Survival of interacting diffusing particles inside a domain with absorbing boundary

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 $-\ln \mathcal{P}\simeq D_{0}T{L}^{d-2}s(n_0)$, 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.

16 pages, 9 figures

References in corpus (4)

• Mapping out of equilibrium into equilibrium in one-dimensional transport models
• Mapping Nonequilibrium onto Equilibrium: The Macroscopic Fluctuations of Simple Transport Models
• Survival of a static target in a gas of diffusing particles with exclusion
• Large fluctuations in diffusion-controlled absorption

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