Hostname: page-component-669899f699-7tmb6 Total loading time: 0 Render date: 2025-04-25T07:08:22.265Z Has data issue: false hasContentIssue false
  • English
  • Français

Collisionless relaxation of a Lynden-Bell plasma

Published online by Cambridge University Press:  16 September 2022

R.J. Ewart Open the ORCID record for R.J. Ewart [Opens in a new window] ,
A. Brown ,
T. Adkins Open the ORCID record for T. Adkins [Opens in a new window]  and
A.A. Schekochihin Open the ORCID record for A.A. Schekochihin [Opens in a new window]
R.J. Ewart*
Affiliation:
Rudolf Peierls Centre for Theoretical Physics, University of Oxford, Oxford OX1 3PU, UK Balliol College, Oxford OX1 3BJ, UK
A. Brown
Affiliation:
Exeter College, Oxford OX1 3DP, UK
T. Adkins
Affiliation:
Rudolf Peierls Centre for Theoretical Physics, University of Oxford, Oxford OX1 3PU, UK Merton College, Oxford OX1 4JD, UK
A.A. Schekochihin
Affiliation:
Rudolf Peierls Centre for Theoretical Physics, University of Oxford, Oxford OX1 3PU, UK Merton College, Oxford OX1 4JD, UK
*
†Email address for correspondence: robert.ewart@physics.ox.ac.uk
Get access Rights & Permissions [Opens in a new window]

Abstract

Plasmas whose Coulomb-collision rates are very small may relax on shorter timescales to non-Maxwellian quasi-equilibria, which, nevertheless, have a universal form, with dependence on initial conditions retained only via an infinite set of Casimir invariants enforcing phase-volume conservation. These are distributions derived by Lynden-Bell (Mon. Not. R. Astron. Soc., vol. 136, 1967, p. 101) via a statistical-mechanical entropy-maximisation procedure, assuming perfect mixing of phase-space elements. To show that these equilibria are reached dynamically, one must derive an effective ‘collisionless collision integral’ for which they are fixed points – unique and inevitable provided the integral has an appropriate H-theorem. We describe how such collision integrals are derived and what assumptions are required for them to have a closed form, how to prove the H-theorems for them, and why, for a system carrying sufficiently large electric-fluctuation energy, collisionless relaxation should be fast. It is suggested that collisionless dynamics may favour maximising entropy locally in phase space before converging to global maximum-entropy states. Relaxation due to interspecies interaction is examined, leading, inter alia, to spontaneous transient generation of electron currents. The formalism also allows efficient recovery of ‘true’ collision integrals for both classical and quantum plasmas.

Keywords

astrophysical plasmasplasma dynamicsplasma nonlinear phenomena
Type
Tutorial
Information
Journal of Plasma Physics , Volume 88 , Issue 5 , October 2022 , 925880501
Check for updates
Copyright
Copyright © The Author(s), 2022. Published by Cambridge University Press

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.)

Article purchase

Temporarily unavailable

References

REFERENCES

Adkins, T. & Schekochihin, A.A. 2018 A solvable model of Vlasov-kinetic plasma turbulence in Fourier–Hermite phase space. J. Plasma Phys. 84, 905840107.CrossRefGoogle Scholar
Arad, I. & Johansson, P.H. 2005 A numerical comparison of theories of violent relaxation. Mon. Not. R. Astron. Soc. 362, 252.CrossRefGoogle Scholar
Balescu, R. 1960 Irreversible processes in ionized gases. Phys. Fluids 3, 52.CrossRefGoogle Scholar
Beraldo e Silva, L., de Siqueira Pedra, W., Sodré, L., Perico, E.L.D. & Lima, M. 2017 The arrow of time in the collapse of collisionless self-gravitating systems: non-validity of the Vlasov-Poisson equation during violent relaxation. Astrophys. J. 846, 125.CrossRefGoogle Scholar
Boltzmann, L. 1896 Vorlesugnen über Gastheorie. J. A. Barth.Google Scholar
Chavanis, P.-H. 2004 Generalized thermodynamics and kinetic equations: Boltzmann, Landau, Kramers and Smoluchowski. Physica A 332, 89.CrossRefGoogle Scholar
Chavanis, P.-H. 2005 Statistical mechanics of geophysical turbulence: application to jovian flows and Jupiter's great red spot. Physica D 200, 257.CrossRefGoogle Scholar
Chavanis, P.-H. 2006 Quasi-stationary states and incomplete violent relaxation in systems with long-range interactions. Physica A 365, 102.CrossRefGoogle Scholar
Chavanis, P.-H. 2022 Kinetic theory of collisionless relaxation for systems with long-range interactions. Physica A, 128089.CrossRefGoogle Scholar
Chavanis, P.H., Sommeria, J. & Robert, R. 1996 Statistical mechanics of two-dimensional vortices and collisionless stellar systems. Astrophys. J. 471, 385.CrossRefGoogle Scholar
Coonin, A.V. 2014 From Marble to Flesh: The Biography of Michelangelo's David. B'Gruppo.Google Scholar
Daligault, J. 2016 On the quantum Landau collision operator and electron collisions in dense plasmas. Phys. Plasmas 23, 032706.CrossRefGoogle Scholar
Danielewicz, P. 1980 Nonrelativistic and relativistic Landau/Fokker-Planck equation for arbitrary statistics. Physica A 100, 167.CrossRefGoogle Scholar
Eyink, G.L. 2018 Cascades and dissipative anomalies in nearly collisionless plasma turbulence. Phys. Rev. X 8, 041020.Google Scholar
Gardner, C.S. 1963 Bound on the energy available from a plasma. Phys. Fluids 6, 839.CrossRefGoogle Scholar
Gibbs, J.W. 1902 Elementary Principles In Statistical Mechanics Developed With Especial Reference To The Rational Foundation Of Thermodynamics. Charles Scribner.Google Scholar
Helander, P. 2017 Available energy and ground states of collisionless plasmas. J. Plasma Phys. 83, 715830401.CrossRefGoogle Scholar
Helander, P. & Sigmar, D.J. 2005 Collisional Transport in Magnetised Plasmas. Cambridge University Press.Google Scholar
Kadomtsev, B.B. 1965 Plasma Turbulence. Academic Press.Google Scholar
Kadomtsev, B.B. & Pogutse, O.P. 1970 Collisionless relaxation in systems with Coulomb interactions. Phys. Rev. Lett. 25, 1155.CrossRefGoogle Scholar
Klimontovich, Y.L. 1967 The Statistical Theory of Non-Equilibrium Processes in a Plasma. MIT Press.Google Scholar
Landau, L.D. 1936 Transport equation in the case of Coulomb interaction. Zh. Eksp. Teor. Fiz. 7, 203.Google Scholar
Lenard, A. 1960 On Bogoliubov's kinetic equation for a spatially homogeneous plasma. Ann. Phys. 10, 390.CrossRefGoogle Scholar
Levin, Y., Pakter, R., Rizzato, F.B., Teles, T.N. & Benetti, F.P.C. 2014 Nonequilibrium statistical mechanics of systems with long-range interactions. Phys. Rep. 535, 1.CrossRefGoogle Scholar
Lorenz, E.N. 1955 Available potential energy and the maintenance of the general circulation. Tellus 7, 157.CrossRefGoogle Scholar
Lynden-Bell, D. 1967 Statistical mechanics of violent relaxation in stellar systems. Mon. Not. R. Astron. Soc. 136, 101.CrossRefGoogle Scholar
Maxwell, J.C. 1860 V. Illustrations of the dynamical theory of gases.—Part I. On the motions and collisions of perfectly elastic spheres. Lond. Edinb. Dublin Philos. Mag. J. Sci. 19, 19.CrossRefGoogle Scholar
Parra, F.I. 2019 Collisional Plasma Physics. Lecture Notes for an Oxford MMathPhys course; URL: http://www-thphys.physics.ox.ac.uk/people/FelixParra/CollisionalPlasmaPhysics/CollisionalPlasmaPhysics.html.Google Scholar
Rightley, S. & Baalrud, S.D. 2021 Kinetic model for electron–ion transport in warm dense matter. Phys. Rev. E 103, 063206.CrossRefGoogle ScholarPubMed
Schekochihin, A.A., Cowley, S.C., Dorland, W., Hammett, G.W., Howes, G.G., Quataert, E. & Tatsuno, T. 2009 Astrophysical gyrokinetics: kinetic and fluid turbulent cascades in magnetized weakly collisional plasmas. Astrophys. J. Suppl. Ser. 182, 310.CrossRefGoogle Scholar
Severne, G. & Luwel, M. 1980 Dynamical theory of collisionless relaxation. Astrophys. Space Sci. 72, 293.CrossRefGoogle Scholar
Shannon, C.E. 1948 A mathematical theory of communication. Bell Syst. Techn. J. 27, 379.CrossRefGoogle Scholar
Spitzer, L. 1967 Physics of Fully Ionized Gases. Interscience Publishers.Google Scholar
Su, C.H. & Oberman, C. 1968 Collisional damping of a plasma echo. Phys. Rev. Lett. 20, 427.CrossRefGoogle Scholar
Swanson, D.G. 2008 Plasma Kinetic Theory. Chapman and Hall.CrossRefGoogle Scholar
Uehling, E.A. & Uhlenbeck, G.E. 1933 Transport phenomena in Einstein-Bose and Fermi-Dirac gases. I. Phys. Rev. 43, 552.CrossRefGoogle Scholar
Zhdankin, V. 2021 Generalized entropy production in collisionless plasma flows and turbulence. Phys. Rev 12, 031011.CrossRefGoogle Scholar