I'm not PBS Space Time, but I'll give it a shot in two parts.

First, let's start with entropy as a measure of disorder. The more disordered a system is, the higher the entropy.

Next let's consider an ordinary elastic or rubber band. When it is in its most relaxed state it has a certain measure of entropy. But as we stretch it, the elastic molecules tend to become less jumbled and more aligned: they have lower entropy because we have done some external work in stretching the band, and that work meets the requirement of the second law of thermodynamics. The more-aligned rubber band molecules pull against one another along the direction of stretching, because they prefer to be in the more jumbled-up, higher-entropy, relaxed state, in line with the second law. This pulling is an example of an entropic force.

Notably, the entropic force increases as we separate the molecules along a single direction: as you stretch an elastic band it gets harder to stretch it further. (It works in more than one dimension too: we can stretch a sheet of rubber, or pulling a ballon in several directions, and so on, and there will be an entropic force felt along all the directions of stretching).

Entropic gravity treats gravitation as an entropic force. Masses want to move towards each other because they have lower entropy when there is less space between them. And as one introduces more and more space between them, it gets harder and harder to separate the masses than a strict inverse square law (a quadratic function) of attraction describes. When masses are very far apart, perhaps many thousands of light-years, the decrease in mutual attraction tends to fall off linearly rather than quadratically. In comparison with General Relativity in the absence of dark matter, it gets harder to separate the edges of dark-matter-free galaxies from their cores. And at larger scales, compared to dark-matter-free General Relativity, it also gets harder to separate galaxies within a cluster from one another.

(One might notice a problem here: galaxy clusters are flinging away from each other according to redshift surveys like SDSS, so one has to break the entropic force at some distance scale, roughly analogous to how one might break an overstretched elastic band or balloon).

Entropic gravity theories (there are numerous individual theories partially written down) aim to eliminate the need for dark matter and dark energy, but also know they must recover the behaviour of General Relativity perfectly within our solar system (where it is very precisely tested and completely accurate to the level of precision) and in various systems inside our galaxy -- eclipsing binaries, including where one or more of the objects are compact or relativistic objects like neutron stars (especially millisecond pulsars) or black holes.

The physical intuition behind entropic gravity is that the vacuum -- a state completely devoid of Standard Model particles -- can be in almost any sort of configuration without real particles appearing. "Vacuum energy" is even used in the standard cosmology as an intuition about the cosmological constant, and particularly in trying to understand cosmic inflation in the very early universe.

In classical General Relativity the entropy of the vacuum is as high as one can get. It is maximally disordered. One way of understanding this is to think about Boltzmann's measure of entropy as the log relationship between the number of microstates and the number of macrostates.

Let's think of how this works. Refining our idea of entropy as a measure of disorder, we can now try to think of it in terms of how sensitively it reacts to a change in the order.

Forgive a small abuse of Boltzmann-entropy terminology here: if we take a macrostate to be one m^3 (cubic metre), and treat every mm^3 (cubic millimetre) inside as a microstate, we can consider systems with different entropies.

If you have a m^3 of water, you can swap any two mm^3 and not make any practical difference to its characteristics at the m^3 scale: it stays liquid, the absorption/emission spectrum doesn't change. Similarly if we instead do this operation on a m^3 box filled with an inert gas. But if we take a m^3 box filled with a curled up human and swap two of the enclosed mm^3 microstates arbitrarily, there is a good chance we quickly end up with a dead human in a box instead of a live human in a box. The mm^3s we swap could be air and heart valve. Or fingernail and alveolus. The result is that the human's emission spectrum -- heat from metabolism -- cools off rapidly. Algor mortis.

We can swap lots of ever-finer sections of the water or inert gas around within the enclosing box and not notice much difference, until we get down to the sub-molecular level. But if we do this with a human or a tree we get a dead organism. As we swap around at the sub-molecular level, in water we end up maybe getting diffuse bubbles of hydrogen peroxide, oxygen, or hydrogen gasses. These have different spectroscopic features. But if we do that to a tree or to a human we are breaking starches apart into sugars, disaccharides into monosaccharides, breaking down fats, and so on. As well as producing bubbles of gasses. Result: dead organism.

If we get down to making many swaps at the subatomic level both water and the inert gas end up undergoing nuclear reactions. In our water, by swapping nucleons about, we may get deuterium or tritium, radioactive isotopes of oxygen, and so on. If we start swapping quarks around maybe we suddenly get antiprotons or antineutrons. And so forth. We have something that is visibly not just one m^3 of water.

But what about vacuum? At the subatomic level, no matter how small you get with the microstates and how vigorously you swap them, a m^3 of classical vacuum remains an m^3 of classical vacuum. You swap the smallest units (much smaller than (planck length)^3) of empty space with similar units of empty space and you still have empty space by any measurement.

This means that classical vacuum has the ultimate entropy. And this is why an expanding universe has enormously-increasing entropy: more and more classical vacuum keeps appearing.

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