Reduced Markovian Descriptions of Brownian Dynamics: Toward an Exact Theory

TL;DR

This paper develops a reduction scheme for Brownian dynamics that improves the accuracy of the Smoluchowski equation in the overdamped regime, using the Dynamic Invariance principle and ensuring consistency with the Fluctuation-Dissipation theorem.

Contribution

It introduces a novel reduction method for Brownian dynamics that provides meaningful corrections to existing models, with explicit calculations and error quantification.

Findings

Explicit reduction error representation

Mobility coefficient derived via Dynamic Invariance

Diffusion coefficient consistent with Fluctuation-Dissipation

Abstract

We outline a reduction scheme for a class of Brownian dynamics which leads to meaningful corrections to the Smoluchowski equation in the overdamped regime. The mobility coefficient of the reduced dynamics is obtained by exploiting the Dynamic Invariance principle, whereas the diffusion coefficient fulfils the Fluctuation-Dissipation theorem. Explicit calculations are carried out in the case of a harmonically bound particle. A quantitative pointwise representation of the reduction error is also provided and connections to both the Maximum Entropy method and the linear response theory are highlighted. Our study paves the way to the development of reduction procedures applicable to a wider class of diffusion processes.