Macroscopic description of particle systems with non-local density-dependent diffusivity

TL;DR

This paper explores macroscopic density equations with non-local, density-dependent diffusivity, revealing differences between interpretations and demonstrating pattern formation under certain conditions, supported by microscopic simulations.

Contribution

It introduces a macroscopic framework for non-local density-dependent diffusion and compares interpretations, highlighting pattern formation and confirming results with microscopic models.

Findings

Differences between Ito and Hanggi-Klimontovich interpretations.

Pattern solutions appear under certain conditions in the Ito interpretation.

Microscopic simulations confirm macroscopic analysis.

Abstract

In this paper we study macroscopic density equations in which the diffusion coefficient depends on a weighted spatial average of the density itself. We show that large differences (not present in the local density-dependence case) appear between the density equations that are derived from different representations of the Langevin equation describing a system of interacting Brownian particles. Linear stability analysis demonstrates that under some circumstances the density equation interpreted like Ito has pattern solutions, which never appear for the Hanggi-Klimontovich interpretation, which is the other one typically appearing in the context of nonlinear diffusion processes. We also introduce a discrete-time microscopic model of particles that confirms the results obtained at the macroscopic density level.