From the BoltzmannH-theorem to Perelman'sW-entropy formula for the Ricci flow

In 1870s, L. Boltzmann proved the famous H-theorem for the Boltzmann equation in the kinetic theory of gas and gave the statistical interpretation of the thermodynamic entropy.In 2002, G. Perelman introduced the notion of W -entropy and proved the W -entropy formula for the Ricci flow.This plays a crucial role in the proof of the no local collapsing theorem and in the final resolution of the Poincaré conjecture and Thurston's geometrization conjecture.In our previous paper [21], the author gave a probabilistic interpretation of the W -entropy using the Boltzmann-Shannon-Nash entropy.In this paper, we make some further efforts for a better understanding of the mysterious W -entropy by comparing the H-theorem for the Boltzmann equation and the Perelman W -entropy formula for the Ricci flow.We also suggest a way to construct the "density of states" measure for which the Boltzmann H-entropy is exactly the W -entropy for the Ricci flow.