Systematic derivation of hybrid coarse-grained models

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.