Full absorption statistics of diffusing particles with exclusion

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.

17 one-column pages, 8 figures, slightly revised version

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• Non equilibrium steady states: fluctuations and large deviations of the density and of the current
• Macroscopic fluctuation theory
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• Steady state statistics of driven diffusions
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• Large fluctuations in diffusion-controlled absorption

Cited by in corpus (5)

• Entropy production and its large deviations in an active lattice gas
• Survival of interacting diffusing particles inside a domain with absorbing boundary
• Fluctuations of a swarm of Brownian bees
• Fluctuations of absorption of interacting diffusing particles by multiple absorbers
• Persistent fluctuations of the swarm size of Brownian bees