Some remarks on free energy and coarse-graining

TL;DR

This paper reviews and extends methods for coarse-graining thermodynamic and dynamical quantities in Langevin systems, focusing on reduced models for specific degrees of freedom, with applications to atomic chains and effective dynamics.

Contribution

It introduces new numerical approaches for coarse-graining in Langevin systems, extending previous results to better compute thermodynamic and dynamical averages.

Findings

Numerical methods for stress-strain relations in atomic chains.

Construction of effective dynamics for coarse-grained variables.

Extension of existing coarse-graining techniques.

Abstract

We present recent results on coarse-graining techniques for thermodynamic quantities (canonical averages) and dynamical quantities (averages of path functionals over solutions of overdamped Langevin equations). The question is how to obtain reduced models to compute such quantities, in the specific case when the functional to be averaged only depends on a few degrees of freedom. We mainly review, numerically illustrate and extend results from [3,18], concerning the computation of the stress-strain relation for one-dimensional chains of atoms, and the construction of an effective dynamics for a scalar coarse-grained variable when the complete system evolves according to the overdamped Langevin equation.