Fluctuations around equilibrium laws in ergodic continuous-time random walks

TL;DR

This paper investigates the large fluctuations in occupation times of ergodic continuous-time random walks near the non-ergodic transition, revealing dual scaling laws and their physical interpretation.

Contribution

It introduces a comprehensive analysis of fluctuation laws, combining infinite and stable densities, and clarifies their physical significance in ergodic and near-critical regimes.

Findings

Large fluctuations exhibit dual time scaling and distribution laws.

Infinite density and Lévy-stable density complement each other and have physical interpretations.

Canonical equilibrium laws govern both mean occupation times and fluctuation distributions.

Abstract

We study occupation time statistics in ergodic continuous-time random walks. Under thermal detailed balance conditions, the average occupation time is given by the Boltzmann-Gibbs canonical law. But close to the non-ergodic phase, the finite-time fluctuations around this mean are large and nontrivial. They exhibit dual time scaling and distribution laws: the infinite density of large fluctuations complements the L\'evy-stable density of bulk fluctuations. Neither of the two should be interpreted as a stand-alone limiting law, as each has its own deficiency: the infinite density has an infinite norm (despite particle conservation), while the stable distribution has an infinite variance (although occupation times are bounded). These unphysical divergences are remedied by consistent use and interpretation of both formulas. Interestingly, while the system's canonical equilibrium laws naturally determine the mean occupation time of the ergodic motion, they also control the infinite and L\'evy-stable densities of fluctuations. The duality of stable and infinite densities is in fact ubiquitous for these dynamics, as it concerns the time averages of general physical observables.