Control-volume representation of molecular dynamics

TL;DR

This paper develops a control-volume framework for molecular dynamics, enabling direct comparison with continuum fluid mechanics and accurate computation of macroscopic properties from molecular simulations.

Contribution

It introduces a novel control-volume formulation for MD based on Irving and Kirkwood's formulas, providing an exact conservative approach for analyzing molecular systems.

Findings

The method accurately computes fluxes and forces across control surfaces.

Numerical experiments demonstrate the approach's effectiveness in equilibrium and non-equilibrium flows.

The formulation naturally relates local pressure calculations from different techniques.

Abstract

A Molecular Dynamics (MD) parallel to the Control Volume (CV) formulation of fluid mechanics is developed by integrating the formulas of Irving and Kirkwood, J. Chem. Phys. 18, 817 (1950) over a finite cubic volume of molecular dimensions. The Lagrangian molecular system is expressed in terms of an Eulerian CV, which yields an equivalent to Reynolds' Transport Theorem for the discrete system. This approach casts the dynamics of the molecular system into a form that can be readily compared to the continuum equations. The MD equations of motion are reinterpreted in terms of a Lagrangian-to-Control-Volume (\CV) conversion function $\vartheta_{i}$, for each molecule $i$. The \CV function and its spatial derivatives are used to express fluxes and relevant forces across the control surfaces. The relationship between the local pressures computed using the Volume Average (VA, Lutsko, J. Appl. Phys 64, 1152 (1988)) techniques and the Method of Planes (MOP, Todd et al, Phys. Rev. E 52, 1627 (1995)) emerges naturally from the treatment. Numerical experiments using the MD CV method are reported for equilibrium and non-equilibrium (start-up Couette flow) model liquids, which demonstrate the advantages of the formulation. The CV formulation of the MD is shown to be exactly conservative, and is therefore ideally suited to obtain macroscopic properties from a discrete system.