Complete Integrability of the Problem of Full Statistics of Nonstationary Mass Transfer in the Simple Inclusion Process

TL;DR

This work determines the exact large-deviation statistics for nonstationary mass transfer in the symmetric inclusion process (SIP) by combining macroscopic fluctuation theory with the inverse scattering method. The authors map the MFT equations to a derivative nonlinear Schrödinger equation via a Hopf-Cole transform and solve the resulting system using a Kaup–Newell–type ISM, obtaining the rate function $s(\kappa,n)$ for $\kappa=K/N$ and $n=N/\sqrt{T}$. The exact solution reveals that SIP interpolates between independent random walkers and the KMP model as $n$ varies, with a finite rate function at the edges $\kappa=\pm 1/2$, contrasting with KMP. The results highlight the power of integrability in OFM problems and provide a framework for exact nonstationary large-deviation statistics in interacting particle systems.

Abstract

The Simple Inclusion Process (SIP) interpolates between two well-known lattice gas models: the independent random walkers and the Kipnis-Marchioro-Presutti model. Here we study large deviations of nonstationary mass transfer in the SIP at long times in one dimension. We suppose that $N\gg 1$ particles start from a single lattice site at the origin, and we are interested in the probability $\mathcal{P}(M,N,T)$ of observing $M$ particles, $0\leq M\leq N$, to the right of the origin at a specified time $T\gg 1$. At large times, the corresponding probability distribution has a large-deviation behavior, $-\ln \mathcal{P}(M,N,T) \simeq \sqrt{T} s(M/N,N/\sqrt{T})$. We determine the rate function $s$ exactly by uncovering and utilizing complete integrability, by the inverse scattering method, of the underlying equations of the macroscopic fluctuation theory. We also analyze different asymptotic limits of the rate function $s$.

Figures (7)

• Figure 1: The Lagrange multiplier $\lambda=\ln(1+\Lambda)$ vs. the relative excess of transferred mass $\kappa=K/N$, found by numerically solving the transcendental equation (\ref{['eq2a']}) for $n=N/\sqrt{T}=1$.
• Figure 2: The rate function $s(\kappa,n)$ vs. the relative excess of transferred mass $\kappa$ for $n=0.1$, $1$ and $10$.
• Figure 3: The rate function $s(\kappa,n)$ vs. the relative excess of transferred mass $\kappa$ for $n=0.1$. Also shown, for the same $n=0.1$, the $\kappa\ll 1$ asymptotic and the rate function for the RWs.
• Figure 4: The rate function $s(\kappa,n)$ vs. relative excess of transferred mass $\kappa$ for $n=50$. Also shown, for the same $n=50$, the $\kappa\ll 1$ asymptotic and the rate function for the KMP model BSM2022a.
• Figure 5: The rate function $s(\kappa,n)$ vs. $\kappa$ for $n=10$. Also shown, for the same $n=10$, the $\kappa\ll 1$ asymptotic and the rate function for the KMP model BSM2022a.