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

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).

18 pages in one-column format, 9 figures

References in corpus (8)

• Non equilibrium steady states: fluctuations and large deviations of the density and of the current
• Nonequilibrium Steady States of Matrix Product Form: A Solver's Guide
• The instanton method and its numerical implementation in fluid mechanics
• Large Deviations of Surface Height in the Kardar-Parisi-Zhang Equation
• Optimal fluctuation approach to a directed polymer in a random medium
• Survival of a static target in a gas of diffusing particles with exclusion
• Survival of interacting diffusing particles inside a domain with absorbing boundary
• Macroscopic fluctuation theory and first-passage properties of surface diffusion

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