Full Statistics of Nonstationary Heat Transfer in the Kipnis-Marchioro-Presutti Model

TL;DR

This paper derives the full probability distribution of non-stationary heat transfer in the KMP model at long times, revealing scaling properties and providing exact solutions using integrability and inverse scattering methods.

Contribution

It provides the first exact solution for the full distribution of heat transfer in the non-stationary KMP model, utilizing integrability and inverse scattering techniques.

Findings

Distribution exhibits long-time scaling similar to single-file diffusion

Exact solution obtained via inverse scattering method

Asymptotic behaviors derived from exact and perturbative methods

Abstract

We investigate non-stationary heat transfer in the Kipnis-Marchioro-Presutti (KMP) lattice gas model at long times in one dimension when starting from a localized heat distribution. At large scales this initial condition can be described as a delta-function, $u(x,t=0)=W \delta(x)$. We characterize the process by the heat, transferred to the right of a specified point $x=X$ by time $T$, $$ J=\int_X^\infty u(x,t=T)\,dx\,, $$ and study the full probability distribution $\mathcal{P}(J,X,T)$. The particular case of $X=0$ has been recently solved [Bettelheim \textit{et al}. Phys. Rev. Lett. \textbf{128}, 130602 (2022)]. At fixed $J$, the distribution $\mathcal{P}$ as a function of $X$ and $T$ has the same long-time dynamical scaling properties as the position of a tracer in a single-file diffusion. Here we evaluate $\mathcal{P}(J,X,T)$ by exploiting the recently uncovered complete integrability of the equations of the macroscopic fluctuation theory (MFT) for the KMP model and using the Zakharov-Shabat inverse scattering method. We also discuss asymptotics of $\mathcal{P}(J,X,T)$ which we extract from the exact solution, and also obtain by applying two different perturbation methods directly to the MFT equations.