Is there a general H-theorem?

Question

In statistical mechanics, Boltzmann showed that for dilute gases the H-function increases. I remember from a lecture that there is no general H-theorem, e.g. for non-dilute gases or in the quantum case but there seem to be also claims to the contrary. So what is the status of the (generalised) H-theorem?

This post imported from StackExchange Physics at 2016-08-02 12:32 (UTC), posted by SE-user corto

Comments

Boltzmann has shown his $H$ function decreases in time (under certain assumptions).

This post imported from StackExchange Physics at 2016-08-02 12:32 (UTC), posted by SE-user Ján Lalinský

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@JánLalinský - I thought Boltzmann defined a priori the $H$-function to decrease because it corresponded to the physical "direction" of time?

This post imported from StackExchange Physics at 2016-08-02 12:32 (UTC), posted by SE-user honeste_vivere

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@honeste_vivere, no, why do you think that? $H$ function is defined as functional of probability distribution. It decreases in some situations and increases in other.

This post imported from StackExchange Physics at 2016-08-02 12:32 (UTC), posted by SE-user Ján Lalinský

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@JánLalinský - I recall reading it in one of Cedric Villani's books and several others. Boltzmann was criticized by some other big name for doing this and his response was for them to propose a situation where $H$ would change in the opposite sense. I forget now but I definitely recall reading something along these lines.

This post imported from StackExchange Physics at 2016-08-02 12:32 (UTC), posted by SE-user honeste_vivere