Hostname: page-component-cd9895bd7-dzt6s Total loading time: 0 Render date: 2024-12-26T10:42:41.957Z Has data issue: false hasContentIssue false
  • English
  • Français

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)
Get access Rights & Permissions [Opens in a new window]

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
Check for updates
Copyright
Copyright © The Author(s), 2020. Published by Cambridge University Press on behalf of The Royal Society of Edinburgh

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Balescu, R.. Equilibrium and nonequilibrium statistical mechanics (New York: Wiley, 1975).Google Scholar
Bannerman, M. and Lue, L.. Exact on-event expressions for discrete potential systems. J. Chem. Phys. 133 (2010), 124506–12.CrossRefGoogle ScholarPubMed
Cecconi, F., Tarazona, P. and Marini-Bettolo-Marconi., U.. Theory of thermostatted inhomogeneous granular fluids: A self-consistent density functional description. J. Chem. Phys. 126 (2007), 1–13.Google Scholar
Cercigani, C.. The Boltzmann equation (New York: Springer, 1988).CrossRefGoogle Scholar
Donev, A., Stillinger, F. H. and Torquato, S.. Neighbor list collision-driven molecular dynamics simulation for nonspherical hard particles. J. Comput. Phys. 202 (2005), 765–793.Google Scholar
Evans, L. C. and Gariepy, R. F.. Measure theory and fine properties of functions (London: Chapman and Hall, 2015).CrossRefGoogle Scholar
Gallagher, I., Saint-Raymond, L. and Texier, B.. From Newton to Boltzmann: hard spheres and short range potentials. Preprint, arXiv:1208.5753, 2012.Google Scholar
Goddard, B. D., Hurst, T. D. and Ocone, R.. Modelling inelastic granular media using Dynamical Density Functional Theory. Preprint, arXiv:2003.07327, 2020.Google Scholar
Goddard, B. D., Kalliadasis, S., Nold, A., Pavliotis, G. A. and Savva., N.. General dynamical density functional theory for classical fluids. Phys. Rev. Lett. 109 (2012), 1–5.CrossRefGoogle ScholarPubMed
Huang, K.. Statistical mechanics (New York: Wiley, 1987).Google Scholar
Lifshitz, E. M. and Pitaevskii, L. P.. Physical kinetics (Oxford: Butterworth Heinemann, 1981).Google Scholar
Lun, C. K. K., Savage, S. B., Jeffrey, D. J. and Chepurniy., N.. Kinetic theories for granular flow: inelastic particles in Couette flow and slightly inelastic particles in a general flowfield. J. Fluid Mech. 140 (1984), 223–256.CrossRefGoogle Scholar
Marsden, J. E., Tromba, A.. Vector calculus (London: Macmillan, 2003).Google Scholar
Pavliotis, G. A., Stuart, A. M.. Multiscale methods: averaging and homogenization (New York: Springer Science & Business Media, 2008).Google Scholar
Radjaï, F., Dubois, F.. Discrete-element modeling of granular materials (New Jersey: Wiley-Iste, 2011).Google Scholar
Risken, H.. Fokker-Planck equation (Berlin Heidelberg: Springer, 1996).CrossRefGoogle Scholar
Royden, H. L.. Real analysis (London: Macmillan, 1988).Google Scholar
Schlick, T.. Molecular modeling and simulation: an interdisciplinary guide (New York: Springer Science & Business Media, 2010).CrossRefGoogle Scholar
Simonella, S.. BBGKY hierarchy for hard sphere systems. Ph.D. thesis. Sapienza University of Rome, 2011.Google Scholar
Wilkinson, M.. On global-in-time chaotic weak solutions of the Liouville equation for hard spheres. Preprint, arXiv:1807.01017, 2018.Google Scholar