The Smoluchowski-Kramers approximation for a system with arbitrary friction depending on both state and distribution

TL;DR

This paper investigates the Smoluchowski-Kramers approximation for stochastic systems with complex friction depending on state and distribution, deriving the limiting dynamics, convergence rate, and extensions to more general interactions.

Contribution

It provides a detailed derivation of the limiting equation with additional drift terms expressed via Lyapunov and Sylvester equations, extending the approximation to broader systems.

Findings

Derived the limiting equation in the small mass limit.

Expressed additional drift terms using Lyapunov and Sylvester equations.

Established the rate of convergence for the approximation.

Abstract

A system of stochastic differential equations describing diffusive phenomena, which has arbitrary friction depending on both state and distribution is investigated. The Smoluchowski-Kramers approximation is seen to describe dynamics in the small mass limit. We obtain the limiting equation and, in particular, the addition drift terms that appear in the limiting equation are expressed in terms of the solutions to the Lyapunov matrix equation and Sylvester matrix equation. Furthermore, we provide the rate of convergence and extend the system to encompass more general interactions and noise.