The number statistics and optimal history of non-equilibrium steady states of mortal diffusing particles

TL;DR

This paper analyzes the number statistics and optimal density histories of non-equilibrium steady states of mortal diffusing particles, including non-interacting and interacting cases, revealing non-Poissonian distributions and their variances.

Contribution

It determines the most likely density histories conditioned on particle number and extends analysis to interacting particle systems with non-Poissonian statistics.

Findings

Total particle number in steady state is Poisson-distributed for non-interacting particles.

Interacting systems exhibit non-Poissonian stationary distributions.

Variances of particle number distributions are explicitly calculated for different models.

Abstract

Suppose that a point-like steady source at $x=0$ injects particles into a half-infinite line. The particles diffuse and die. At long times a non-equilibrium steady state sets in, and we assume that it involves many particles. If the particles are non-interacting, their total number $N$ in the steady state is Poisson-distributed with mean $\bar{N}$ predicted from a deterministic reaction-diffusion equation. Here we determine the most likely density history of this driven system conditional on observing a given $N$. We also consider two prototypical examples of \emph{interacting} diffusing particles: (i) a family of mortal diffusive lattice gases with constant diffusivity (as illustrated by the simple symmetric exclusion process with mortal particles), and (ii) random walkers that can annihilate in pairs. In both examples we calculate the variances of the (non-Poissonian) stationary distributions of $N$.