Smoothness of the diffusion coefficients for particle systems in continuous space

TL;DR

This paper proves that for certain particle systems in continuous space, the diffusion matrix smoothly depends on particle density, with explicit derivative descriptions using correctors, advancing understanding of particle system behavior.

Contribution

It establishes the infinite differentiability of the diffusion matrix as a function of density and provides explicit derivative formulas for these matrices.

Findings

Diffusion matrix is infinitely differentiable in particle density.

Explicit formulas for derivatives of the diffusion matrix.

Method applicable to local interaction particle systems.

Abstract

For a class of particle systems in continuous space with local interactions, we show that the asymptotic diffusion matrix is an infinitely differentiable function of the density of particles. Our method allows us to identify relatively explicit descriptions of the derivatives of the diffusion matrix in terms of correctors.