Collision statistics for random flights with anisotropic scattering and absorption

TL;DR

This paper derives explicit formulas for collision statistics in random walks with anisotropic scattering and absorption, applicable to arbitrary domains and boundary conditions, especially when the diffusion limit is not reached.

Contribution

It provides new explicit formulas linking collision moments to the equilibrium distribution for a broad class of random walks with anisotropic scattering and absorption.

Findings

Formulas applicable to arbitrary domains and boundary conditions.

Analytical calculations demonstrated for 1D exponential displacements.

Assessment of hitting statistics when displacements are comparable to domain size.

Abstract

For a broad class of random walks with anisotropic scattering kernel and absorption, we derive explicit formulas that allow expressing the moments of the collision number $n_V$ performed in a volume $V$ as a function of the particle equilibrium distribution. Our results apply to arbitrary domains $V$ and boundary conditions, and allow assessing the hitting statistics for systems where the typical displacements are comparable to the domain size, so that the diffusion limit is possibly not attained. An example is discussed for one-dimensional (1d) random flights with exponential displacements, where analytical calculations can be carried out.