Large Deviations for the density and the current in Non-Equilibrium-Steady-States 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.

v2=revised version (14 pages)

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