Stochastic PDE approach to fluctuating interfaces

TL;DR

This paper introduces a new class of stochastic partial differential equations motivated by microscopic particle systems, analyzing interface fluctuations and deriving a nonlinear SPDE to describe these phenomena.

Contribution

It proposes a novel SPDE framework for fluctuating interfaces, derived heuristically from particle systems with new scaling parameters and assumptions.

Findings

Gaussian fluctuation of the interface observed

Derived nonlinear SPDE for interface fluctuation

Analysis based on the Boltzmann-Gibbs principle

Abstract

We propose a new type of SPDEs, singular or with regularized noises, motivated by a study of the fluctuation of the density field in a microscopic interacting particle system. They include a large scaling parameter $N$, which is the ratio of macroscopic to microscopic size, and another scaling parameter $K=K(N)$, which controls the formation of the interface of size $K^{-1/2}$ in the density field. They are derived heuristically from the particle system, assuming the validity of the so-called ``Boltzmann-Gibbs principle", that is, a combination of the local ensemble average due to the local ergodicity and its asymptotic expansion. We study a simple situation where the interface is flat and immobile. Under making a proper stretch to the normal direction to the interface, we observe a Gaussian fluctuation of the interface. We also heuristically derive a nonlinear SPDE which describes the fluctuation of the interface.