A derivation of the Liouville equation for hard particle dynamics with non-conservative interactions

TL;DR

This paper derives a modified Liouville equation for systems of particles with non-conservative, instantaneous interactions, addressing challenges in modeling such dynamics accurately.

Contribution

It introduces a weak formulation of the Liouville equation that accounts for non-conservative and instantaneous particle interactions.

Findings

Derived a weak formulation suitable for non-conservative interactions

Addressed exclusion of non-physical particle positions during collisions

Provided an example illustrating the formulation

Abstract

The Liouville equation is of fundamental importance in the derivation of continuum models for physical systems which are approximated by interacting particles. However, when particles undergo instantaneous interactions such as collisions, the derivation of the Liouville equation must be adapted to exclude non-physical particle positions, and include the effect of instantaneous interactions. We present the weak formulation of the Liouville equation for interacting particles with general particle dynamics and interactions, and discuss the results using an example.