## From the Boltzmann \$H\$-theorem to Perelman's \$W\$-entropy formula for the Ricci flow    [PDF]

Xiang-Dong Li
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.
View original: http://arxiv.org/abs/1303.5193