From the Boltzmann \(H\)-theorem to Perelman's \(W\)-entropy formula for the Ricci flow

Abstract

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 Poincar\'e conjecture and Thurston's geometrization conjecture. In our previous paper \cite{Li11a}, 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.