Validity of Boltzmanns Equation and $H$-function theorem?

Question

A while ago I came across a resource (which I have forgotten) on the validity of Boltzmann's equation. It talked about the fact that the Boltzmann's equation is valid at the extrema of the $H$-function. In the discussion there was a graph that looked similar to the following

but better drawn (clearly). With the dots indicating some of the locations where the Boltzmann equation holds.

This may not be the exact theory, but it is along the right lines. Does anyone know if it has a name and what the theory actually states? (Even better if you can name the actual source I was looking at.)

Rob Tan · Answer

Very late answer, but it may be Kerson Huang "Statistical Mechanics" fig. 4.7 pag. 89 (in second edition).
The dots are to evidence the local maxima in time, related to states of molecular chaos of the gas. The theory states that a system with a distribution function $f$ that satisfies Boltzmann's transport equation tends to diminish its $H(f)$ with time (ideally going towards Maxwell-Bolzmann distribution in which $H$ is a minimum and $text{d}H/text{d}t=0$): at the same time, the theory behind is intrinsecally statistical so you should expect some fluctuations of your quantity, that in fact are present. That's mainly all I know of the subject, hope this helps someone.

Validity of Boltzmanns Equation and $H$-function theorem?

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