Phase Transitions, Logarithmic Sobolev Inequalities, and Uniform-in-Time Propagation of Chaos for Weakly Interacting Diffusions (Scott A. Smith)

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05 29, 2023

  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 large N limit of the constant in the logarithmic Sobolev inequality (LSI) for the N-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.

   

  Publication:

  Communications in Mathematical Physics (2023). https://doi.org/10.1007/s00220-023-04659-z

   

  Author:

  Matías G. Delgadino

  Department of Mathematics, The University of Texas at Austin, Austin, USA

  

  Rishabh S. Gvalani

  Max-Planck-Institut für Mathematik in den Naturwissenschaften, Leipzig, Germany

  

  Grigorios A. Pavliotis

  Department of Mathematics, Imperial College London, London, UK

  

  Scott A. Smith

  Academy of Mathematics and Systems Sciences, Chinese Academy of Sciences, Beijing, China

  Email: ssmith@amss.ac.cn

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