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 $R= T^{-1}\int_{0}^{T}ρ\left(x=0,t\right)\,dt$ of the density $ρ\left(x=0,t\right)$ 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 $ρ(x,t)$ 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.

Figures (3)

• Figure 1: Solid lines: $s(R)/n_0$ vs. $R/n_0$ for the annealed (a) and quenched (b) initial conditions. The dashed and dotted lines in the main plots and the dashed lines in the insets correspond to the $R \ll n_0$, $R \simeq n_0$ and $R \gg n_0$ asymptotic behaviors, respectively, see Eqs. \ref{['sOfRApprox']} and \ref{['sOfRApproxQuenched']}.
• Figure 2: The optimal density history of the noninteracting RWs, for the annealed initial condition, as a function of $x$ at times $t=0, 0.1$ and $0.5$ (solid, dashed and dotted lines, respectively) for $\Lambda = 1$ (a) and $\Lambda = -1$ (b). At times $1/2 \le t \le 1$, $q(x,t)$ is given by the time-reversal symmetry relation $q(x,t) = q(x,1-t)$. Insets: The optimal densities at the origin, $q_0(t) = q(x=0,t)$ (solid lines) and their time average values $R$ (dashed lines).
• Figure 3: The optimal density history of the noninteracting RWs, for the quenched initial condition, as a function of $x$ at times $t=0, 0.25$, $0.5$ and $1$ (solid, dashed, dotted, and dash-dotted lines, respectively) for $\Lambda = 1$ (a) and $\Lambda = -1$ (b). Insets: The optimal densities at the origin, $q_0(t) = q(x=0,t)$ (solid lines) and their time average values $R$ (dashed lines).