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.