Vol. 11, No. 6, 2018

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ISSN: 1948-206X (e-only)
ISSN: 2157-5045 (print)
Propagation of chaos, Wasserstein gradient flows and toric Kähler–Einstein metrics

Robert J. Berman and Magnus Önnheim

Vol. 11 (2018), No. 6, 1343–1380
Abstract

Motivated by a probabilistic approach to Kähler–Einstein metrics we consider a general nonequilibrium statistical mechanics model in Euclidean space consisting of the stochastic gradient flow of a given (possibly singular) quasiconvex N-particle interaction energy. We show that a deterministic “macroscopic” evolution equation emerges in the large N-limit of many particles. This is a strengthening of previous results which required a uniform two-sided bound on the Hessian of the interaction energy. The proof uses the theory of weak gradient flows on the Wasserstein space. Applied to the setting of permanental point processes at “negative temperature”, the corresponding limiting evolution equation yields a drift-diffusion equation, coupled to the Monge–Ampère operator, whose static solutions correspond to toric Kähler–Einstein metrics. This drift-diffusion equation is the gradient flow on the Wasserstein space of probability measures of the K-energy functional in Kähler geometry and it can be seen as a fully nonlinear version of various extensively studied dissipative evolution equations and conservation laws, including the Keller–Segel equation and Burger’s equation. In a companion paper, applications to singular pair interactions in one dimension are given.

Keywords
statistical mechanics, Kähler–Einstein metrics, propagation of chaos, Langvin equation
Mathematical Subject Classification 2010
Primary: 00A05
Milestones
Received: 28 June 2016
Revised: 9 October 2017
Accepted: 12 January 2018
Published: 3 May 2018
Authors
Robert J. Berman
Department of Mathematical Sciences
Chalmers University of Technology
Göteborg
Sweden
Magnus Önnheim
Department of Mathematical Sciences
Chalmers University of Technology and University of Götenburg
Göteborg
Sweden