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In the preface to Landau and Lifshitz's Statistical Physics, they comment the following

In the discussion of the foundations of classical statistical physics, we consider from the start the statistical distribution for small parts (subsystems) of systems, not for entire closed systems. This is in accordance with the fundamental problems and aims of physical statistics, and allows a complete avoidance of the problem of the ergodic and similar hypotheses, which in fact is not important as regards these aims.

After explaining that they will consider (macroscopical) subsystems of the whole closed system, they claim

A fundamental feature of this approach is the fact that, because of the extreme complexity of the external interactions with the other parts of the system, during a sufficiently long time the subsystem considered will be many times in every possible state.

But this is very similar to the ergodic hypothesis, is it not? From my Stat Physics lecture notes,

Given a sufficiently large time, any closed system will approach arbitrarily closely any point in phase space

I don't understand how considering subsystems overcomes the need for this hypothesis, neither what advantage it has. It seems to me that the authors are using the complexity of the interactions between the subsystem and the rest of the system to grant credibility to the claim that the subsystem will eventually have populated the entire phase space. But why isn't this equally credible to stating the ergodic hypothesis?

Qmechanic
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Lourenco Entrudo
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2 Answers2

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The problem with the ergodic hypothesis is that it is just that: a hypothesis, and plenty of effort has been spent on trying to show whether it is a reasonable guess or indeed whether it is true, for the case of a closed system. Landau and Lifshitz are deftly sidestepping all that argumentation about closed systems by saying let's not bother even trying to treat a closed system. Let's allow our system to exchange energy with other systems (or, in their language, let's treat subsystems). This will be sufficient for all purposes of their treatment. But you are right, they do still wish to claim that such a subsystem will eventually explore all the available states, and they do make such a claim as a hypothesis or axiom. The difference is that to argue for the reasonableness of their hypothesis they can appeal to the complexity of the interactions at the boundary, rather than worry about some symmetry or other in a closed system which might make the ergodic hypothesis inapplicable.

Andrew Steane
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There seems to be some ambiguity about what they mean by a "closed" system to begin with. Usually, a closed system can perfectly exchange energy with its surrounding insofar as its matter content remains unchanged. At least that is one accepted definition of the term "closed" in thermodynamics. Alternatively, some may refer to "closed" as meaning isolated with no exchange of energy whatsoever and no change in the matter content either, which is what they seem to use.

Furthermore, there seems to be the possible tacit presupposition that the subsystems may be in some sort of equilibrium with each other, otherwise one would have to specify exactly which part of the system they are looking at and with which Gibbs ensemble they wish to characterise it and with which corresponding parameter values. This is something that does happen in seemingly 'equilibrated' systems whereby the observed stationary statistics are better explained by conceiving subsystems each obeying a canonical ensemble law for example, but with different inverse temperature values giving rise to a compound statistics which is non-Gibbsian. This kind of analyses has given rise to a theory called super-statistics which has been very fruitful to interpret astrophysical data for example.

Now, someone mentioned the FPUT problem in the comments. But what what is interesting with this problem is that it does not matter whether you are looking a subpart of the system or not, in the 'non-ergodic' regime people will observe energy localisation both in physical space and mode space to the extent that equipartition is never observed for any mode for certain initial conditions. Worse even, the time it takes for the FPUT model to relax to equipartition appears to increase as a power law with the system size.

Finally, I would personally take issue with the, often mentioned, statement that

A fundamental feature of this approach is the fact that, because of the extreme complexity of the external interactions with the other parts of the system, during a sufficiently long time the subsystem considered will be many times in every possible state.

I don't believe that, apart from faith or pure hypothesising, there is any reason to make such a blanket assertion for any possible physical system (see discussion with FPUT above). For one thing, for macroscopic systems recurrence times can be arbitrarily large and randomly so and way larger than any typical observation time. For another, it is strange to request this of all states since it will entail the subsystems cycling back multiple times to a possibly lower entropy state (in the sense of Boltzmann say).

Consequently, I agree with the OP that Landau and Lifshitz are attempting to sweet-talk the readers into making them believe that they don't need to make any hypotheses, while they actually make a ton of them, many of which corresponding to one version of ergodicity or another.

gatsu
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