## 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}∫_{0}^{T}ρx=0,tdt of the density ρx=0,t 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 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 ρ(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.

Original language | American English |
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Article number | 129616 |

Journal | Physica A: Statistical Mechanics and its Applications |

Volume | 639 |

DOIs | |

State | Published - 1 Apr 2024 |

### Bibliographical note

Publisher Copyright:© 2024 Elsevier B.V.

## Keywords

- Large deviation theory
- Lattice gases
- Macroscopic fluctuation theory
- Stochastic processes