Full statistics of regularized local energy density in a freely expanding Kipnis-Marchioro-Presutti gas

We combine the Macroscopic Fluctuation Theory and the Inverse Scattering Method to determine the full long-time statistics of the energy density u ( x , t ) averaged over a given spatial interval, U = 1 2 L ∫ − L L d x u ( x , t ) , in a freely expanding Kipnis-Marchioro-Presutti (KMP) lattice gas on the line, following the release at t = 0 of a finite amount of energy at the origin. In particular, we show that, as time t goes to infinity at fixed L, the large deviation function of U approaches a universal, L-independent form when expressed in terms of the energy content of the interval | x | < L . A key part of the solution is the determination of the most likely configuration of the energy density at time t, conditional on U.