Collision densities and mean residence times for $d$-dimensional exponential flights

TL;DR

This paper analyzes exponential flights, a stochastic process relevant to various transport phenomena, providing new exact and asymptotic results for their behavior in multiple dimensions, including stationary densities and residence times.

Contribution

It introduces a general framework for d-dimensional exponential flights and derives novel exact and asymptotic results, including the stationary density in 2D and mean residence times.

Findings

Derived the stationary probability density for 2D systems.

Obtained asymptotic results for exponential flights.

Supported findings with Monte Carlo simulations.

Abstract

In this paper we analyze some aspects of {\em exponential flights}, a stochastic process that governs the evolution of many random transport phenomena, such as neutron propagation, chemical/biological species migration, or electron motion. We introduce a general framework for $d$-dimensional setups, and emphasize that exponential flights represent a deceivingly simple system, where in most cases closed-form formulas can hardly be obtained. We derive a number of novel exact (where possible) or asymptotic results, among which the stationary probability density for 2d systems, a long-standing issue in Physics, and the mean residence time in a given volume. Bounded or unbounded, as well as scattering or absorbing domains are examined, and Monte Carlo simulations are performed so as to support our findings.