Effusion of stochastic processes on a line

TL;DR

This paper investigates the leakage of particles modeled by Gaussian processes from a confined region, deriving new statistical results for particle counts and their correlations, extending previous Brownian motion studies.

Contribution

It provides new analytical results for the statistics of leaked particles for arbitrary Gaussian processes, including joint distributions and the effects of initial correlations.

Findings

Joint two-time distribution is a bivariate Poisson for annealed initial conditions.

Variance of particle number exhibits strong memory effects for Gaussian processes.

Initial correlations significantly influence particle leakage statistics.

Abstract

We consider the problem of leakage or effusion of an ensemble of independent stochastic processes from a region where they are initially randomly distributed. The case of Brownian motion, initially confined to the left half line with uniform density and leaking into the positive half line is an example which has been extensively studied in the literature. Here we derive new results for the average number and variance of the number of leaked particles for arbitrary Gaussian processes initially confined to the negative half line and also derive its joint two-time probability distribution, both for the annealed and the quenched initial conditions. For the annealed case, we show that the two-time joint distribution is a bivariate Poisson distribution. We also discuss the role of correlations in the initial particle positions on the statistics of the number of particles on the positive half line. We show that the strong memory effects in the variance of the particle number on the positive real axis for Brownian particles, seen in recent studies, persist for arbitrary Gaussian processes and also at the level of two-time correlation functions.