Full absorption statistics of diffusing particles with exclusion

TL;DR

This paper analyzes the probability distribution of particles absorbed by a spherical absorber in a lattice gas with exclusion, revealing non-Poissonian behavior and non-monotonic density profiles at large absorption numbers.

Contribution

It applies macroscopic fluctuation theory to derive the full absorption statistics for a lattice gas with exclusion, extending understanding beyond the Poisson approximation.

Findings

For small densities, absorption follows a Poisson distribution.

At large N, the probability decays as exp(-n_0 N^2 /  N).

Conditional density profiles can be non-monotonic for large N and low densities.

Abstract

Suppose that an infinite lattice gas of constant density $n_0$, whose dynamics are described by the symmetric simple exclusion process, is brought in contact with a spherical absorber of radius $R$. Employing the macroscopic fluctuation theory and assuming the additivity principle, we evaluate the probability distribution ${\mathcal P}(N)$ that $N$ particles are absorbed during a long time $T$. The limit of $N=0$ corresponds to the survival problem, whereas $N\gg \bar{N}$ describes the opposite extreme. Here $\bar{N}=4\pi R D_0 n_0 T$ is the \emph{average} number of absorbed particles (in three dimensions), and $D_0$ is the gas diffusivity. For $n_0\ll 1$ the exclusion effects are negligible, and ${\mathcal P}(N)$ can be approximated, for not too large $N$, by the Poisson distribution with mean $\bar{N}$. For finite $n_0$, ${\mathcal P}(N)$ is non-Poissonian. We show that $-\ln{\mathcal P}(N) \simeq n_0 N^2/\bar{N}$ at $N\gg \bar{N}$. At sufficiently large $N$ and $n_0<1/2$ the most likely density profile of the gas, conditional on the absorption of $N$ particles, is non-monotonic in space. We also establish a close connection between this problem and that of statistics of current in finite open systems.