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

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.

4 pages, 3 figures

References in corpus (10)

• Dynamic Density Functional Theory of Fluids
• Langevin Equation for the Density of a System of Interacting Langevin Processes
• Dynamical density functional theory for interacting Brownian particles: stochastic or deterministic?
• Non-local Interaction Effects on Pattern Formation in Population Dynamics
• Clustering, advection and patterns in a model of population dynamics with neighborhood-dependent rates
• Aggregation of finite size particles with variable mobility
• Fluctuations impact on a pattern-forming model of population dynamics with non-local interactions
• Nonlocal competition and logistic growth: patterns, defects and fronts
• Diffusion, peer pressure and tailed distributions
• Self-propelled non-linearly diffusing particles. Aggregation and continuum description

Cited by in corpus (7)

• Classical dynamical density functional theory: from fundamentals to applications
• Non-Markovian Levy diffusion in nonhomogeneous media
• Crystallization and melting of bacteria colonies and Brownian Bugs
• Pattern Formation in Populations with Density-Dependent Movement and Two Interaction Scales
• Single-species fragmentation: the role of density-dependent feedbacks
• Turing instability in oscillator chains with non-local coupling
• Influence of density-dependent diffusion on pattern formation in a refuge