Phase-space structures in quantum-plasma wave turbulence
aa r X i v : . [ phy s i c s . p l a s m - ph ] S e p Phase-space structures in quantum-plasma wave turbulence
F. Haas, ∗ B. Eliasson,
1, 2
P. K. Shukla, † and G. Manfredi Institut f¨ur Theoretische Physik IV, Ruhr-Universit¨at BochumD-44780 Bochum, Germany Department of Physics, Ume˚a University, SE-901 87, Ume˚a, Sweden Institut de Physique et Chimie des Mat´eriaux de StrasbourgBP43, F-67034 Strasbourg, France
Abstract
The quasilinear theory of the Wigner-Poisson system in one spatial dimension is examined. Con-servation laws and properties of the stationary solutions are determined. Quantum effects areshown to manifest themselves in transient periodic oscillations of the averaged Wigner functionin velocity space. The quantum quasilinear theory is checked against numerical simulations ofthe bump-on-tail and the two-stream instabilities. The predicted wavelength of the oscillations invelocity space agrees well with the numerical results.
PACS numbers: 52.25Dg, 52.35.Mw, 52.35Ra ∗ Universidade do Vale do Rio dos Sinos - UNISINOS, Av. Unisinos 950, 93022-000, S˜ao Leopoldo, RS,Brazil † Department of Physics, Ume˚a University, SE-901 87, Ume˚a, Sweden; GOLP / Instituto de Plasmas e Fus˜aoNuclear, Instituto Superior T´ecnico, Universidade T´ecnica de Lisboa, 1049-001 Lisboa, Portugal; SUPA,Department of Physics, University of Strathclyde, Glasgow, G40NG, UK; School of Physics, University ofKwazulu-Natal, Durban 4000, South Africa. . INTRODUCTION Quantum plasmas have attracted a renewed attention in recent years. The inclusion ofquantum terms in the plasma fluid equations – such as quantum diffraction effects, modifiedequations of state [1, 2], and spin degrees of freedom [3] – leads to a variety of new physicalphenomena. Recent advances include linear and nonlinear quantum ion-acoustic waves ina dense magnetized electron-positron-ion plasma [4], the formation of vortices in quantumplasmas [5], the quantum Weibel and filamentation instabilities [6]–[10], the structure of weakshocks in quantum plasmas [11], the nonlinear theory of a quantum diode in a dense quantummagnetoplasma [12], quantum ion-acoustic waves in single-walled carbon nanotubes [13],the many-electron dynamics in nanometric devices such as quantum wells [14, 15], theparametric study of nonlinear electrostatic waves in two-dimensional quantum dusty plasmas[16], stimulated scattering instabilities of electromagnetic waves in an ultracold quantumplasma [17], and the propagation of waves and instabilities in quantum plasmas with spinand magnetization effects [18, 19].However, to date, only few works have investigated the important question of quan-tum plasma turbulence. The most notable exception is the paper by Shaikh and Shukla[20], where simulations of the two- and three-dimensional coupled Schr¨odinger and Pois-son equations – with parameters representative of the next-generation laser-solid interactionexperiments, as well as of dense astrophysical objects – were carried out. In that work,new aspects of the dual cascade in two-dimensional electron plasma wave turbulence atnanometric scales were identified. Nevertheless, the quantum-plasma wave turbulence re-mains a largely unexplored field of research. A reasonable strategy to attack these problemswould consist in extending well-known techniques issued from the theory of classical plasmaturbulence in order to include quantum effects.In this context, the simplest approach is given by the weak turbulence kinetic equationsfirst derived in Refs. [21]–[24], the so-called quasilinear theory. In quasilinear theory, thenon-oscillating part of the distribution function is flattened in the resonant region of velocityspace. It is interesting to carry over the basic techniques of the classical quasilinear theory tothe kinetic models of quantum plasmas. The resulting quantum quasilinear equations wouldbe a useful tool for the study of quantum plasma (weak) turbulence, and for quantum plas-mas in general. In this regard, it is natural to initially restrict the analysis to the quasilinear2elaxation of a one-dimensional quantum plasma in the electrostatic approximation.Some earlier works explored the similarities between the classical plasma and thequantum-mechanical treatment of a radiation field [25]–[29]. In those papers, one of theaims was to obtain information on the classical plasma through a quantum-mechanicallanguage. For instance, the relaxation of an instability can be viewed as the spontaneousemission of “quanta” of a radiation field described by some quasilinear-type equations. How-ever, the application of the quasilinear method to a truly quantum plasma governed by theWigner-Poisson system (i.e. the quantum analogue of the Vlasov-Poisson system) seems tobe restricted to the work of Vedenov [30]. Surprisingly, there has been no systematic analysisof the consequences of the quasilinear theory in the Wigner-Poisson case. The present workis a first attempt in this direction.This manuscript is organized in the following fashion. In Sect. 2, the quantum quasilinearequations are derived from the Wigner-Poisson system. In comparison with the classicalquasilinear equations, the quantum model exhibits a finite-difference structure. The basicproperties of the quantum quasilinear theory are discussed in Sect. 3, where we derive someconservation laws, as well as an appropriate H-theorem for the quasilinear equations. Theexistence of an entropy-like quantity is used to prove that the averaged Wigner functionrelaxes to a plateau, just like in the classical case. However, the distinctive feature of thequantum quasilinear equations is the existence of a transient periodic structure in velocityspace, as shown in Sect. 4. In Sect. 5, the theory is checked against numerical simulationsof the bump-on-tail and two-stream instabilities, with good agreement with the predictions.Conclusions are drawn in Sect. 6.
II. QUANTUM QUASILINEAR EQUATIONS
The quasilinear equations for the Wigner-Poisson system were derived long ago by Ve-denov [30], without fully exploring their consequences. For completeness, the derivationprocedure is reproduced here. The Wigner equation reads ∂f∂t + v ∂f∂x = Z dv ′ K ( v − v ′ , x, t ) f ( x, v ′ , t ) , (1)where K ( v − v ′ , x, t ) = − iem π ¯ h Z dλ e im ( v − v ′ ) λ/ ¯ h [ φ ( x + λ/ , t ) − φ ( x − λ/ , t )] . (2)3ere, f ( x, v, t ) is the Wigner pseudo-distribution in one spatial dimension, with position x ,velocity v , and time t . Also, ¯ h = h/ π is the scaled Planck’s constant, e is the absolutevalue of the electron charge, and m is the electron mass. The electrostatic potential φ ( x, t )satisfies the Poisson equation, ∂ φ∂x = eε (cid:18)Z dvf − n (cid:19) , (3)where ε is the vacuum permittivity and n a fixed, neutralizing ionic background. Periodicboundary conditions are assumed, with periodicity length L . Accordingly, for any quantity A = A ( x, v, t ), the spatial average is h A ( x, v, t ) i = 1 L Z L/ − L/ dxA ( x, v, t ) . (4)In particular, it is useful to define F ( v, t ) = h f ( x, v, t ) i and to restrict to h φ ( x, t ) i = 0.The quasilinear theory proposes a perturbation solution of the form f = F ( v, t ) + f ( x, v, t ) , φ = φ ( x, t ) , (5)for small f and φ . After averaging the Wigner equation, taking into account that h φ ( x, t ) i = 0, we obtain from (1) ∂F∂t = iem π ¯ h Z dλdv ′ e im ( v − v ′ ) λ/ ¯ h (cid:28)(cid:20) φ (cid:16) x − λ , t (cid:17) − φ (cid:16) x + λ , t (cid:17)(cid:21) f ( x, v ′ , t ) (cid:29) . (6)In the quasilinear theory, the right-hand side of Eq. (6) is evaluated using the results of thelinear theory. This implies, in particular, that the mode coupling effects are not taken intoaccount. Notice that F changes slowly, since ∂F/∂t is a second order quantity. Thus, for toostrong damping or instability, the quasilinear theory is no longer valid. Also, the trappingeffect can be included only in the framework of a fully nonlinear theory. Generally speaking,the conditions of validity of the quasilinear theory are still the subject of hot debates at theclassical level [31].We now introduce the spatial Fourier transformsˆ f k ( v, t ) = 1 √ π Z L/ − L/ dx e − ikx f ( x, v, t ) , (7)ˆ φ k ( t ) = 1 √ π Z L/ − L/ dx e − ikx φ ( x, t ) , (8)4ith the corresponding inverse transforms f ( x, v, t ) = √ πL X k e ikx ˆ f k ( v, t ) , (9) φ ( x, t ) = √ πL X k e ikx ˆ φ k ( t ) , (10)where k = 2 πn/L , n = 0 , ± , ± , ... Linearizing the Wigner equation, Fourier transformingit in space and Laplace transforming it in time, we obtainˆ f k ( v, t ) = em ˆ φ k ( t )2 π ¯ h ( ω k − kv ) Z dλ dv ′ e im ( v − v ′ ) λ/ ¯ h ( e ikλ/ − e − ikλ/ ) F ( v ′ , t ) . (11)In Eq. (11), since we are interested only in the long-lived collective oscillations, the initialperturbation f ( x, v,
0) was neglected. Finally, ω k stands for the allowable frequency modes,obtained from the the well-known [32] quantum dispersion relation D ( k, ω ) = 1 − ω p n Z L dv F ( v )( ω k − kv ) − ¯ h k / m = 0 , (12)which is assumed to be adiabatically valid, where L denotes the Landau contour, and ω p =( n e /mε ) / is the electron plasma frequency; for brevity, we omit to write out the secondargument t of F from here and onward.Since f and φ are real, we have the parity propertiesˆ f , − k ( v, t ) = ˆ f ∗ k ( v, t ) , ˆ φ , − k ( t ) = ˆ φ ∗ k ( t ) , ω − k = − ω ∗ k , (13)which will be used in the remainder of this paper. By defining ω k = Ω k + iγ k , (14)where Ω k and γ k are real, the following properties also holdΩ − k = − Ω k , γ − k = γ k . (15)By using Eqs. (11) and (13) into (6), and carrying out straightforward calculations, weobtain ∂F∂t = 2 πie L ¯ h X k =0 | ˆ φ k ( t ) | (cid:20) F ( v + ¯ hk/m ) − F ( v ) ω k − kv − ¯ hk / m + F ( v − ¯ hk/m ) − F ( v ) ω k − kv + ¯ hk / m (cid:21) . (16)5quation (16) can be put in a convenient form by using the parity properties (13)–(15) andthat | γ k | is small so that γ k / [(Ω k − kv ± ¯ hk / m ) + γ k ] ≃ πδ (Ω k − kv ± ¯ hk / m ), where δ is Dirac’s delta function. Hence we express Eq. (16) as ∂F∂t = 4 π mω p n L ¯ h X k> ˆ ε k ( t ) k nh F (cid:16) v + ¯ hkm (cid:17) − F ( v ) i δ (cid:16) Ω k − kv − ¯ hk m (cid:17) + h F (cid:16) v − ¯ hkm (cid:17) − F ( v ) i δ (cid:16) Ω k − kv + ¯ hk m (cid:17)o , (17)in which the growth rate does not appear explicitly, and where we have defined the spectraldensity of the electrostatic field fluctuations asˆ ε k ( t ) = ε L k | ˆ φ k ( t ) | . (18)The presence of the delta functions in Eq. (17) emphasizes the fact that only particlessatisfying the resonance condition Ω k − kv ± ¯ hk / m = 0 are taken into account, whereas F ( v, t ) remains unchanged in the non-resonant region of velocity space.The time variation of the spectral density is given in the same way as in the Vlasov-Poisson case, i.e. ∂ ˆ ε k ∂t = 2 γ k ˆ ε k . (19)Equations (17) and (19) are the quasilinear equations for the Wigner-Poisson system. Inthe formal classical limit ¯ h →
0, they reduce to the well-known quasilinear equations for theVlasov-Poisson system.
III. PROPERTIES OF THE QUANTUM QUASILINEAR EQUATIONS
Taking velocity moments of Eq. (17), several conservation laws can be easily derived.For instance, we obtain the conservation of the number of particles, ddt Z dv F = 0 , (20)and of the linear momentum ddt Z dv mv F = 0 , (21)where the dispersion relation (12) and the parity properties of the spectral density have beenused. In addition, the total energy is also invariant ddt Z dv mv F + 2 πL X k ˆ ε k ! = 0 . (22)6ntermediate steps to derive Eq. (22) require the use of the parity properties as well as thesecond quasilinear equation (19). In addition, a useful approximation is to consider Ω k ≃ ω p for k >
0, jointly with γ k ≃ πmω p n ¯ hk (cid:20) F (cid:18) ω p k + ¯ hk m (cid:19) − F (cid:18) ω p k − ¯ hk m (cid:19)(cid:21) . (23)In order to gain insight into the asymptotic behavior of F ( v, t ), it is interesting to lookfor an entropy-like quantity. Following Ref. [33], we consider the quantity R F dv , which,from Eq. (17), can be proved to obey the equation ddt Z dvF = − π mω p n L ¯ h X k> ˆ ε k ( t ) k (cid:20) F (cid:18) Ω k k + ¯ hk m (cid:19) − F (cid:18) Ω k k − ¯ hk m (cid:19)(cid:21) ≤ , (24)where the last inequality follows since all terms in the right-hand side are non-positive. Thisresult constitutes a sort of H-theorem for the averaged distribution function.The time derivative of the non-negative quantity in the left-hand side of Eq. (24) isalways non-positive, so that asymptotically we have ddt R dvF →
0, and F (cid:18) Ω k k + ¯ hk m (cid:19) − F (cid:18) Ω k k − ¯ hk m (cid:19) = 0 , (25)for all wavenumbers where the spectral density is non-zero. Equation (25) also resemblesthe basic equation obtained when applying the Nyquist method to the stability analysis ofthe Wigner-Poisson system [34].Equation (25) is a finite-difference-like version of the classical plateau condition [ F ′ ( v ) =0] in the region where the spectral density is not zero. The finite-difference structure ofEq. (25) favors the appearance of oscillations in velocity space, which are not present in theclassical case. These oscillations are analyzed in detail in the next section. IV. TRANSIENT QUANTUM OSCILLATIONS IN VELOCITY SPACE
In the derivation at the end of Sect. 3, it was implicitly assumed that the spectral densityis not zero in a broad region of momentum space. In contrast, let us see what happens inthe idealized situation where the spectral density ˆ ε k is strongly peaked at a single mode K . Denoting the associated spectral density by ˆ ε ( t ) and using the growth rate (23), the7uantum quasilinear equations read ∂F∂t = mLω p n ¯ h ˆ ε ( t ) nh F (cid:16) v + ¯ hKm (cid:17) − F ( v ) i δ (cid:16) ω p − Kv − ¯ hK m (cid:17) (26)+ h F (cid:16) v − ¯ hKm (cid:17) − F ( v ) i δ (cid:16) ω p − Kv + ¯ hK m (cid:17)o , and d ˆ ε ( t ) dt = πmω p n ¯ hK ˆ ε ( t ) (cid:20) F (cid:18) ω p K + ¯ hK m (cid:19) − F (cid:18) ω p K − ¯ hK m (cid:19)(cid:21) , (27)where, for simplicity, the approximation Ω K = ω p was also adopted.A particular class of stationary solutions of Eqs. (26)-(27) is given by any function F ( v )that is a periodic in velocity space, with period ¯ hK/m : F ( v + ¯ hK/ m ) − F ( v − ¯ hK/ m ) = 0 . (28)Assuming F ( v ) ∼ exp( iαv ) in Eq. (28), with α to be determined, we easily obtain thecharacteristic equation sin( α ¯ hK/ m ) = 0. Hence, the general (exact) equilibrium solutionis the linear combination F ( v ) = a + ∞ X n =1 a n cos (cid:18) πnvλ v (cid:19) + ∞ X n =1 b n sin (cid:18) πnvλ v (cid:19) , (29)where a n , b n are arbitrary real constants. Notice the singular character of the quantum oscil-lations, whose “wavelength” of the fundamental mode ( n = 1) in velocity space, λ v = ¯ hK/m ,tends to zero as ¯ h →
0. The solution given by Eq. (29) represents periodic oscillations invelocity space. However, this is necessarily a transient solution that cannot be sustainedfor long times. Indeed, strictly speaking, Eq. (29) applies only at the resonance, which isa set of measure zero on the velocity axis. The generation of harmonics with wavenumbers k = K (which is forbidden in the quasilinear theory) would ultimately lead to a broad energyspectrum.When many wavenumbers are present, the simple periodic solution given in Eq. (28) doesnot hold anymore, because different values of k induce different velocity-space wavelengths λ v . Only the solution F ( v ) = const . holds independently of k . Therefore, we expect that aplateau will eventually appear on a finite region in velocity space. Nevertheless, Eq. (29)provides an estimate for the characteristic oscillation length of the averaged Wigner functionin velocity space, near resonance. 8 IG. 1: Initial velocity distribution used in the simulations of the bump-on-tail instability, bothfor the Vlasov and Wigner cases.
The above arguments also suggest that monochromatic waves are the best candidates fordisplaying such quantum oscillations. In contrast, if the energy spectrum is broad from thevery start, the formation of a plateau is likely to be favored over the appearance of peri-odic oscillations. In the following section, these predictions will be compared to numericalsimulations of the Wigner-Poisson system.
V. NUMERICAL SIMULATIONS
We have performed numerical simulations of the Vlasov and Wigner equations using aphase-space code based on a splitting method [35]. In the Wigner equation, the accelerationterm (2) is a convolution product in velocity space and is therefore calculated numericallyby Fourier transforming it in velocity space.To study the differences in the nonlinear evolution of the Wigner and Vlasov equations,we have simulated the well-known bump-on-tail instability, whereby a high-velocity beamis used to destabilize a Maxwellian equilibrium. We use the initial condition f = (1 + δ )( n / √ πv th )[0 . − v / v th ) + 0 . − v − . v th ) /v th )], where δ represents randomfluctuations of order 10 − that help seed the instability (see Fig. 1). Here v th = p k B T e /m is the electron thermal speed. We use periodic boundary conditions with spatial period L = 40 πλ De , where λ De = v th /ω p is the Debye length. Three simulations were performed,with different values of the normalized Planck constant, defined as H = ¯ hω p /mv th : H = 0(Vlasov), H = 1, and H = 2. 9 IG. 2: Simulations of the Wigner-Poisson and Vlasov-Poisson systems, at time ω p t = 200, for H =0 (Vlasov, left frame), H = 1 (middle frame), and H = 2 (right frame). Initially monochromaticspectrum. Top panels: electron distribution function f ( x, v ) in phase space. Bottom panels:spatially averaged electron distribution function F ( v ) in velocity space. In order to highlight the transient oscillations in velocity space, we first perturb theabove equilibrium with a monochromatic wave having λ De k = 0 .
25 (i.e., a wavelength of8 πλ De ). Figures 2 shows the results from simulations of the Wigner-Poisson and Vlasov-Poisson systems. In both simulations, due to the bump-on-tail instability, electrostaticwaves develop nonlinearly and create periodic trapped-particle islands (electron holes) withthe wavenumber k = 0 .
25. The theory described in the previous sections predicts theformation of velocity-space oscillations in the Wigner evolution, which should be absent inthe classical (Vlasov) simulations. This is the case in the results presented in Fig. 2, wherethe oscillations are clearly visible.In order to estimate their wavelength, a zoom on the spatially averaged electron distri-bution function F ( v ) is shown in Fig. 3. According to the quasilinear theory, the velocitywavelength should be equal to λ v = ¯ hk/m for the fundamental mode with n = 1 [see Eq.(29)]. In our units, this yields λ v /v th = Hkλ De , which is equal to 0.25 for H = 1 and to 0.5for H = 2. The wavelengths observed in the simulations are slightly smaller: λ v /v th ≃ . IG. 3: Zoom on the spatially averaged electron distribution function F ( v ) in velocity space forthe same cases as in Fig. 2. and 0.35 for H = 1 and H = 2, respectively. However: (i) the oscillations are absent inthe Vlasov case, as expected, (ii) the order of magnitude of the wavelength is correct, and(iii) the wavelength is proportional to H , in accordance with the quasilinear theory. Theslight discrepancy in the observed value of λ v may have at least two origins. First, spatialwavenumbers different from 0 .
25 can be excited due to the nonlinear mode coupling. Second,other modes with n > . ≤ k ≤ . ω p t = 500 the Wigner solution still displays11 IG. 4: Simulations of the Wigner-Poisson and Vlasov-Poisson systems, for ω p t = 500, for H = 0(Vlasov, left panel) and H = 1 (Wigner, right panel). Initially broad wavenumber spectrum. Toppanels: electron distribution function f ( x, v ) in phase space. Bottom panels: spatially averagedelectron distribution function F ( v ) in velocity space. some oscillatory behavior in velocity space, which is absent in the Vlasov evolution.Another set of simulations were performed for the case of a two-stream instability. Theinitial distribution function is composed of two Maxwellians with thermal speed v th , eachcentered at v = ± v th (see Fig. 5). Only the fundamental mode of the system, with thewavenumber K = 0 . λ − De , is excited, and it grows exponentially due to instability. Threesimulations were performed, with different values of the normalized Planck constant: H = 0(Vlasov), H = 1, and H = 2. The expected oscillation wavelength in velocity space is λ v = ¯ hK/m . For the three simulated cases, it should take the values λ v = 0, λ v = 0 . v th ,and λ v = 0 . v th . A zoom of the averaged Wigner function F ( v, t ) around the velocity v = 0is plotted in Fig. 6 for the three cases, at time ω p t = 110. The velocity-space oscillationsare absent from the Vlasov simulation. In the Wigner cases, their wavelength is rather closeto the theoretical value; in particular, it appears to grow with the scaled Planck constant,12 IG. 5: Initial velocity distribution used in the simulations of the two-stream instability, both forthe Vlasov and Wigner cases. as expected from the theory. As in the bump-on-tail case, the oscillations tend to disappearover longer times.
VI. CONCLUSION
In this paper, the quasilinear theory for the Wigner-Poisson system was revisited. Con-servation laws and the asymptotic solution were established. Distinctive quantum effectsappear in the form of a transient oscillatory behavior in velocity space. Such quantum ef-fects are favored when the energy spectrum is restricted to few modes. For longer times, theplasma tends to become classical – due to the spatial harmonic generation and the modecouplings – at least as far as the averaged Wigner function F ( v, t ) is concerned. Thus,just as in the classical case, F ( v, t ) evolves asymptotically toward a plateau in the resonantregion of velocity space. It would be interesting to investigate whether monochromaticityenhances quantum effects in plasmas in general, which would represent a useful property foran experimental validation of the quantum plasma models. Acknowledgments
This work was partially supported by the Alexander von Humboldt Foundation and by13
IG. 6: Zoom of the averaged Wigner distribution F ( v ) for the two-stream instability at time ω p t = 110. Top frame: H = 0 (Vlasov); middle panel: H = 1; bottom panel: H = 2. the Swedish Research Council. [1] F. Haas, G. Manfredi, and M. R. Feix, Phys. Rev. E , 2763 (2000).[2] G. Manfredi and F. Haas, Phys. Rev. B , 075316 (2001).[3] M. Marklund and G. Brodin, Phys. Rev. Lett. , 025001 (2007).
4] P. K. Shukla and B. Eliasson, Phys. Rev. Lett. , 245001 (2006); S. A. Khan and W. Masood,Phys. Plasmas , 062301 (2008).[5] Q. Haque and H. Saleem, Phys. Plasmas , 064504 (2008).[6] F. Haas and M. Lazar, Phys. Rev. E , 046404 (2008).[7] F. Haas, Phys. Plasmas , 022104 (2008).[8] L. N Tsintsadze and P. K. Shukla, J. Plasma Phys. , 431 (2008).[9] F. Haas, P. K. Shukla, and B. Eliasson, Nonlinear saturation of the Weibel instability in a denseFermi plasma , J. Plasma Phys. , in press, 2008; doi: doi: 10.1017/S0022377808007368.[10] A. Bret, Phys. Plasmas , 084503 (2007).[11] V. Bychkov, M. Modestov, and M. Marklund, Phys. Plasmas , 032309 (2008).[12] P. K. Shukla and B. Eliasson, Phys. Rev. Lett. , 036801 (2008).[13] L. Wei and Y. N. Wang, Phys. Rev. B , 193407 (2007).[14] G. Manfredi and P.-A. Hervieux, Appl. Phys. Lett. , 061108 (2007).[15] G. Manfredi and P.-A. Hervieux, Phys. Rev. Lett. , 190404 (2006).[16] S. Ali, W. M. Moslem, I. Kourakis, and P. K. Shukla, New J. Phys. , 023007 (2008).[17] P. K. Shukla and L. Stenflo, Phys. Plasmas , 044505 (2006).[18] G. Brodin, M. Marklund, and G. Manfredi, Phys. Rev. Lett. , 175001 (2008).[19] M. Marklund, B. Eliasson, and P. K. Shukla, Phys. Rev. E , 067401 (2007); P. K. Shukla,Phys. Lett. A , 312 (2007).[20] D. Shaikh and P. K. Shukla, Phys. Rev. Lett. , 125002 (2007); New J. Phys. , 083007(2008).[21] A. A. Vedenov, E. P. Velikhov, and R. Z. Sagdeev, Nucl. Fusion , 82 (1961).[22] Yu. A. Romanov and G. F. Filippov, Sov. Phys. JETP , 87 (1961).[23] A. A. Vedenov, E. P. Velikhov, and R. Z. Sagdeev, Nucl. Fusion Suppl. , 465 (1962).[24] W. E. Drummond and D. Pines, Nucl. Fusion Suppl. , 1049 (1962).[25] D. Pines and J. R. Schrieffer, Phys. Rev. , 804 (1962).[26] N. Matsudaira, Phys. Fluids , 539 (1966).[27] V. Arunasalam, Phys. Rev. , 102 (1966).[28] V. Arunasalam, Phys. Rev. A , 1353 (1973).[29] B. N. Breizman and J. Weiland, Eur. J. Phys. , 222 (1986).[30] A. A. Vedenov, Dokl. Akad. Nauk SSSR , 334 (1962).
31] G. Laval and D. Pesme, Plasma Phys. Control. Fusion , A239 (1999).[32] Yu. L. Klimontovich and V. P. Silin, in Plasma Physics , ed. J. Drummond (McGraw-Hill, NewYork, 1961).[33] G. Manfredi and M. R. Feix, Phys. Rev. E , 4665 (2000).[34] F. Haas, G. Manfredi, and J. Goedert, Phys. Rev. E , 026413 (2001).[35] N. Suh, M. R. Feix, and P. Bertrand, J. Comput. Phys. , 403 (1991).[36] W. L. Kruer, J. M. Dawson, R. N. Sudan, Phys. Rev. Lett. , 838 (1969).[37] M. Albrecht-Marc, A. Ghizzo, T. W. Johnston, T. R´eveill´e, D. Del Sarto, and P. Bertrand,Phys. Plasmas , 072704 (2007)., 072704 (2007).