An invariance property of diffusive random walks

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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 trajectories 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 (Case K. M. and Zweifel P. F., Linear Transport Theory (Addison-Wesley) 1967), leading to broad physics and biology applications.