On the kinetic description of the objective molecular dynamics

TL;DR

This paper introduces a multiscale framework for objective molecular dynamics, linking microscopic interactions to kinetic and hydrodynamic models, and simplifies these models using symmetry properties.

Contribution

It develops a hierarchical approach connecting objective molecular dynamics to kinetic and hydrodynamic equations, revealing simplified models due to symmetry.

Findings

Mean-field scaling simplifies molecular interaction rules.

Boltzmann scaling leads to homo-energetic Boltzmann equation.

Derivation of Euler and Navier-Stokes systems from OMD symmetry.

Abstract

In this paper, we develop a multiscale hierarchy framework for objective molecular dynamics (OMD), a reduced order molecular dynamics with a certain symmetry, that connects it to the statistical kinetic equation, and the macroscopic hydrodynamic model. In the mesoscopic regime, we exploit two interaction scalings that lead, respectively, to either a mean-field type or to a Boltzmann type equation. It turns out that, under the special symmetry of OMD, the mean-field scaling results in vastly simplified dynamics that extinguishes the underlying molecular interaction rule, whereas the Boltzmann scaling yields a meaningful reduced model called the homo-energetic Boltzmann equation. At the macroscopic level, we derive the corresponding Euler and Navier-Stokes systems by conducting a detailed asymptotic analysis. The symmetry again significantly reduces the complexity of the resulting hydrodynamic systems.