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

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 $N$.

11 pages in one-column format, 1 figure, revised and extended version

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• Optimal paths of non-equilibrium stochastic fields: the Kardar-Parisi-Zhang interface as a test case
• First-passage properties of mortal random walks: ballistic behavior, effective reduction of dimensionality, and scaling functions for hierarchical graphs
• Mortal Brownian motion: three short stories
• Persistent fluctuations of the swarm size of Brownian bees
• Competition in a system of Brownian particles: Encouraging achievers
• Probabilities of moderately atypical fluctuations of the size of a swarm of Brownian Bees
• Negative large deviations of the front velocity of $N$-particle branching Brownian motion