Vlasov dynamics of 1D models with long-range interactions

4 MB·English

by  Pierre de Buyl

Universit´e Libre de Bruxelles
Center for Nonlinear Phenomena and Complex Systems
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Vlasov dynamics of 1D models with
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Pierre de Buyl
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Academic year 2009-2010
Dissertation presented in partial fulfilment of the requirements for
the degree of Doctor of Philosophy in Physics
Supervisor : Prof. Pierre Gaspard
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Thoughtcrime does not entail death: thoughtcrime is death.
Georges Orwell, Nineteen Eighty Four
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`a Sophie,
pour toujours.
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Abstract
Gravitational and electrostatic interactions are fundamental examples of sys-
tems with long-range interactions. Equilibrium properties of simple models
with long-range interactions are well understood and exhibit exotic behaviors :
negative specific heat and inequivalence of statistical ensembles for instance.
The understanding of the dynamical evolution in the case of long-range in-
teracting systems still represents a theoretical challenge. Phenomena such as
out-of-equilibrium phase transitions or quasi-stationary states have been found
even in simple models.
Thepurposeofthepresentthesisistoinvestigatethedynamicalpropertiesof
systems with long-range interactions, specializing on one-dimensional models.
The appropriate kinetic description for these systems is the Vlasov equation.
A statistical theory devised by D. Lynden-Bell is adequate to predict in some
situationstheoutcomeofthedynamics. Acomplementarynumericalsimulation
tool for the Vlasov equation is developed.
A detailed study of the out-of-equilibrium phase transition occuring in the
Free-Electron Laser is performed and the transition is analyzed with the help
of Lynden-Bell’s theory. Then, the presence of stretching and folding in phase
space for the Hamiltonian Mean-Field model is studied and quantified from
the point of view of fluid dynamics. Finally, a system of uncoupled pendula
for which the asymptotic states are similar to the ones of the Hamiltonian
Mean-Fieldmodel is introduced. Its asymptotic evolutionis predicted via both
Lynden-Bell’s theory and an exact computation. This system displays a fast
initial evolution similar to the violent relaxation found for interacting systems.
Moreover,an out-of-equilibrium phase transition is found if one imposes a self-
consistent condition on the system.
In summary, the present thesis discusses original results related to the oc-
curenceofquasi-stationarystatesandout-of-equilibriumphasetransitionsin1D
models with long-range interaction. The findings regarding the Free-Electron
Laser are of importance in the perspective of experimental realizations of the
aforementioned phenomena.
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R´esum´e
Les interactions gravitationnelles et´electrostatiques sont deux exemples fonda-
mentauxdesyst`emeseninteractiondelongueport´ee. Lespropri´et´esd’´equilibre
de mod`eles simples en interaction de longue port´ee sont bien comprises et
r´ev`elentdescomportemensexotiques: capacit´esp´ecifiquen´egativeetin´equivalence
des ensembles statistiques par exemple.
Lacompr´ehensiondel’´evolutiondynamiquedanslecasdesyst`emeseninter-
action de longue port´ee repr´esente encore actuellement un d´efi th´eorique. Des
mod`eles simples pr´esentent des propri´et´es telles que des transitions de phase
hors d’´equilibre ou des ´etats quasi-stationnaires.
Lebutdelapr´esenteth`eseestd’´etudierlespropri´et´esdynamiquesdesyst`emes
en interaction de longue port´ee pour des mod`eles `a une dimension. La de-
scription cin´etique ad´equate est donn´ee par l’´equation de Vlasov. Une th´eorie
statistique propos´ee par D. Lynden-Bell est appropri´ee pour pr´edire dans cer-
taines situations l’aboutissementde la dynamique. Un outil de simulationpour
l’´equation de Vlasov compl`ete cette approche.
Une´etuded´etaill´eedelatransitiondephasedansleLaser`aElectronsLibres
est pr´esent´ee et la transition est analys´ee `a l’aide de la th´eorie de Lynden-Bell.
Ensuite, la pr´esence d’´etirement et de repliement est ´etudi´ee dans le mod`ele
Hamiltonian Mean-Field en analogie avec la dynamique des fluides. Enfin, un
syst`eme de pendules d´ecoupl´es dont les ´etats asymptotiques sont similaires `a
ceux du mod`ele Hamiltonian Mean-Fieldest introduit. Son´evolutionasympto-
tique est pr´edite par la th´eorie de Lynden-Bell et par une approche exacte. Ce
syst`eme pr´esente une ´evolution initiale rapide similaire `a la relaxation violente
pr´esente dans des mod`eles plus compliqu´es. De plus, une transition de phase
hors d’´equilibre est trouv´ee si une condition d’auto-consistence est impos´ee.
En r´esum´e, la pr´esente th`ese comporte des r´esultats originaux li´es `a la
pr´esence d’´etats quasi-stationnaires et de transitions de phase hors d’´equilibre
dansdesmod`elesunidimensionnelseninteractiondelongueport´ee. Lesr´esultats
concernant le Laser `a Electrons Libres offrent une perspective de r´ealisation
exp´erimentale des ph´enom`enes d´ecrits dans cette th`ese.
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Copyright notice
The present thesis is copyrighted to Pierre de Buyl.
Mostofthe resultsandfiguresinchapter4 haveappearedinde Buyl, Com-
mun. Nonlin. Sci. Numer. Simulat. doi:10.1016/j.cnsns.2009.08.020 (2009).
Most of the results and figures in chapter 5 have appeared in de Buyl et al,
Phys. Rev. ST Accel. Beams 12 060704 (2009).
List of publications
Someofthe resultspresentedinthis thesishavebeenpresentedinthe following
publications :
Out-of-equilibrium mean-field dynamics of a model for wave-particle in-
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teraction
P. de Buyl, D. Fanelli, R. Bachelard and G. De Ninno
PhysicalReviewSpecialTopics-AcceleratorsandBeams12060704(2009)
Numerical resolution of the Vlasov equation for the Hamiltonian Mean-
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Field model
P. de Buyl
Communications in Nonlinear Science and Numerical Simulation (2009)
doi:10.1016/j.cnsns.2009.08.020
Transition de phases hors-d’´equilibre dans le Laser `a Electrons Libres
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P. de Buyl, R. Bachelard,M.-E. Couprie, G. De Ninno and D. Fanelli
`eme
“Comptes-Rendusdela12 RencontreduNon-Lin´eaire”,Non-Lin´eaire
Publications, 2009
Deep saturation dynamics in a Free-Electron Laser
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R. Bachelard, M.-E. Couprie, P. de Buyl, G. De Ninno and D. Fanelli
Proceedings of the FEL09 Conference, Liverpool, 2009
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Table of contents
Remerciements 14
1 Introduction 17
The long-range character of the interaction modifies significantly the macro-
scopic behaviour of physical systems. We present general properties of such
systems. The reasons motivating their investigations are observations in the
contextofgalacticdynamicsandinsystemsofchargedparticles. Wespecialize
ontheparticulartopicofthecollisionlessevolutionin1Dmodelsofsystemswith
long-range interactions. The Vlasovian description of these systems has been
thefocusofrecentinterest,inducingtheneedforanumericalimplementation.
Definitions: systemswithlong-rangeinteractions,continuum limit.
1.1 Motivations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
1.2 Examples of systems with long-range interactions . . . . . . . . . 18
1.3 Thermodynamical behaviour of long-range interacting systems . 21
1.4 Dynamical properties. . . . . . . . . . . . . . . . . . . . . . . . . 23
1.5 Outline of the thesis . . . . . . . . . . . . . . . . . . . . . . . . . 27
2 The Vlasov equation 29
WeintroducetheVlasovequationasthecontinuumlimitinsystemswithlong-
rangeinteractions. WediscussitsderivationviatheBBGKYhierarchyofkinetic
theoryandtheconservationpropertiesofVlasovdynamics. WepresentLynden-
Bell’stheoryforthestatistical mechanicsofviolentrelaxation.
Definitions:kineticdescription,filamentation,microscopicdescription,coarse-
graineddistributionfunction,violentrelaxation.
2.1 The phase space description . . . . . . . . . . . . . . . . . . . . . 30
2.2 Kinetic equations . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
2.2.1 Formalism . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
2.2.2 Boltzmann’s equation . . . . . . . . . . . . . . . . . . . . 33
2.2.3 The Vlasov equation . . . . . . . . . . . . . . . . . . . . . 34
2.2.4 Properties of the Vlasov equation . . . . . . . . . . . . . . 34
2.2.5 Filamentation . . . . . . . . . . . . . . . . . . . . . . . . . 35
2.3 The statistical theory of Lynden-Bell . . . . . . . . . . . . . . . . 35
2.3.1 Violent relaxation . . . . . . . . . . . . . . . . . . . . . . 36
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