entropy increase it predicts so well arises because one assumes one can ignore correlations and similar 'hidden information' at times after the initial time.  If you start with a non-equilibrium state and try to use the Fluctuation Theorem to predict backwards how that non-equilibrium state came to be, it will predict that it arose spontaneously as a fluctuation from an equilibium state -- because those are the only possibilities it knows about.
The cause of time asymmetry in the observed universe must be looked for elsewhere.
Jheald 12:28, 21 October 2005 (UTC)
thread_title:  Fluctuation theorem 

message: But I have to say it is a pretty hard book to digest in a short while.  I get the feeling that I am going to be reading it for quite some time,
because I find the section on quantum mechanics particularly difficult, and yet required to understand some of his most profound ideas.
I found an article on Boltzmann_entropy which I am starting to understand is related to H.
The citations I found from Tolman so far, don't need to go in the first paragraph if this makes the first paragraph seem to wordy,
but I think the actual page numbers and wording might actually belong someplace at least in the article?
One thing that I have found in my experience to be true is that citations take more work than typing, so maybe the main contribution
here is providing actual inline page numbers to a source for some of the material.
::Um, I don't want to sound overbearing, and it's really good the way you're working through this; but can I suggest that you wait until you understand what is in the article first, so that you're confident you have a solid grasp of the big picture first, before you start trying to rewrite it?  
::While you're getting there, questions like "I don't understand this" or "This doesn't seem quite right to me" or "Why is the article saying that" are maybe more useful than trying to rewrite it as you're going along.
::If you have access to a good university library, you also might find that some more modern "introduction to statistical mechanics" books may give a quicker, more accessible presentation.  It's always good to have a look at what more than one source of exposition is saying (and the references would be good for the article, too!)  Jheald (talk) 21:35, 24 February 2009 (UTC)
replies: text: I'm not saying there isn't a lot wrong with this article, because there most definitely is.  But I do think the best way to fix it is from the big picture first; i.e. to do some fairly comprehensive reading first, and work out what the most important things we want to say are first, rather than starting in on it piecemeal.  Jheald (talk) 22:30, 24 February 2009 (UTC)
replies: text: I agree to this methodology.  However this suggests a more major undertaking than I had originally planned.  I had at first thought to work on providing inline citations to the existing article with possibly some slight rewordings to bring the wikipedia text into agreement with those sources.  That is OK though because the more I learn about H theorem the more interesting it becomes.  Also the concept of the H theorem is deeper and more complicated than I first thought it would be!  I especially appreciate your help in working through this. I am sure that there are both passive current and future readers of this discussion who are, or will be equally grateful for your contributions.  
Hobojaks (talk) 21:08, 25 February 2009 (UTC)
thread_title: Been reading Tolman

message: In classical thermodynamics, the H-theorem was a thermodynamic principle first introduced by Boltzmann in 1872.  The H-theorem demonstrates the tendency of the molecules of a system to approach their equilibrium Maxwell-Boltzmann distribution Toleman 1938 pg 134. The quantity called H by Boltzmann can be regraded as an appropriate measure of the extent to which the condition of the system deviates from that corresponding to equilibriumtoleman 1938 p 134.  This quantity H has a tendency to decrease with time to a minimum and thus for the system to approach its equilibrium condition.Toleman 1938 p 134. 
The H-theorem has been restated in quantum mechanical form.Tolman 1938 chapter XII The Quantum Mechanical H-Theorem.
The decrease in the quantity H is related to the increase in the entropy
.  For example the approach to equilibrium of an ideal gas in an irreversible process, can be demonstrated using an approach based on the Boltzmann equation.
It appears to predict an irreversible increase in entropy, despite microscopically reversible dynamics.  This has led to much discussion.
ergodic hypothesisTolemam 1938 pg 65  —Preceding unsigned comment added by 76.191.171.210 (talk) 07:23, 24 February 2009 (UTC)
thread_title: Some material for possible inclusion

message: It would seem to me that equations 47.5 to 47.9 of on page 135 - 136 of tolemans book in the section titled definition of the quantity H are very important.  When I get  chance I will start to transcribe these here for possible inclusion.  —Preceding unsigned comment added by Hobojaks (talk • contribs) 21:14, 24 February 2009 (UTC)
replies: text: They are all variants on H = \sum p_i \log p_i, which is really the key idea.  Best to go with that, I think.  Jheald (talk) 22:36, 24 February 2009 (UTC)

text: Something like H or \overline{\overline{H}}? Physchim62 (talk) 12:29, 26 February 2009 (UTC)
replies: text: Not just that, but it gives a nice general notation based on transition probabilities between states, into which terms the classical versions can easily be cast as well. Jheald (talk) 11:26, 1 March 2009 (UTC)

text: I would use H, without the double bar, for the first time you are introducing the system.  -- After all, it's only Tolman who uses the double bar, it's not "standard" notation.  For other uses, we could than qualify H in difference ways, eg perhaps with a subscript, if we really want to emphasise that it is being calculated for a different probability distribution.  Jheald (talk) 11:26, 1 March 2009 (UTC)
thread_title:  Some important formula 

message: Hobojaks (talk) 19:20, 26 February 2009 (UTC)
thread_title: Notes

message: 1)Complete,  I am changing the order of the quantum mechanical version of H theorem, with the other section.
As per discussion this is the more relevant definition of H theorem.  This may require some cleanup.
(2)Complete, I am moving the discussion of Shannon entropy into the first quantum mechanical section of the 
article.
(3)Completed, I am pasting in the inline citation from Toleman for the first formula on entropy as
suggested by J-heald.
(4)First stage now complete,These changes being made, my work will be done with the quantum mechanical section, and I also
plan to leave the first paragraph of the article alone for a while to let these changes cool down for a while.
(5) I would then like to reconstruct the section on the original Boltzmann H theorem along the lines as it was developed by Toleman, providing the proper inline citations.  With a note about the formula offered by Rief.
:5(a) My first step will be to copy formulas 47.5 to 47.9 found in Toleman page 135.
Toleman is pretty good about tracing the historical development of H theorem.  I plan on making it clear that this is Toleman's development of the concept, leaving any parts of Boltzman's original formulation, not covered by Toleman a bow for another Ulysses.  The two developments are close but the extent that Boltzman's development differs from Toleman's so far I have only Toleman's account.
I now have all the fomulas i wanted to add to the article except for one that I am still working on the proper math tags for.
See the really ugly period at the start of the lower part of the formula that I used to get the equals signs to line up????
\gamma \,
H= \sum( n_i \log n_i -  n_i \log \delta v_\gamma)Toleman 1938 pg. 135 formula 47.7
.\,\,\,\,\,= \sum n_i \log n_i + constToleman 1938 pg. 135 formula 47.7
I plan on changing the formula current formula for H per unit volume, to make it more in line with the format and definitions of Reif with the appropriate inline citations.  Or possibly doing away with it all together?  But this would require more discussion.  Can anybody provide an inline source citation for this formula?
(6)This will leave the article on H-theorem with still more work to do, but I feel like maybe it will be someone else's turn to take it from there for a while, and I plan on a cooling off period where I might participate in discussion, but not do any more editing of the article for a while.  Hobojaks (talk) 20:36, 28 February 2009 (UTC)
replies: text: No time now, but I'll try and comment in more detail on Monday.  Jheald (talk) 11:28, 1 March 2009 (UTC)
thread_title:  Plan for a little editing. 

message: I am not sure who is the "source" being cited but the criticism clearly shows that its author has zero understanding for the proof presented in the article - and zero understanding for thermodynamics in general.
1) It's the very point of the proof that one divides the evolution into infinitesimal steps that are long enough that the "thermalization" occurs in the interval, and it produces nonzero probabilities for indistinguishable microstates. On the other hand, because this time can still be essentially arbitrarily short for macroscopic systems, the first-order perturbation theory becomes actually exact, so there is no error introduced by the first-order approximation of the Fermi's rule. At any rate, even if a higher-order corrected rule were used instead, one could still prove the inequality.
2) The derivation clearly talks about the probabilities of all possible microstates of the system. It makes no assumption about the number of "particles" or other degrees of freedom that describe the system. So in this sense, the derivation is absolutely exact and surely doesn't neglect any interactions. Only the "critic" is introducing some assumptions about the number of particles and the organization of the Hilbert space as a tensor product. The proof is doing nothing of the sort and is completely general in this sense.
3) It's completely absurd that an isolated system with many (N) particles will "sit in one microstate". Quite on the contrary, it will try to evolve into an equal mixture of completely all microstates with equal probabilities. That's what the proof shows - and what every sane person knows from the 2nd law of thermodynamics and everyday life, anyway.--Lumidek (talk) 09:27, 3 April 2009 (UTC)
thread_title:  The "Criticism" section is complete nonsense 

message: 1) If Boltzmann proved the H-theorem, why in the world does the article start off with the quantum version of it? First the classical treatment should be given, followed by the quantum version.
2) The article says to look up the Wikipedia entry on Shannon's information entropy in order to better understand the definition of H. This is ridiculous! The H-theorem article should not only be self-contained, but it should be able to explain the H-theorems's foundations better than any other article, since after all, it's about the H-theorem!
3) There are misplaced interjections, such as mentioning "a recent controversy called the Black hole information paradox." at the end of the Quantum mechanical H-theorem section, and a discussion about Loschmidt's paradox and molecular chaos at the end of the Boltzmann's H-theorem section. These should be discussed appropriately in the Analysis section or in their own section. 
4) The Analysis section quotes various researchers without references and makes a personal attack on Gull in the paragraph "(It may be interesting ...". Overall it reads like an incoherent rant and rambling. 
5) The article is full of sentences devoid of any information, like:
"This has led to much discussion."
"and thus a better feel for the meaning of H."
" something must be wrong"
" and therefore begs the question." (even linking to begs the question)
"But this would not be very interesting."
"With such an assumption, one moves firmly into the domain of predictive physics: if the assumption goes wrong, it may produce predictions which are systematically and reproducibly wrong."  — Preceding unsigned comment added by Michael assis (talk • contribs) 20:32, 10 November 2012 (UTC)
thread_title:  Terribly written article! 

message: ″In classical statistical mechanics, the H-theorem, introduced by Ludwig Boltzmann in 1872, describes the tendency to increase in the quantity H (defined below) in a nearly-ideal gas of molecules.″
Shouldn't the H quantity increase? Also
"As this quantity H was meant to represent the entropy of thermodynamics, the H-theorem was an early demonstration of the power of statistical mechanics as it claimed to derive the second law of thermodynamics—a statement about fundamentally irreversible processes—from reversible microscopic mechanics."
If I'm not wrong it should be noted somewhere that H should be the negative entropy, because it leads to a lot of confusion otherwise (also considering that in many contexts the symbol H represents the common entropy).
Correct me if I'm wrong, I'm not an expert on the subject.
~~maioenrico~~  — Preceding unsigned comment added by 79.45.212.156 (talk) 15:00, 13 December 2015 (UTC)
thread_title:  Isn't the introduction wrong? 

message: Hello fellow Wikipedians,
I have just modified one external link on H-theorem. Please take a moment to review my edit. If you have any questions, or need the bot to ignore the links, or the page altogether, please visit this simple FaQ for additional information. I made the following changes:
*Added archive https://web.archive.org/web/20120204035512/http://www.ucl.ac.uk/~ucesjph/reality/entropy/text.html to http://www.ucl.ac.uk/~ucesjph/reality/entropy/text.html
When you have finished reviewing my changes, you may follow the instructions on the template below to fix any issues with the URLs.
Cheers.—InternetArchiveBot (Report bug) 13:34, 11 December 2017 (UTC)
thread_title:  External links modified