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.
Highlights
An interesting feature of many partial differential equations (PDEs) describing dissipative mechanisms in particle systems is that they can be seen as gradient flows of an associated free-energy functional
This is the case of the linear Fokker–Planck equation [27], which describes the evolution of the probability of one or many Brownian independent particles, and many other nonlinear Fokker–Planck equations including nonlinear diffusions and McKean–Vlasov like equations [3, 22, 35, 38, 42]
If we consider N Brownian particles moving under an external potential V (x), their evolution can be described by the following stochastic differential equation (SDE):
Summary
An interesting feature of many partial differential equations (PDEs) describing dissipative mechanisms in particle systems is that they can be seen as gradient flows (or steepest descents) of an associated free-energy functional. It has been shown by mean field or matched asymptotic expansions directly on the PDE level that the macroscopic model preserves the structure and that interactions appeared as a quadratic term in the free energy. This embeds into a more abstract setting of evolutionary convergence of gradient flows, see [4, 30, 32, 37, 39] All these papers are working on the lowest-order limit, while we seek to derive a first-order expansion in terms of the small volume of particles (note that the lowest order in our case is a linear Fokker–Planck equation for single particles that can be obtained ).
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