Smoluchowski-Kramers approximation in the case of variable friction

TL;DR

This paper investigates the small mass limit of Langevin equations with variable friction, proposing a modified approximation approach due to the non-existence of classical limits, and explores applications with complex coefficients.

Contribution

It introduces a modified Smoluchowski-Kramers approximation for variable friction cases where classical limits fail, and applies it to problems with oscillating or discontinuous coefficients.

Findings

Modified approximation successfully captures the small mass limit.

Application to systems with fast oscillations demonstrates effectiveness.

Provides theoretical foundation for complex coefficient scenarios.

Abstract

We consider the small mass asymptotics (Smoluchowski-Kramers approximation) for the Langevin equation with a variable friction coefficient. The limit of the solution in the classical sense does not exist in this case. We study a modification of the Smoluchowski-Kramers approximation. Some applications of the Smoluchowski-Kramers approximation to problems with fast oscillating or discontinuous coefficients are considered.