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On the Size of Chaos via Glauber Calculus in the Classical Mean-Field Dynamics

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Abstract

We consider a system of classical particles, interacting via a smooth, long-range potential, in the mean-field regime, and we optimally analyze the propagation of chaos in form of sharp estimates on many-particle correlation functions. While approaches based on the BBGKY hierarchy are doomed by uncontrolled losses of derivatives, we propose a novel non-hierarchical approach that focusses on the empirical measure of the system and exploits discrete stochastic calculus with respect to initial data in form of higher-order Poincaré inequalities for cumulants. This main result allows to rigorously truncate the BBGKY hierarchy to an arbitrary precision on the mean-field timescale, thus justifying the Bogolyubov corrections to mean field. As corollaries, we also deduce a quantitative central limit theorem for fluctuations of the empirical measure, and we discuss the Lenard–Balescu limit for a spatially homogeneous system away from thermal equilibrium.

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Notes

  1. More precisely, for \(H:{\mathbb {D}}^m\rightarrow {\mathbb {R}}\), we write \({{\text {Sym}}}(H)(z_1,\ldots ,z_m)=\frac{1}{m!}\sum _{\sigma \in {\mathcal {S}}_m}H(z_{\sigma (1)},\ldots ,z_{\sigma (m)})\), where \({\mathcal {S}}_m\) denotes the set of all permutations of the set [m].

  2. More explicitly, \({\mathcal {G}}^\circ \) is the centered Gaussian field characterized by its variance structure \({\text {Var}}\!\left[ {\int _{\mathbb {D}}\phi \, {\mathcal {G}}^\circ }\right] =\int _{\mathbb {D}}\phi ^2F^\circ -(\int _{\mathbb {D}}\phi F^\circ )^2\) for all \(\phi \in C^\infty _c({\mathbb {D}})\).

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Acknowledgements

The author wishes to warmly thank François Golse, Laure Saint-Raymond, and Sergio Simonella for motivating discussions. His work is supported by the CNRS-Momentum program.

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  1. Laboratoire de Mathématiques d’Orsay, CNRS, Université Paris-Saclay, 91400, Orsay, France

    Mitia Duerinckx

  2. Département de Mathématique, Université Libre de Bruxelles, 1050, Brussels, Belgium

    Mitia Duerinckx

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  1. Mitia Duerinckx
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Correspondence to Mitia Duerinckx.

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Communicated by C. Mouhot

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Duerinckx, M. On the Size of Chaos via Glauber Calculus in the Classical Mean-Field Dynamics. Commun. Math. Phys. 382, 613–653 (2021). https://doi.org/10.1007/s00220-021-03978-3

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