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

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 -lnP(I) similar to IlnI, 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).