Exact and asymtotic formulas for overdamped Brownian dynamics

TL;DR

This paper derives exact and asymptotic formulas for dynamical correlations in overdamped Brownian motion, generalizing known sum rules and relating static correlations, with tests on specific models.

Contribution

It introduces new formulas connecting dynamical and static correlations in overdamped Brownian dynamics, including a generalized sum rule and small-$k$ asymptotics.

Findings

Derived a generalized $f$-sum rule for overdamped Brownian motion.

Established a relation between static current-current and density-density correlations.

Provided asymptotic behavior of the dynamical structure factor at small wave numbers.

Abstract

Exact and asymptotic formulas relating to dynamical correlations for overdamped Brownian motion are obtained. These formulas include a generalization of the $f$-sum rule from the theory of quantum fluids, a formula relating the static current-current correlation to the static density-density correlation, and an asymptotic formula for the small-$k$ behaviour of the dynamical structure factor. Known exact evaluations of the dynamical density-density correlation in some special models are used to illustrate and test the formulas.