2013/03/21 by Xiang-Dong Li, Xiang‐Dong Li, Li, Xiang-Dong · 1 citation
Mathematics · Physics and Astronomy · #Advanced Thermodynamics and Statistical Mechanics #Boltzmann constant #Boltzmann equation #Boltzmann's entropy formula #Configuration entropy #Conjecture #Differential Geometry (math.DG) #Entropy (arrow of time) #Entropy in thermodynamics and information theory #FOS: Mathematics #FOS: Physical sciences #Geometric Analysis and Curvature Flows #Geometry #H-theorem #Joint quantum entropy #Mathematical Physics (math-ph) #Mathematical physics #Mathematics #Maximum entropy thermodynamics #Physics #Principle of maximum entropy #Pure mathematics #Ricci curvature #Ricci flow #Statistical physics #Statistics #Thermodynamics #math-ph #math.DG #math.MP
paper · pdf · doi:10.48550/arxiv.1303.5193
published in arXiv (Cornell University) (Cornell University)
arxiv created 2013/03/21 · openalex publication_date 2013/03/21 · arxiv updated 2013/03/22 · openalex created_date 2022/08/14 · openalex updated_date 2026/08/05
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 \citeLi11a, 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.