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
