From Generalized Langevin Equations to Brownian Dynamics and Embedded Brownian Dynamics

TL;DR

This paper derives simplified stochastic models from generalized Langevin equations, explores their accuracy, and introduces embedded Brownian dynamics to better approximate complex nonlocal systems while preserving fluctuation-dissipation relations.

Contribution

It presents a novel reduction of generalized Langevin equations to coordinate-only models and introduces embedded Brownian dynamics that maintain fluctuation-dissipation properties.

Findings

The accuracy of Brownian dynamics depends on the kernel's intrinsic frequency.

Embedding into extended dynamics improves approximation when standard Brownian dynamics is insufficient.

The second fluctuation-dissipation theorem is exactly satisfied in the embedded models.

Abstract

We present the reduction of generalized Langevin equations to a coordinate-only stochastic model, which in its exact form, involves a forcing term with memory and a general Gaussian noise. It will be shown that a similar fluctuation-dissipation theorem still holds at this level. We study the approximation by the typical Brownian dynamics as a first approximation. Our numerical test indicates how the intrinsic frequency of the kernel function influences the accuracy of this approximation. In the case when such an approximate is inadequate, further approximations can be derived by embedding the nonlocal model into an extended dynamics without memory. By imposing noises in the auxiliary variables, we show how the second fluctuation-dissipation theorem is still exactly satisfied.