We analyze nonlinear aspects of the self-consistent wave–particle interaction using Hamiltonian dynamics in the single wave model, where the wave is modified due to the particle dynamics. This interaction plays an important role in the emergence of plasma instabilities and turbulence. The simplest case, where one particle is coupled with one wave , is completely integrable, and the nonlinear effects reduce to the wave potential pulsating while the particle either remains trapped or circulates forever. On increasing the number of particles (, ), integrability is lost and chaos develops. Our analyses identify the two standard ways for chaos to appear and grow (the homoclinic tangle born from a separatrix, and the resonance overlap near an elliptic fixed point). Moreover, a strong form of chaos occurs when the energy is high enough for the wave amplitude to vanish occasionally.
REFERENCES
1.
Y.
Elskens
and D.
Escande
, Microscopic Dynamics of Plasmas and Chaos
(IOP Publishing
, Bristol
, 2003
).2.
D. F.
Escande
and Y.
Elskens
, “Microscopic dynamics of plasmas and chaos: The wave-particle interaction paradigm
,” Plasma Phys. Control. Fusion
45
, A115
–124
(2003
). 3.
4.
5.
C. F. F.
Karney
and A.
Bers
, “Stochastic ion heating by a perpendicularly propagating electrostatic wave
,” Phys. Rev. Lett.
39
, 550
–554
(1977
). 6.
G. R.
Smith
and N.
Pereira
, “Phase-locked particle motion in a large-amplitude plasma wave
,” Phys. Fluids
21
, 2253
–2262
(1978
). 7.
8.
S.
Ichimaru
, Statistical Plasma Physics, Vol. I: Basic Principles
(CRC Press
, Boca Raton
, 2018
).9.
Y.
Elskens
, “Irreversible behaviours in Vlasov equation and many-body Hamiltonian dynamics: Landau damping, chaos and granularity,” in Topics in Kinetic Theory, Toronto, 24 March–2 April 2004, Fields Institute Communications Series Vol. 46, edited by T. Passot, C. Sulem, and P. L. Sulem (American Mathematical Society, Providence, 2005), pp. 89–108.10.
N.
Besse
, Y.
Elskens
, D. F.
Escande
, and P.
Bertrand
, “Validity of quasilinear theory: Refutations and new numerical confirmation
,” Plasma Phys. Control. Fusion
53
, 025012
(2011
). 11.
Y.
Elskens
, “Gaussian convergence for stochastic acceleration of particles in the dense spectrum limit
,” J. Stat. Phys.
148
, 591
–605
(2012
). 12.
A. V.
Timofeev
, “Cyclotron oscillations of plasma in an inhomogeneous magnetic field
,” Sov. Phys. Uspekhi
16
, 445
(1974
). 13.
D. F.
Escande
, D.
Bénisti
, Y.
Elskens
, D.
Zarzoso
, and F.
Doveil
, “Basic microscopic plasma physics from -body mechanics
,” Rev. Mod. Plasma Phys.
2
, 9
(2018
). 14.
F.
Doveil
, D. F.
Escande
, and A.
Macor
, “Experimental observation of nonlinear synchronization due to a single wave
,” Phys. Rev. Lett.
94
, 085003
(2005
). 15.
F. F.
Chen
, Introduction to Plasma Physics and Controlled Fusion
(Plenum Press
, New York
, 1984
).Vol. 1.16.
W.
Herr
, “Introduction to Landau damping
,” in Advanced Accelerator Physics, Trondheim, 19–29 August 2013
, edited by W. Herr (CERN
, Geneva
, 2014
). 17.
J.
He
, L.
Wang
, C.
Tu
, E.
Marsch
, and Q.
Zong
, “Evidence of Landau and cyclotron resonance between protons and kinetic waves in solar wind turbulence
,” Astrophys. J. Lett.
800
, L31
(2015
). 18.
C. H. K.
Chen
, K. G.
Klein
, and G. G.
Howes
, “Evidence for electron landau damping in space plasma turbulence
,” Nat. Commun.
10
, 1
–8
(2019
). 19.
20.
D. D.
Ryutov
, “Landau damping: Half a century with the great discovery
,” Plasma Phys. Control. Fusion
41
(3A
), A1
–A12
(1999
). 21.
P.
Stubbe
and A. I.
Sukhorukov
, “On the physics of Landau damping
,” Phys. Plasmas
6
, 2976
–2988
(1999
). 22.
Y.
Elskens
, D. F.
Escande
, and F.
Doveil
, “Vlasov equation and -body dynamics: How central is particle dynamics to our understanding of plasmas?
,” Eur. Phys. J. D
68
, 1
–7
(2014
). 23.
D. F.
Escande
, “Stochasticity in classical Hamiltonian systems: Universal aspects
,” Phys. Rep.
121
, 165
–261
(1985
). 24.
M. C.
de Sousa
, F. M.
Steffens
, R.
Pakter
, and F. B.
Rizzato
, “Standard map in magnetized relativistic systems: Fixed points and regular acceleration
,” Phys. Rev. E
82
, 026402
(2010
). 25.
Y. H.
Ichikawa
, T.
Kamimura
, and C. F. F.
Karney
, “Stochastic motion of particles in tandem mirror devices
,” Physica D
6
, 233
–240
(1983
). 26.
H. E.
Mynick
and A. N.
Kaufman
, “Soluble theory of nonlinear beam-plasma interaction
,” Phys. Fluids
21
, 653
–663
(1978
). 27.
I. N.
Onishchenko
, A. R.
Linetskii
, N. G.
Matsiborko
, V. D.
Shapiro
, and V. I.
Shevchenko
, “Contribution to the nonlinear theory of excitation of a monochromatic plasma wave by an electron beam
,” Soviet Phys. JETP
11
, 281
–285
(1971
).28.
T. M.
O’Neil
, J. H.
Winfrey
, and J. H.
Malmberg
, “Nonlinear interaction of a small cold beam and a plasma
,” Phys. Fluids
14
, 1204
–1212
(1971
). 29.
J. L.
Tennyson
, J. D.
Meiss
, and P. J.
Morrison
, “Self-consistent chaos in the beam-plasma instability
,” Physica D
71
, 1
–17
(1994
). 30.
A.
Antoniazzi
, Y.
Elskens
, D.
Fanelli
, and S.
Ruffo
, “Statistical mechanics and Vlasov equation allow for a simplified Hamiltonian description of single-pass free electron laser saturated dynamics
,” Eur. Phys. J. B
50
, 603
–611
(2006
). 31.
J. C.
Adam
, G.
Laval
, and I.
Mendonça
, “Time-dependent nonlinear Langmuir waves
,” Phys. Fluids
24
, 260
–267
(1981
). 32.
M.-C.
Firpo
and Y.
Elskens
, “Kinetic limit of N-body description of wave-particle self-consistent interaction
,” J. Stat. Phys.
93
, 193
–209
(1998
). 33.
N.
Carlevaro
, G.
Montani
, and D.
Terzani
, “On the viability of the single-wave model for the beam plasma instability
,” Europhys. Lett.
115
, 45004
(2016
). 34.
M.-C.
Firpo
and Y.
Elskens
, “Phase transition in the collisionless damping regime for wave-particle interaction
,” Phys. Rev. Lett.
84
, 3318
–3321
(2000
). 35.
N. A.
Yampolsky
and N. J.
Fisch
, “Simplified model of nonlinear Landau damping
,” Phys. Plasmas
16
, 072104
(2009
). 36.
Z.
Huang
and K.-J.
Kim
, “Review of x-ray free-electron laser theory
,” Phys. Rev. ST Accel. Beams
10
, 034801
(2007
). 37.
D.
del Castillo-Negrete
and M.-C.
Firpo
, “Coherent structures and self-consistent transport in a mean field Hamiltonian model
,” Chaos
12
, 496
–507
(2002
). 38.
D.
Testa
, A.
Fasoli
, D.
Borba
, M.
de Baar
, M.
Bigi
, J.
Brzozowski
, P.
de Vries
, JET-EFDA Contributors
et al., “Alfvén mode stability and wave-particle interaction in the JET tokamak: Prospects for scenario development and control schemes in burning plasma experiments
,” Plasma Phys. Control. Fusion
46
, S59
–S79
(2004
). 39.
E.
Hairer
, C.
Lubich
, and G.
Wanner
, Geometric Numerical Integration: Structure-preserving Algorithms for Ordinary Differential Equations
(Springer
, Berlin
, 2006
).40.
G. M.
Zaslavsky
, Hamiltonian Chaos and Fractional Dynamics
(Oxford University Press
, New York
, 2005
).41.
J. D.
Crawford
and A.
Jayaraman
, “First principles justification of a “single wave model” for electrostatic instabilities
,” Phys. Plasmas
6
, 666
–673
(1999
). 42.
D.
Farina
, F.
Casagrande
, U.
Colombo
, and R.
Pozzoli
, “Hamiltonian analysis of the transition to the high-gain regime in a Compton free-electron-laser amplifier
,” Phys. Rev. E
49
, 1603
–1609
(1994
). 43.
D.
del Castillo-Negrete
, “Dynamics and self-consistent chaos in a mean field Hamiltonian model,” in Dynamics and Thermodynamics of Systems with Long-Range Interactions, Les Houches, 18–22 February 2002, edited by T. Dauxois, S. Ruffo, E. Arimondo, and M. Wilkens (Springer, Berlin, 2002), pp. 407–436.44.
A. V.
Bolsinov
, A. V.
Borisov
, and I. S.
Mamaev
, “Topology and stability of integrable systems
,” Russian Math. Surveys
65
, 259
–318
(2010
). 45.
A. H.
Boozer
, “Arnold diffusion and adiabatic invariants
,” Phys. Lett. A
185
, 423
–427
(1994
). 46.
C. R.
Menyuk
, “Particle motion in the field of a modulated wave
,” Phys. Rev. A
31
, 3282
–3290
(1985
). 47.
Y.
Elskens
, “Finite- dynamics admit no travelling-waves solutions for the Hamiltonian model and single-wave collisionless plasma model,” in ESAIM: Proceedings (EDP Sciences, 2001), Vol. 10, pp. 221–215.48.
D. F.
Escande
, S.
Zekri
, and Y.
Elskens
, “Intuitive and rigorous microscopic description of spontaneous emission and Landau damping of Langmuir waves through classical mechanics
,” Phys. Plasmas
3
, 3534
–3539
(1996
). 49.
50.
V.
Kozlov
, “Integrability and non-integrability in Hamiltonian mechanics
,” Russ. Math. Surveys
38
, 1
–76
(1983
). 51.
J.
Guckenheimer
and P.
Holmes
, Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields
(Springer
, New York
, 1983
).52.
E.
Ott
, Chaos in Dynamical Systems
(Cambridge University Press
, 2002
).53.
A. J.
Lichtenberg
and M. A.
Lieberman
, Regular and Stochastic Motion
(Springer
, New York
, 1983
).54.
R.
Pakter
and G.
Corso
, “Improving regular acceleration in the nonlinear interaction of particles and waves
,” Phys. Plasmas
2
, 4312
–4324
(1995
). 55.
M.
Hénon
and C.
Heiles
, “The applicability of the third integral of motion: Some numerical experiments
,” Astron. J.
69
, 73
(1964
). 56.
J. H.
Lowenstein
, Essentials of Hamiltonian Dynamics
(Cambridge University Press
, Cambridge
, 2012
).57.
Y.
Elskens
and C.
Firpo
, “Kinetic theory and large-N limit for wave-particle self-consistent interaction
,” Phys. Scripta
T75
, 169
–172
(1998
). © 2021 Author(s). Published under an exclusive license by AIP Publishing.
2021
Author(s)
You do not currently have access to this content.
