H-theorem and conservation of the Gibbs entropy

[alexV]

My understanding of the Boltzmann's H-theorem is that if a set of a large number of colliding bolls is not in the thermodynamical equilibrium (i.e. the probability distribution function W doesn't obey the Maxwell distribution), its entropy will grow (without supplying heat) until the equilibrium is reached. On the other hand, the Gibbs entropy defined as the integral of W*logW over all phase space is a constant of motion regardless of the system being in thermodynamic equilibrium or not (the latter is a direct consequence of the Liouville equation). How does this two statements reconcile?

 

[anuttarasammyak]

On the other hand, the Gibbs entropy defined as the integral of W*logW over all phase space is a constant of motion regardless of the system being in thermodynamic equilibrium or not

Not an answer to your question, but please tell me where can I learn it. I want to know the two ways of entropy
$$k_B \ln W, -k_B \Sigma_i p_i \ln p_i$$
clearly.

 

[alexV]

Although my question is not about difference between definition of entropies, I can still provide a few references:
1. E.T. Jaynes, "Gibbs vs Boltzmann Entropies", Am. J. Physics, v. 33, 391 (1965); doi: 10.1119/1.1971557
2. R.H. Swendsen, J.-S. Wang, " The Gibbs "volume" entropy is incorrect", atXiv: 1506.0691 1v1 [cond-mat.stat_mech] 23 Jun 2015.
3. P. Buonsante, R. Franzosi, A. Smerzi, "On the dispute between Boltzmann and Gibbs entropy", Annals of Physics, 375 (2016), 414-434; doi 10.1016/j.aop.2016.10.017.
These papers, however, seems to be concerned with calculation of entropies at the thermodynamic equilibrium; this has no bearing to my question.

 

[anuttarasammyak]

On the other hand, the Gibbs entropy defined as the integral of W*logW over all phase space is a constant of motion regardless of the system being in thermodynamic equilibrium or not (the latter is a direct consequence of the Liouville equation).

Gibbs entropy undertakes coarse graining by finite small volumes of phase space, not integral but sum, I think. Volume unit of h^3N from QM corresponds to it.