Time Reversal of the Overdamped Langevin Equation and Fixman's Law

TL;DR

This paper explores the time reversal properties of the overdamped Langevin equation, revealing a new relation called Fixman's Law, which impacts understanding of diffusion and hydrodynamic interactions in complex systems.

Contribution

It introduces a generalized Fixman Law for systems with memory effects and clarifies the causal structure of dissipation in Langevin dynamics.

Findings

Derived the time reversal form of the overdamped Langevin equation.

Quantitatively calibrated Kirkwood's approximation for polymer hydrodynamics.

Established the generalized Fixman Law for systems with memory kernels.

Abstract

We discuss how the first order Langevin equation for the overdamped dynamics of an interacting system has a natural time reversal of simple but surprising form, with consequences for correlation functions. This leads to the correlation of interactions as a strictly restraining term in the time-dependent diffusion tensor of the system, deriving the relation first suggested by Fixman. Applying this to the time-dependent diffusion of dilute polymer coils leads to the quantitative calibration of Kirkwood's approximation for their hydrodynamic radius. We find the generalized ``Fixman Law" for dissipation with a memory kernel, which has revealing causal structure, and we also discuss the case of the second order Langevin Equation.