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, $$U=\frac{1}{2L}{\int }_{-L}^{L}dxu(x,t),$$ 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 $\mathrm{\mid }x\mathrm{\mid }<L$. A key part of the solution is the determination of the most likely configuration of the energy density at time $t$, conditional on $U$.