Phase Transitions, Logarithmic Sobolev Inequalities, and Uniform-in-Time Propagation of Chaos for Weakly Interacting Diffusions
Phase Transitions, Logarithmic Sobolev Inequalities, and Uniform-in-Time Propagation of Chaos for Weakly Interacting Diffusions
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DOI：
10.1007/s00220-023-04659-z
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发表时间：
2021-12
影响因子：
2.4
通讯作者：
M. Delgadino;Rishabh S. Gvalani;G. Pavliotis;Scott A. Smith
中科院分区：
文献类型：
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作者：
M. Delgadino;Rishabh S. Gvalani;G. Pavliotis;Scott A. Smith
在这篇文章中，我们研究了约束势和相互作用势不一定是凸的弱相互作用扩散的平均场极限。本文研究了N粒子系统对数Sobolev不等式中常数的大N极限与平均场极限相变的存在与否之间的关系。我们表明，LSI常数的非简并意味着均匀的时间传播的混沌和高斯的波动在平衡。作为副产品，我们的分析，我们提供了简洁的，据我们所知，新的证据的广义形式的Talagrand的不平等和定量传播的混乱，采用技术从理论的梯度流，特别是黎曼演算空间的概率措施。
In this article, we study the mean field limit of weakly interacting diffusions for confining and interaction potentials that are not necessarily convex. We explore the relationship between the largeNlimit of the constant in the logarithmic Sobolev inequality (LSI) for theN-particle system and the presence or absence of phase transitions for the mean field limit. We show that the non-degeneracy of the LSI constant implies uniform-in-time propagation of chaos and Gaussianity of the fluctuations at equilibrium. As byproducts of our analysis, we provide concise and, to our knowledge, new proofs of a generalised form of Talagrand’s inequality and of quantitative propagation of chaos by employing techniques from the theory of gradient flows, specifically the Riemannian calculus on the space of probability measures.