The crossover from the Macroscopic Fluctuation Theory to the Kardar-Parisi-Zhang equation controls the large deviations beyond Einstein's diffusion

TL;DR

This paper explores the transition between macroscopic fluctuation theory and the KPZ equation in 1D diffusive systems, revealing how large deviations and extremal diffusion behaviors evolve beyond classical Einstein diffusion.

Contribution

It introduces a nonlinear system interpolating between MFT and KPZ, solves it with inverse scattering, and characterizes the large deviation rate function for atypical diffusion in random fields.

Findings

Derived the crossover rate function for large deviations.

Revealed the interpolation between MFT and KPZ limits.

Connected the results to Fredholm determinant asymptotics.

Abstract

We study the crossover from the macroscopic fluctuation theory (MFT) which describes 1D stochastic diffusive systems at late times, to the weak noise theory (WNT) which describes the Kardar-Parisi-Zhang (KPZ) equation at early times. We focus on the example of the diffusion in a time-dependent random field, observed in an atypical direction which induces an asymmetry. The crossover is described by a non-linear system which interpolates between the derivative and the standard non-linear Schrodinger equations in imaginary time. We solve this system using the inverse scattering method for mixed-time boundary conditions introduced by us to solve the WNT. We obtain the rate function which describes the large deviations of the sample-to-sample fluctuations of the cumulative distribution of the tracer position. It exhibits a crossover as the asymmetry is varied, recovering both MFT and KPZ limits. We sketch how it is consistent with extracting the asymptotics of a Fredholm determinant formula, recently derived for sticky Brownian motions. The crossover mechanism studied here should generalize to a larger class of models described by the MFT. Our results apply to study extremal diffusion beyond Einstein's theory.