Dynamical properties of coarse-grained linear SDEs

TL;DR

This paper investigates the dynamical properties of coarse-grained linear stochastic differential equations, revealing limitations of standard models in capturing dynamics and proposing simple augmentations for improvement.

Contribution

It provides a detailed analysis of coarse-graining approaches in linear SDEs, highlighting their static accuracy and dynamical limitations, and suggests modifications to improve dynamical predictions.

Findings

Standard coarse-graining captures static statistics well.

Dynamical statistics like mean-squared displacement are systematically biased.

Simple model augmentations can improve dynamical accuracy.

Abstract

Coarse-graining or model reduction is a term describing a range of approaches used to extend the time-scale of molecular simulations by reducing the number of degrees of freedom. In the context of molecular simulation, standard coarse-graining approaches approximate the potential of mean force and use this to drive an effective Markovian model. To gain insight into this process, the simple case of a quadratic energy is studied in an overdamped setting. A hierarchy of reduced models is derived and analysed, and the merits of these different coarse-graining approaches are discussed. In particular, while standard recipes for model reduction accurately capture static equilibrium statistics, it is shown that dynamical statistics such as the mean-squared displacement display systematic error, even when a system exhibits large time-scale separation. In the linear setting studied, it is demonstrated both analytically and numerically that such models can be augmented in a simple way to better capture dynamical statistics.