Entropy Anomaly in Langevin-Kramers Dynamics with a Temperature Gradient, Matrix Drag, and Magnetic Field

TL;DR

This paper rigorously analyzes entropy production in Langevin-Kramers systems with temperature gradients, magnetic fields, and anisotropic coefficients, revealing an explicit formula for entropy anomalies in the overdamped limit.

Contribution

It extends previous theories by deriving a comprehensive formula for entropy production in complex Langevin systems with magnetic and anisotropic effects.

Findings

Derived an explicit formula for anomalous entropy production.

Developed a homogenization theory for integral processes involving position and velocity.

Provided bounds on the convergence rate of entropy in the overdamped limit.

Abstract

We investigate entropy production in the small-mass (or overdamped) limit of Langevin-Kramers dynamics. The results generalize previous works to provide a rigorous derivation that covers systems with magnetic field as well as anisotropic (i.e. matrix-valued) drag and diffusion coefficients that satisfy a fluctuation-dissipation relation with state-dependent temperature. In particular, we derive an explicit formula for the anomalous entropy production which can be estimated from simulated paths of the overdamped system. As a part of this work, we develop a theory for homogenizing a class of integral processes involving the position and scaled-velocity variables. This allows us to rigorously identify the limit of the entropy produced in the environment, including a bound on the convergence rate.