The Interplay between Memory and Potentials of Mean Force: A Discussion on the Structure of Equations of Motion for Coarse Grained Observables

TL;DR

This paper critically examines the derivation of the generalized Langevin equation for coarse-grained systems, highlighting limitations in common approximations and clarifying the conditions under which potentials of mean force and memory terms are valid.

Contribution

It provides a detailed analysis of the assumptions behind the generalized Langevin equation derivation and discusses the validity of using potentials of mean force in modeling coarse-grained dynamics.

Findings

Common simplified structures are not exact representations of microscopic dynamics.

The potential of mean force and linear memory terms are not always justifiable.

The fluctuation-dissipation relation may not hold in all cases.

Abstract

The underdamped, non-linear, generalized Langevin equation is widely used to model coarse-grained dynamics of soft and biological materials. By means of a projection operator formalism, we show under which approximations this equation can be obtained from the Hamiltonian dynamics of the underlying microscopic system and in which cases it makes sense to introduce a potential of mean force. We discuss shortcomings of previous derivations presented in the literature and demonstrate the implications of our derivation for the structure of memory terms and their connection to generalized fluctuation-dissipation relations. We show, in particular, that the widely used, simple structure which contains a potential of mean force, a memory term which is linear in the observable, and a fluctuating force which is related to the memory term by a fluctuation-dissipation relation, is neither exact nor can it, in general, be derived as a controlled approximation to the exact dynamics.