Eur. Phys. J. B (2013) 86: 456
https://doi.org/10.1140/epjb/e2013-40838-y

Regular Article

Persistent random walk with exclusion

¹ Dipartimento di Fisica, Università di Firenze and INFN, 50019 Sesto F.no ( FI), Italy
² Dipartimento di Sistemi e Informatica and INFN, Università Di Firenze, Via S. Marta 3, 50139 Florence, Italy
³ Centre de Biophysique Moléculaire (CBM), CNRS-UPR 4301 and Université d’Orléans, Département de Physique, 45071 Orléans Cedex, France

^a e-mail: Francesco.Piazza@cnrs-orleans.fr

Received: 16 September 2013
Published online: 4 November 2013

Abstract

Modelling the propagation of a pulse in a dense 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.