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.

 

[Demystifier]

The key is to distinguish fine grained Gibbs entropy from coarse grained Gibbs entropy. The former is a constant of motion. The latter isn't.

 

[Demystifier]

The picture illustrates how the fine grained volume (in phase space) remains the same during the evolution, while the coarse grained volume increases. Thinking of Gibbs entropy as the logarithm of volume in phase space, this explains how the fine grained entropy remains the same, while the coarse grained entropy increases.

 

[Demystifier]

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.

See e.g. https://research.engineering.nyu.edu/~jbain/tcs_seminar/lectures/04.Stat_Mech_Entropy.pdf