Large Deviations for the density and the current in Non-Equilibrium-Steady-States on disordered rings

TL;DR

This paper applies the Level 2.5 large deviations framework to disordered rings with independent particles, analyzing the density and current fluctuations explicitly for directed and non-directed models, revealing detailed large deviation properties.

Contribution

It provides explicit large deviations analysis for density and current in disordered ring models, including the directed trap model and non-directed models, using the Level 2.5 framework.

Findings

Explicit contraction formulas for the directed trap model.

Analysis of tail behavior of current large deviations.

Comparison with Fokker-Planck dynamics on disordered rings.

Abstract

The so-called 'Level 2.5' general result for the large deviations of the joint probability of the density and of the currents for Markov Jump processes is applied to the case of $N$ independent particles on a ring with random transition rates. We first focus on the Directed Trap model, where the contractions needed to obtain the large deviations properties of the density alone and of the current alone can be explicitly written in each disordered sample, and where the deformed Markov operator needed to evaluate the generating function of the current can be also explicitly analyzed via its highest eigenvalue and the corresponding left and right eigenvectors. We then turn to the non-directed model, where the tails for large currents $ j \to \pm \infty$ of the rate function for the current alone can still be studied explicitly, either via contraction or via the deformed Markov operator method. We mention the differences with the large deviations properties of the Fokker-Planck dynamics on disordered rings.