Reduced Markovian Descriptions of Brownian Dynamics: Toward an Exact Theory

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.

Automated summary

This paper presents a reduction scheme for overdamped Brownian dynamics and proposes meaningful corrections to the Smoluchowski equation. The mobility coefficient is obtained using the Dynamic Invariance principle, while the diffusion coefficient satisfies the Fluctuation-Dissipation theorem. The study also provides a quantitative representation of the reduction error and highlights connections to the Maximum Entropy method and linear response theory.