Exclusion processes: short range correlations induced by adhesion and contact interactions

TL;DR

This paper investigates how local interactions like adhesion and contact influence short-range correlations in exclusion processes, revealing wave-like correlation dynamics and developing hydrodynamic models relevant for biological migration.

Contribution

It introduces a detailed analysis of contact interactions in exclusion processes, linking microscopic correlations to macroscopic hydrodynamic limits, especially for biological cell migration.

Findings

Correlation waves travel with velocity ~ t^(-1/2) during space invasion.

Contact interactions affect the shape and height of correlation waves.

Hydrodynamic equations capture self-similar density and correlation evolution.

Abstract

We analyze the out-of-equilibrium behavior of exclusion processes where agents interact with their nearest neighbors, and we study the short-range correlations which develop because of the exclusion and other contact interactions. The form of interactions we focus on, including adhesion and contact-preserving interactions, is especially relevant for migration processes of living cells. We show the local agent density and nearest-neighbor two-point correlations resulting from simulations on two dimensional lattices in the transient regime where agents invade an initially empty space from a source and in the stationary regime between a source and a sink. We compare the results of simulations with the corresponding quantities derived from the master equation of the exclusion processes, and in both cases, we show that, during the invasion of space by agents, a wave of correlations travels with velocity v(t) ~ t^(-1/2). The relative placement of this wave to the agent density front and the time dependence of its height may be used to discriminate between different forms of contact interactions or to quantitatively estimate the intensity of interactions. We discuss, in the stationary density profile between a full and an empty reservoir of agents, the presence of a discontinuity close to the empty reservoir. Then, we develop a method for deriving approximate hydrodynamic limits of the processes. From the resulting systems of partial differential equations, we recover the self-similar behavior of the agent density and correlations during space invasion.