Whitham Approach to Certain Large Fluctuation Problems in Statistical Mechanics

TL;DR

This paper explores the connection between the Whitham limit of macroscopic fluctuation theory and the inverse scattering method, enabling exact solutions to certain large deviation problems in statistical mechanics.

Contribution

It establishes a link between the strongly non-linear limit of fluctuation theory and inverse scattering, facilitating exact solutions via Riemann-Hilbert problem techniques.

Findings

The inverse scattering problem can be solved using steepest descent in the Whitham limit.

The equations simplify to allow exact solutions in the strongly non-linear regime.

This approach aids in solving large deviation problems precisely.

Abstract

We show the relationship between the strongly non-linear limit (also termed the dispersionless or the Whitham limit) of the macroscopic fluctuation theory of certain statistical models and the inverse scattering method. We show that in the strongly non-linear limit the inverse scattering problem can be solved using the steepest descent method of the associated Riemann--Hilbert problem. The importance of establishing this connection, is that the equations in the strongly non-linear limit can often be solved exactly by simple means, the connection then provides a limit in which one can solve the inverse scattering problem, thus aiding potentially the exact solution of a particular large deviation problem.