Residence time and collision statistics for exponential flights: the rod problem revisited

TL;DR

This paper develops a general method using Feynman-Kac formalism to analyze residence time and collision statistics of particles performing exponential flights, exemplified through the 1D rod problem, with implications for Monte Carlo simulations.

Contribution

It introduces a new analytical approach for calculating moments of collision counts and residence times in exponential flight processes, applied to the rod problem.

Findings

Explicit expressions for collision and residence time moments in 1D systems.

Relevance of results for improving Monte Carlo estimators.

Demonstration of the method's applicability to transport phenomena.

Abstract

Many random transport phenomena, such as radiation propagation, chemical/biological species migration, or electron motion, can be described in terms of particles performing {\em exponential flights}. For such processes, we sketch a general approach (based on the Feynman-Kac formalism) that is amenable to explicit expressions for the moments of the number of collisions and the residence time that the walker spends in a given volume as a function of the particle equilibrium distribution. We then illustrate the proposed method in the case of the so-called {\em rod problem} (a 1d system), and discuss the relevance of the obtained results in the context of Monte Carlo estimators.