Fractional Macroscopic Fluctuation Theory for a Superdiffusive Ginzburg-Landau dynamics

TL;DR

This paper studies a boundary-driven Ginzburg-Landau model with long-range interactions, deriving a fractional heat equation for macroscopic behavior and establishing large deviations principles for the system's fluctuations.

Contribution

It introduces a fractional macroscopic fluctuation framework for superdiffusive Ginzburg-Landau dynamics with explicit large deviations results.

Findings

Macroscopic evolution governed by fractional heat equation

Stationary profile characterized by fractional Laplace equation

Explicit semi-closed form for stationary large deviations rate function

Abstract

We investigate a boundary-driven Ginzburg-Landau dynamics with long-range interactions. In the hydrodynamic limit, the macroscopic evolution is governed by a fractional heat equation with Dirichlet boundary conditions, while the corresponding stationary profile is characterized by a fractional Laplace equation. We establish a dynamical large deviations principle for the empirical measure and derive the associated stationary large deviations principle for the non-equilibrium steady state, which can be computed semi-explicitly. We further show that the stationary rate function coincides with the quasi-potential associated with the dynamical large deviations functional.