Full Statistics of Regularized Local Energy Density in a Freely Expanding Kipnis-Marchioro-Presutti Gas

TL;DR

This paper combines advanced theoretical methods to analyze the long-time statistical behavior of energy density in a freely expanding KMP lattice gas, revealing universal large deviation properties independent of interval size.

Contribution

It introduces a novel combination of Macroscopic Fluctuation Theory and Inverse Scattering Method to derive the full statistics of energy density in an expanding lattice gas.

Findings

Large deviation function becomes universal at long times.

Derived the most likely energy density configuration conditioned on total energy.

Established the asymptotic form of energy fluctuations in the system.

Abstract

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, $$U =\frac{1}{2L}\int_{-L}^{L}dx\, u(x,t),$$ 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 $U$.