Statistics of large currents in the Kipnis-Marchioro-Presutti model in a ring geometry

TL;DR

This paper analytically investigates the large current fluctuations in the KMP model on a ring, revealing a traveling wave profile for supercritical currents and deriving the probability distribution, with results confirmed by simulations.

Contribution

It introduces an analytical solution for the optimal density profile as a traveling wave in the KMP model for supercritical currents, extending understanding of current fluctuations.

Findings

Traveling wave solution for supercritical currents in the KMP model.

Asymptotic behavior of current distribution near and far from critical current.

Good agreement between analytical results and simulations.

Abstract

We use the macroscopic fluctuation theory to determine the statistics of large currents in the Kipnis-Marchioro-Presutti (KMP) model in a ring geometry. About 10 years ago this simple setting was instrumental in identifying a breakdown of the additivity principle in a class of lattice gases at currents exceeding a critical value. Building on earlier work, we assume that, for supercritical currents, the optimal density profile, conditioned on the given current, has the form of a traveling wave (TW). For the KMP model we find this TW analytically, in terms of elliptic functions, for any supercritical current $I$. Using this TW solution, we evaluate, up to a pre-exponential factor, the probability distribution $P(I)$. We obtain simple asymptotics of the TW and of $P(I)$ for currents close to the critical current, and for currents much larger than the critical current. In the latter case we show that $-\ln P (I) \sim I\ln I$, whereas the optimal density profile acquires a soliton-like shape. Our analytic results are in a very good agreement with Monte-Carlo simulations and numerical solutions of Hurtado and Garrido (2011).