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arXiv · 1303.5193

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

Xiang-Dong Li
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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 Poincaré 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.

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Xiang-Dong Li. 2013-03-21. From the Boltzmann $H$-theorem to Perelman's $W$-entropy formula for the Ricci flow. https://arxiv.org/abs/1303.5193

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