Systematic derivation of hybrid coarse-grained models

TL;DR

This paper systematically derives hybrid coarse-grained models in molecular dynamics using the Mori-Zwanzig projection, addressing key issues like atom-bead coexistence, mapping choices, and effective potential approximation, supported by numerical simulations.

Contribution

It extends the Mori-Zwanzig approach to hybrid multi-scale systems, providing detailed derivations and analysis of approximation errors and dissipative effects.

Findings

Non-constant dissipative terms are essential for accurate macroscopic modeling.

The approach reduces the need for ad-hoc calibration of potentials and thermostats.

Numerical simulations validate the theoretical derivations and highlight key model features.

Abstract

Significant efforts have been devoted in the last decade towards improving the predictivity of coarse-grained models in molecular dynamics simulations and providing a rigorous justification of their use, through a combination of theoretical studies and data-driven approaches. One of the most promising research effort is the (re-)discovery of the Mori-Zwanzig projection as a generic, yet systematic, theoretical tool for deriving coarse-grained models. Despite its clean mathematical formulation and generality, there are still many open questions about its applicability and assumptions. In this work, we propose a detailed derivation of a hybrid multi-scale system, generalising and further investigating the approach developed in [Espa\~{n}ol, P., EPL, 88, 40008 (2009)]. Issues such as the general co-existence of atoms (fully-resolved degrees of freedom) and beads (larger coarse-grained units), the role of the fine-to-coarse mapping chosen, and the approximation of effective potentials are discussed. The concept of an approximate projection is introduced along with a discussion of its use as measure of the error committed with the approximation of the true interactions among the beads. The theoretical discussion is supported by numerical simulations of a monodimensional non-linear periodic benchmark system with an open-source parallel Julia code, easily extensible to arbitrary potential models and fine-to-coarse mapping functions. The results presented highlight the importance of introducing, in the macroscopic model, a non-constant dissipative term, given by the Mori-Zwanzig approach, to correctly reproduce the reference fine-grained results without requiring \emph{ad-hoc} calibration of interaction potentials and thermostats.