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

Part of: Two-phase and multiphase flows Dynamical problems

Published online by Cambridge University Press:  28 December 2020

Benjamin D. Goddard ,
Tim D. Hurst  and
Mark Wilkinson
Benjamin D. Goddard
Affiliation:
School of Mathematics and the Maxwell Institute for Mathematical Sciences, University of Edinburgh, EdinburghEH9 3FD, UK (b.goddard@ed.ac.uk)
Tim D. Hurst
Affiliation:
School of Mathematics and the Maxwell Institute for Mathematical Sciences, University of Edinburgh, EdinburghEH9 3FD, UK (b.goddard@ed.ac.uk)
Mark Wilkinson
Affiliation:
Department of Mathematics, Nottingham Trent University, Nottingham, NG11 8NS, UK (mark.wilkinson@ntu.ac.uk)
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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 two examples.

Keywords

Liouville equationHard particlesBBGKY hierarchyInelastic collisions

MSC classification

Primary: 70F35: Collision of rigid or pseudo-rigid bodies
Secondary: 74H20: Existence of solutions 76T25: Granular flows
Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 151 , Issue 3 , June 2021 , pp. 1040 - 1074
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Copyright
Copyright © The Author(s), 2020. Published by Cambridge University Press on behalf of The Royal Society of Edinburgh

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