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

TL;DR

This work addresses the full nonstationary heat-transfer statistics in the Kipnis–Marchioro–Presutti (KMP) lattice gas by combining macroscopic fluctuation theory (MFT) with the Zakharov–Shabat inverse scattering method (ISM) for a derivative nonlinear Schrödinger (DNLS)–type system. By recasting the MFT equations into an integrable DNLS framework, the authors derive a parametric representation of the rate function $s(j)$ through the scattering data, yielding explicit expressions for $j(\lambda)$ and $s(\lambda)$ and revealing a nontrivial time-reversal symmetry. They obtain small- and large-$j$ asymptotics, and provide numerical optimal paths $u(x,t)$, validated by Monte Carlo simulations of the microscopic KMP model. This constitutes the first exact non-steady-state large-deviation result for current statistics in a lattice gas with quenched initial conditions and opens avenues for applying ISM to other fluctuating quantities and models such as SSEP. The work highlights the power of integrating OFM/MFT with integrable systems to solve challenging non-equilibrium fluctuation problems.

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 Schrödinger 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.

Figures (5)

• Figure 1: Monte-Carlo simulation of the KMP model with $W=1$. Plotted is the simulated temperature profile $u$ as a function of $x$ at time $t=1.5 \times {10}^4$ (bars), its spatial average over each 50 consecutive lattice sites (solid line) and the theoretical Gaussian profile (\ref{['meanfield']}) (dashed line).
• Figure 2: Analytical results for $Q_+(k)$, described by Eqs. (\ref{['basically the result']}), (\ref{['Phi']}) and (\ref{['vpmfinal']}) (solid lines), versus numerical results (dashed lines) for $\lambda=1$, or $j=0.09568\dots$. The symmetric and antisymmetric curves show $\text{Re}\,Q_+(k)$ and $\text{Im}\,Q_+(k)$, respectively.
• Figure 3: The exact rate function $s(j)$, given by Eqs. (\ref{['jlambda']}) and \ref{['slambdasimpler']} (solid line) and two asymptotes: $s(|j|\ll 1)=\sqrt{8\pi} j^2$ and Eq. (\ref{['slargej']}) (dashed lines). Symbols: properly rescaled data from $10^6$ direct Monte-Carlo simulations of the microscopic KMP model for $T=10^2$, see SM for details.
• Figure 4: Exact $\Phi(\lambda,k)$, given by Eq. (\ref{['b3']}) (black solid line), and the asymptotes $\Phi(\lambda,k) \simeq -1/(2\pi)$ and Eq. (\ref{['b4']}) (blue and magenta dashed lines, respectively), for $\lambda=10^6$. The region of $k\lesssim 1/\lambda$ cannot be seen on this scale, and its contribution to the integral is negligible.
• Figure 5: The optimal temperature profile $u(x,t)$ for $\lambda=10$ (corresponding to $j \simeq 0.38$) at times $1/4$, $1/2$, $3/4$ and $1$. Noticeable is a shock-like singularity of $u$ at $x=0$ and $t=1$.