Coarse graining of a Fokker-Planck equation with excluded volume effects preserving the gradient-flow structure

Abstract

The propagation of gradient flow structures from microscopic to macroscopic models is a topic of high current interest. In this paper we discuss this propagation in a model for the diffusion of particles interacting via hard-core exclusion or short-range repulsive potentials. We formulate the microscopic model as a high-dimensional gradient flow in the Wasserstein metric for an appropriate free-energy functional. Then we use the JKO approach to identify the asymptotics of the metric and the free-energy functional beyond the lowest order for single particle densities in the limit of small particle volumes by matched asymptotic expansions. While we use a propagation of chaos assumption at far distances, we consider correlations at small distance in the expansion. In this way we obtain a clear picture of the emergence of a macroscopic gradient structure incorporating corrections in the free energy functional due to the volume exclusion.