Inverse Scattering Method Solves the Problem of Full Statistics of Nonstationary Heat Transfer in the Kipnis-Marchioro-Presutti Model

TL;DR

This paper applies the inverse scattering method to solve the integrable equations of macroscopic fluctuation theory, deriving exact large deviation functions for nonstationary heat transfer in the KMP model, revealing new symmetry properties.

Contribution

It introduces an exact solution approach for the full statistics of heat transfer in the KMP model using inverse scattering, connecting macroscopic fluctuation theory with integrable systems.

Findings

Explicit large deviation function for heat transfer derived

Uncovered a nontrivial symmetry relation in the statistics

Solved equations related to the derivative nonlinear Schrödinger equation

Abstract

We determine the full statistics of nonstationary heat transfer in the Kipnis-Marchioro-Presutti lattice gas model at long times by uncovering and exploiting complete integrability of the underlying equations of the macroscopic fluctuation theory. These equations are closely related to the derivative nonlinear Schr\"{o}dinger equation (DNLS), and we solve them by the Zakharov-Shabat inverse scattering method (ISM) adapted by Kaup and Newell (1978) for the DNLS. We obtain explicit results for the exact large deviation function of the transferred heat for an initially localized heat pulse, where we uncover a nontrivial symmetry relation.