In Volume 1 Chapter 39 of the Feynman Lectures on Physics, Feynman derives the ideal gas law from Newton's laws of motion. But then on page 41-1, he puts a caveat to the derivation he has just completed (italics in original):
Incidentally, when we say that the mean kinetic energy of the particle is $\frac{3}{2}kT$, we claim to have derived this from Newton's laws…. and it is most interesting that we can apparently get so much from so little … How do we get so much out? The answer is that we have been perpetually making a certain important assumption, which is that if a given system is in thermal equilibrium at some temperature, it will also be in thermal equilibrium with anything else at the same temperature. For instance, if we wanted to see how a particle would move if it was really colliding with water, we could imagine that there was a gas present, composed of another kind of particle, little fine pellets that (we suppose) do not interact with water, but only hit the particle with "hard" collisons. Suppose that the particle has a prong sticking out of it; all our pellets have to do is hit the prong. We know all about this imaginary gas of pellets at temperature $T$ – it is an ideal gas. Water is complicated, but an ideal gas is simple. Now, our particle has to be in equilibrium with the gas of pellets. Therefore, the mean motion of the particle must be what we get for gaseous collisions, because if it were not moving at the right speed relative to the water but, say, was moving faster, that would mean that the pellets would pick up energy from it and get hotter than the water. But we had started them at the same temperature, and we assume that if a thing is once in equilibrium it stays in equilibrium – parts of it do not get hotter and other parts colder, spontaneously. This proposition is true and can be proved from the laws of mechanics, but the proof is very complicated and can be established only by using advanced mechanics. It is much easier to prove in quantum mechanics than it is in classical mechanics. It was first proved by Boltzmann, but now we simply take it to be true, and then we can argue that our particle has to have $\frac{3}{2}kT$ of energy if it is hit with artificial pellets, so it also must have $\frac{3}{2}kT$ when it is being hit with water at the same temperature and we take away the pellets; so it is $\frac{3}{2}kT$. It is a strange line of argument, but perfectly valid.
My question is, what is the proposition by Boltzmann that Feynman is referring to? I can think of three possibilities:
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It could refer to the Equipartition Theorem, which was independently proven by Maxwell and Boltzmann, since the title of the section is "The Equipartition of Energy".
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It could refer to the Zeroth Law of Thermodynamics, because "it will also be in thermal equilibrium with anything else" (especially when he emphasizes the "anything else") sounds like the fact that thermal equilibrium is an equivalence relation.
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It could refer to Boltzmann's H-theorem, which was the derivation of the second law of thermodynamics from Newton's laws, because "if a thing is once in equilibrium it stays in equilibrium – parts of it do not get hotter and other parts colder, spontaneously" sounds like some formulations of the second law.
So which of the three is it, if it's any of the three, and how would you prove it from Newton's laws?
Best Answer
I am very confident that Feynman is talking about the zeroth law of thermodynamics here.
Feynman is discussing the classical Brownian motion experiment of pollen grains swimming in water in terms of kinetic theory. However, kinetic theory is only valid in ideal gases, while in liquids there are many additional hydrodynamic effects. Feynman argues this doesn't really matter, since if the grain is in equilibrium with water, it might as well be in equilibrium with "a gas of pellets". This argument is based on the zeroth law of thermodynamics.
The zeroth law of thermodynamics, introduced by Boltzmann in his book on "Theory of heat", is indeed very complicated to proof from mechanical theory. A thorough mathematical proof has been published in 2012. I am not an expert enough to judge whether this proof is comprehensive or if it will be developed further. Also, I am not aware of the proof by Ludwig Boltzmann that Feynman mentions. A proof derived from quantum mechanics, on the other hand, is relatively straightforward and well known since the 1980s (see Gorini et al. 1984).
I don't think Feynman refers to the equipartition theorem or the H-theorem here. Feynman has already touched on the equipartition theorem in Chapter 39-4. In chapter 41-1, he goes into further detail and gives additional examples. He also mentions that "if a thing is once in equilibrium it stays in equilibrium". This observation is closely related with entropy and the second law of thermodynamics, but it is also the basic definition of equilibrium: The properties do not change over time. If they would, it wouldn't be an equilibrium. Feynman uses this for this ad hoc derivation of the zeroth law, but it is not the assumption he tacitly made earlier.