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.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Notes
- 1.
- 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.
References
L. Boltzmann, Weitere Studien über das Wärmegleichgewicht unter Gasmolekülen, Wien. Ber. 66, 275 (1872).
A. Campa, T. Dauxois, S. Ruffo, Statistical mechanics and dynamics of solvable models with long-range interactions, Phys. Rep. 480, 57-159 (2009) [arXiv:0907.0323].
A. Campa, T. Dauxois, D. Fanelli, S. Ruffo, Physics of Long-Range Interacting Systems (Oxford University Press, Oxford, England 2014).
Y. Elskens, D. Escande, F. Doveil, Vlasov equation and \(N\)-body dynamics, Eur. Phys. J. D68, 218 (2014) [arxiv:1403.0056].
H. Haug, Statistische Physik (Vieweg, Braunschweig, Germany 1997).
K. Huang, Statistical Mechanics, 2 \(^{\rm nd}\) ed. (Wiley, New York, USA 1987); pp. 53ff.
M. Henkel, Phenomenology of local scale-invariance: from conformal invariance to dynamical scaling, Nucl. Phys. B641, 405 (2002) [hep-th/0205256].
M. Henkel, M. Pleimling, Non-equilibrium phase transitions vol. 2: ageing and dynamical scaling far from equilibrium (Springer, Heidelberg, Germany 2010).
M. Hénon, Vlasov equation ?, Astron. Astrophys. 114, 211 (1982).
J.H. Jeans, On the theory of star-streaming and the structure of the universe, Monthly Notices Roy. Astron. Soc. 76, 70 (1915).
H.-J. Kreuzer, Nonequilibrium thermodynamics and its statistical foundations (Oxford University Press, Oxford, England 1981); ch. 7.
H. Mo, F. van den Bosch, S. White, Galaxy formation and evolution (Cambridge University Press, Cambridge, England 2010).
F. Pegoraro, F. Califano, G. Manfredi, P.J. Morrison, Theory and applications of the Vlassov equation, Eur. Phys. J. D69, 68 (2015) [arXiv:1502.03768].
S. Stoimenov and M. Henkel, From conformal invariance towards dynamical symmetries of the collisionless Boltzmann equation, Symmetry 7, 1595 (2015) [arXiv:1509.00434].
C. Vilani, Particle systems and non-linear Landau damping, Phys. Plasmas 21, 030901 (2014).
A.A. Vlasov, On vibration properties of electron gas (in Russian), Sov. Phys. JETP, 8, 291 (1938).
Acknowledgements
Most of this work was done during the visits of S.S. at Université de Lorraine Nancy and of M.H. at XI\(^\mathrm{th}\) International workshop “Lie theories and its applications in physics”. These visits was supported by PHC Rila. M.H. was partly supported by the Collège Doctoral Nancy-Leipzig-Coventry (Systèmes complexes à l’équilibre et hors équilibre) of UFA-DFH. S.S. has received partial support from the Bulgarian National Science Fund Grant DFNI-T02/6.
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2016 Springer Nature Singapore Pte Ltd.
About this paper
Cite this paper
Stoimenov, S., Henkel, M. (2016). Conformal Invariance of the 1D Collisionless Boltzmann Equation. In: Dobrev, V. (eds) Lie Theory and Its Applications in Physics. LT 2015. Springer Proceedings in Mathematics & Statistics, vol 191. Springer, Singapore. https://doi.org/10.1007/978-981-10-2636-2_33
Download citation
DOI: https://doi.org/10.1007/978-981-10-2636-2_33
Published:
Publisher Name: Springer, Singapore
Print ISBN: 978-981-10-2635-5
Online ISBN: 978-981-10-2636-2
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)