Macroscopic fluctuation theory of local time in lattice gases

Abstract

The local time in an ensemble of particles measures the amount of time the particles spend in the vicinity of a given point in space. Here we study fluctuations of the empirical time average $R=T^{-1}{\int }_0^T\rho \left(x=0,t\right)dt$ of the density $\rho \left(x=0,t\right)$ at the origin (so that $R$ is the local time spent at the origin, rescaled by $T$) for an initially uniform one-dimensional diffusive lattice gas. We consider both the quenched and annealed initial conditions and employ the Macroscopic Fluctuation Theory (MFT). For a gas of non-interacting random walkers (RWs) the MFT yields exact large-deviation functions of $R$, which are closely related to the ones recently obtained by Burenev \textit{et al.} (2023) using microscopic calculations for non-interacting Brownian particles. Our MFT calculations, however, additionally yield the most likely history of the gas density $\rho (x,t)$ conditioned on a given value of $R$. Furthermore, we calculate the variance of the local time fluctuations for arbitrary particle- or energy-conserving diffusive lattice gases. Better known examples of such systems include the simple symmetric exclusion process, the Kipnis-Marchioro-Presutti model and the symmetric zero-range process. Our results for the non-interacting RWs can be readily extended to a step-like initial condition for the density.

26 pages, 3 figures

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• Short-time large deviations of first-passage functionals for high-order stochastic processes