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–Lie Algebras Canonically Associated to Probability Measures on with All Moments Read chapter Read first chapter

2016 | OriginalPaper | Chapter

Conformal Invariance of the 1D Collisionless Boltzmann Equation

Authors : Stoimen Stoimenov, Malte Henkel

Published in: Lie Theory and Its Applications in Physics

Publisher: Springer Singapore

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Abstract

Dynamical symmetries of the collisionless Boltzmann transport equation, with an external driving force, are derived in \(d=1\) spatial dimensions. Both positions and velocities are considered as independent variables. The Lie algebra of dynamical symmetries is isomorphic to the 2D projective conformal algebra, but we find new non-standard representations. Several examples with explicit external forces are presented.

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previous chapter Quantum Plactic and Pseudo-Plactic Algebras
next chapter On Reducibility Criterions for Scalar Generalized Verma Modules Associated to Maximal Parabolic Subalgebras
Footnotes
1
In plasma physics, the CBE is often called the Vlasov equation [16], although its application to galactical dynamics by Jeans occurred more than 20 years earlier [10].
 
2
This paper contains the main results of our original work [14], presented by the first author at the LT-11 conference.
 
3
The usual form of space translations does not work [14]. \(Y_{-1}\) is found (i) as a symmetry of the CBE and (ii) it forms a closed Lie algebra with the other basic generators \(X_{-1,0}\). The ansatz (13) is a particular solution the differential equation following from this. It leads to a Boltzmann operator \({\hat{B}}= -\mu X_{-1}-Y_{-1}\) linear in the generators. We believe this to be a natural auxiliary hypothesis.
 
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H.-J. Kreuzer, Nonequilibrium thermodynamics and its statistical foundations (Oxford University Press, Oxford, England 1981); ch. 7.
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S. Stoimenov and M. Henkel, From conformal invariance towards dynamical symmetries of the collisionless Boltzmann equation, Symmetry 7, 1595 (2015) [arXiv:​1509.​00434].
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Metadata
Title
Conformal Invariance of the 1D Collisionless Boltzmann Equation
Authors
Stoimen Stoimenov
Malte Henkel
Copyright Year
2016
Publisher
Springer Singapore
Book
Lie Theory and Its Applications in Physics

Print ISBN: 978-981-10-2635-5

Electronic ISBN: 978-981-10-2636-2

Copyright Year: 2016

DOI
https://doi.org/10.1007/978-981-10-2636-2_33

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