Collision number statistics for transport processes

TL;DR

This paper develops a generalized method to compute the moments of particle collision counts in various transport processes, extending the Kac formula and applicable to diverse domain types.

Contribution

It introduces a unified framework for calculating collision number moments using convolutions of equilibrium distributions, generalizing the Kac formula for residence times.

Findings

Derived explicit formulas for collision moments in different domains

Extended Kac formula to a broader class of transport processes

Validated approach with practical examples in various geometries

Abstract

Many physical observables can be represented as a particle spending some random time within a given domain. For a broad class of transport-dominated processes, we detail how it is possible to express the moments of the number of particle collisions in an arbitrary volume in terms of repeated convolutions of the ensemble equilibrium distribution. This approach is shown to generalize the celebrated Kac formula for the moments of residence times, which is recovered in the diffusion limit. Some practical applications are illustrated for bounded, unbounded and absorbing domains.