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Phase Transitions, Logarithmic Sobolev Inequalities, and Uniform-in-Time Propagation of Chaos for Weakly Interacting Diffusions

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Abstract

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.

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Acknowledgements

The authors would like to thank Martin Hairer and Luigia Ripani for useful discussions during the course of this work. G.A.P. was partially supported by the EPSRC through the grant number EP/P031587/1 and by JPMorgan Chase & Co under J.P. Morgan A.I. Research Awards 2019 and 2021. M.G.D. was partially supported by CNPq-Brazil (#308800/2019-2), Instituto Serrapilheira, NSF-DMS-2205937 and NSF-DMS RTG 1840314. S.A.S acknowledges financial support from the National Key R &D Program of China (No. 2022YFA1006300) and the Chinese Academy of Sciences.

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Authors and Affiliations

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

    Matías G. Delgadino

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

    Rishabh S. Gvalani

  3. Department of Mathematics, Imperial College London, London, UK

    Grigorios A. Pavliotis

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

    Scott A. Smith

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  1. Matías G. Delgadino
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  2. Rishabh S. Gvalani
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  3. Grigorios A. Pavliotis
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Correspondence to Matías G. Delgadino.

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Delgadino, M.G., Gvalani, R.S., Pavliotis, G.A. et al. Phase Transitions, Logarithmic Sobolev Inequalities, and Uniform-in-Time Propagation of Chaos for Weakly Interacting Diffusions. Commun. Math. Phys. 401, 275–323 (2023). https://doi.org/10.1007/s00220-023-04659-z

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