Asymptotic Analysis of the Green-Kubo Formula

TL;DR

This paper provides an in-depth asymptotic analysis of the Green-Kubo formula for self-diffusion, including alternative representations, effects of irreversibility, and various limits, supported by examples.

Contribution

It introduces a new Poisson equation-based representation and analyzes the impact of microscopic irreversibility on diffusion coefficients.

Findings

Derived an alternative Green-Kubo representation using Poisson equations.

Established a Stieltjes integral formula for diffusion tensor components.

Analyzed asymptotic limits and effects of irreversible dynamics on diffusion.

Abstract

A detailed study of various distinguished limits of the Green-Kubo formula for the self-diffusion coefficient is presented in this paper. First, an alternative representation of the Green-Kubo formula in terms of the solution of a Poisson equation is derived when the microscopic dynamics is Markovian. Then, the techniques developed in \cite{golden2, AvelMajda91} are used to obtain a Stieltjes integral representation formula for the symmetric and antisymmetric parts of the diffusion tensor. The effect of irreversible microscopic dynamics on the diffusion coefficient is analyzed and various asymptotic limits of physical interest are studied. Several examples are presented that confirm the findings of our theory.