From the Boltzmann $H$-theorem to Perelman's $W$-entropy formula for the Ricci flow
Abstract
In 1870s, L. Boltzmann proved the famous -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 -entropy and proved the -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 -entropy using the Boltzmann-Shannon-Nash entropy. In this paper, we make some further efforts for a better understanding of the mysterious -entropy by comparing the -theorem for the Boltzmann equation and the Perelman -entropy formula for the Ricci flow. We also suggest a way to construct the "density of states" measure for which the Boltzmann -entropy is exactly the -entropy for the Ricci flow.
Keywords
Cite
@article{arxiv.1303.5193,
title = {From the Boltzmann $H$-theorem to Perelman's $W$-entropy formula for the Ricci flow},
author = {Xiang-Dong Li},
journal= {arXiv preprint arXiv:1303.5193},
year = {2013}
}