Entropy, from Boltzmann H-theorem to Perelman's W-formula for Ricci flow

In 1872-1877, 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 Poincare conjecture and Thurstons geometrization conjecture. In our previous work in 2007, inspired by Perelman's original work, we gave a probabilistic interpretation of the W-entropy using the BoltzmannShannon-Nash entropy. In this talk, 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 Ricci flow.

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