Position-dependent memory kernel in generalized Langevin equations: theory and numerical estimation

TL;DR

This paper rigorously derives position-dependent memory kernels in generalized Langevin equations from the Mori-Zwanzig formalism, providing a theoretical foundation and numerical methods for coarse-grained molecular dynamics modeling.

Contribution

It introduces a formal derivation of position-dependent memory kernels and a numerical approach for their estimation from all-atom simulations.

Findings

Derived Volterra equations for the memory kernel

Established fluctuation-dissipation relation for the model

Provided a numerical scheme for kernel estimation

Abstract

Generalized Langevin equations with non-linear forces and position-dependent linear friction memory kernels, such as commonly used to describe the effective dynamics of coarse-grained variables in molecular dynamics, are rigorously derived within the Mori-Zwanzig formalism. A fluctuation-dissipation theorem relating the properties of the noise to the memory kernel is shown. The derivation also yields Volterra-type equations for the kernel, which can be used for a numerical parametrization of the model from all-atom simulations.