Persistent random walk with exclusion

TL;DR

This paper extends the persistent random walk model to include spatial exclusion effects, deriving nonlinear mean-field equations and analyzing the transition from ballistic to diffusive behavior in dense environments.

Contribution

It introduces a novel non-ideal persistent random walk model with exclusion constraints and derives analytical expressions for the effective velocity and MSD evolution.

Findings

MSD transitions from ballistic to diffusive behavior.

Effective velocity depends on density and initial pulse shape.

Normal diffusion is recovered at long times after a density-dependent crossover.

Abstract

Modelling the propagation of a pulse in a dense {\em milieu} poses fundamental challenges at the theoretical and applied levels. To this aim, in this paper we generalize the telegraph equation to non-ideal conditions by extending the concept of persistent random walk to account for spatial exclusion effects. This is achieved by introducing an explicit constraint in the hopping rates, that weights the occupancy of the target sites. We derive the mean-field equations, which display nonlinear terms that are important at high density. We compute the evolution of the mean square displacement (MSD) for pulses belonging to a specific class of spatially symmetric initial conditions. The MSD still displays a transition from ballistic to diffusive behaviour. We derive an analytical formula for the effective velocity of the ballistic stage, which is shown to depend in a nontrivial fashion upon both the density (area) and the shape of the initial pulse. After a density-dependent crossover time, nonlinear terms become negligible and normal diffusive behaviour is recovered at long times.