An invariance property of diffusive random walks

Abstract

Starting from a simple animal-biology example, a general, somewhat counter-intuitive property of diffusion random walks is presented. It is shown that for any (non-homogeneous) purely diffusing system, under any isotropic uniform incidence, the average length of trajectories through the system (the average length of the random walk trajecto-ries from entry point to first exit point) is independent of the characteristics of the diffusion process and therefore depends only on the geometry of the system. This exact invariance property may be seen as a generalization to diffusion of the well known mean-chord-length property [1], leading to broad physics and biology applications. Let us first consider a practical animal-biology example that was at the origin of the theoretical work reported hereafter. It is well established, for diverse species, that the spontaneous displacement of insects such as ants, on an horizontal planar surface, may be accurately mod-eled as a constant-speed diffusion random-walk [2-4]. We assume that a circle of radius R is drawn on the surface and we measure the average time that ants spend inside the circle when they enter it (see Fig. 1). The last assumption is that measurements are performed a long time after ants were dropped on the surface, so that the memory of their initial position is lost : this is enough to ensure that no specific direction is favored and therefore that when ants encounter the circle, their incident directions are distributed isotropically. Simple Monte Carlo simulations of such experiments (see Fig. 2) indicate without any doubt that the average encounter time (time between entry into the circle and first exit), for a fixed velocity, depends only of the circle radius. It is independent of the characteristics of the diffusion walk : the mean free path $\lambda$ (average distance between two scattering events, i.e. between two direction-changes), and the single-scattering phase function p(u s , u i) (probability density function of the scattering direction u s for an incident direction u i). Furthermore, this average time scales as R/v, which means that the average trajectory-length < L > scales as R. The average trajectory-length would therefore be the same for different experiments with any insect species-or for any type of diffusive corpuscular motion. There are two reasons why this observation may initially sound counter-intuitive. The first reason is that the average length of diffusion trajectories between two points is known to scale approximately as d 2 /$\lambda$ where d is the distance between the two points (1). For shorter mean (1)This scaling would be exact for an infinite domain in the limit d >> $\lambda$ [5] c EDP Sciences