# Continuum Mechanics as a Computable Coarse-Grained Picture of Molecular Dynamics

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## Abstract

In this paper, I determine the minimum amount of information that continuum mechanics needs to obtain from Newtonian molecular dynamics, in order to avail itself of stress-strain responses uniformly valid for a vast range of macroscopic regimes, being quantitatively determined by microscopic physical properties. Described from the opposite, bottom-up point of view, the procedure I put forward uses the basic kinematic and dynamical machinery of continuum mechanics to upscale molecular dynamics to the macroscopic level in a practicable and efficient way.

## Keywords

Molecular dynamics Continuum mechanics Space-time averaging Streaming velocity Thermal velocity Stress Hierarchical multiscale modelling## Mathematics Subject Classification

82C22 82C21 74A25 74A10 82C80## Notes

### Acknowledgements

This paper is the provisional completion of ruminations and lucubrations that started about a decade ago and intensified in the last two-three years. I wish to acknowledge my debt towards my former students and postdocs, M. Ribezzi-Crivellari, M. Paoluzzi and M. Minozzi, who collaborated with me on this project in the now remote years of its long incubation. Without the ongoing interactions—sometimes harsh, always instructive—with S. Bonella, G. Ciccotti, M. Ferrario, and P. Podio-Guidugli, my ideas would have never developed to the present stage—much less would they have been written down.

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