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

TL;DR

This paper explores the deep analogy between Boltzmann's $H$-theorem and Perelman's $W$-entropy for Ricci flow, offering new insights and a potential way to construct a 'density of states' measure linking the two concepts.

Contribution

It compares the $H$-theorem and $W$-entropy, providing a probabilistic interpretation and proposing a method to construct a measure connecting the two entropies.

Findings

Comparison between Boltzmann $H$-theorem and Perelman's $W$-entropy.

Probabilistic interpretation of $W$-entropy using Boltzmann-Shannon-Nash entropy.

Proposal of a way to construct a 'density of states' measure for the $W$-entropy.

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.