  Motivated by a probabilistic approach to Kahler-Einstein metrics we consider
a general non-equilibrium statistical mechanics model in Euclidean space
consisting of the stochastic gradient flow of a given (possibly singular)
quasi-convex 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-Ampere operator, whose static solutions correspond to toric
Kahler-Einstein metrics. This drift-diffusion equation is the gradient flow on
the Wasserstein space of probability measures of the K-energy functional in
Kahler geometry and it can be seen as a fully non-linear version of various
extensively studied dissipative evolution equations and conservations laws,
including the Keller-Segel equation and Burger's equation. We also obtain a
real probabilistic (and tropical) analog of the complex geometric
Yau-Tian-Donaldson conjecture in this setting. In a companion paper
applications to singular pair interactions are given.
