Survival of interacting diffusing particles inside a domain with absorbing boundary

TL;DR

This paper uses macroscopic fluctuation theory to analyze the probability that no particles are absorbed in a diffusive gas within a domain over a long time, revealing stationary density profiles and deriving explicit formulas in various dimensions.

Contribution

It provides a theoretical framework for calculating the survival probability of interacting diffusive particles with absorbing boundaries in any dimension, including explicit solutions for the SSEP model.

Findings

The survival probability decays exponentially with time for a broad class of gases.

Explicit stationary density profiles are derived for 1D and certain higher-dimensional domains.

Analytical expressions for the action function s(n_0) are obtained near close packing for any shape and dimension.

Abstract

Suppose that a $d$-dimensional domain is filled with a gas of (in general, interacting) diffusive particles with density $n_0$. A particle is absorbed whenever it reaches the domain boundary. Employing macroscopic fluctuation theory, we evaluate the probability ${\mathcal P}$ that no particles are absorbed during a long time $T$. We argue that the most likely gas density profile, conditional on this event, is stationary throughout most of the time $T$. As a result, ${\mathcal P}$ decays exponentially with $T$ for a whole class of interacting diffusive gases in any dimension. For $d=1$ the stationary gas density profile and ${\mathcal P}$ can be found analytically. In higher dimensions we focus on the simple symmetric exclusion process (SSEP) and show that $-\ln {\mathcal P}\simeq D_0TL^{d-2} \,s(n_0)$, where $D_0$ is the gas diffusivity, and $L$ is the linear size of the system. We calculate the rescaled action $s(n_0)$ for $d=1$, for rectangular domains in $d=2$, and for spherical domains. Near close packing of the SSEP $s(n_0)$ can be found analytically for domains of any shape and in any dimension.