The General Solution to Vlasov Equation and Linear Landau Damping

Deng Zhou Institute of Plasma Physics, Chinese Academy of Sciences Hefei 230031, P. R. China

ABSTRACT

A general solution to linearized Vlasov equation for an electron electrostatic wave in a homogeneous unmagnetized plasma is derived. The quasi-linear diffusion coefficient resulting from this solution is a continuous function of at ( ) = 0 in contrast to that derived from the traditional Vlasov treatment. The general solution is also equivalent to the Landau’s treatment of the plasma normal oscillations,휔 and퐼푚 hence휔 leads to the well-known Landau damping.

Linear electron plasma waves in a collisionless plasma can be obtained by solving the linearized Vlasov equation together with the Poisson equation, which was first treated by 1 Vlasov . The dispersion relation is given by / ( , ) = 1 + = 0 2 (1) 푒 +∞ 휕푓0 휕푣 where and are respectively the frequency and the wave number，other symbols are 퐷 휔 푘 푚푘휖0 ∫−∞ 휔−푘푣 푑푣 obvious. ( , ) is usually called the plasma dielectric function. From Eq. (1), we can get the usual휔 Langmuir푘 wave. However, it is inadequate to treat all the effects of thermal particles. 퐷We휔 notice푘 from Eq. (1) that there is a singularity at = / in the integral and ( , ) is not a continuous function of at ( ) = 0. To overcome these insufficiencies, Landau solved the Vlasov푣 equation휔 푘 as an initial-value 퐷problem휔 푘 and introduced the deformed integration휔 퐼푚 route휔 in the plasma dielectric function to make sure that ( , ) is an analytical function in the whole complex plane of . Landau’s treatment leads to the famous landau damping2, which was later demonstrated in experiments3. The introduction퐷 휔 푘 on the two kinetic treatments of plasma waves can be휔 found in most plasma textbooks, such as Ref.[4]. An overview by Ryutov summarized the studies of Landau damping by the end of twentieth century5. The Landau damping derived by Landau is a pure achievement of applied mathematics. On it’s physical explanation different people have different points of view ( see, for example, Drummond6 and references therein ). On a weak nonlinear level, a quasi-linear approach is adopted to describe the interaction between waves and particles7,8. The averaged background particle distribution may experience diffusion in the velocity space due to the wave-particle interaction. Taking the one dimensional electrostatic case as an example, one gets distribution evolution equation = (2) 0 0 with the휕 푓quasi휕-linear휕푓 diffusion coefficient 휕푡 휕푣 풟 휕푣 (3) ( ) 훾 where ( ) is the real2 (imaginary)2 component of . Eq. (3) is derived from the Vlasov’s 풟 ∝ Ω−푘푣 +훾 solution for unstable modes, i. e. > 0. When the resonance broadening disappears Ω 훾 휔 훾 one meets a difficulty in extending (3) to the < 0 cases. We note that in the limit 0, Eq. (3) reduces to ( ) ( ) (4) 훾 훾 → where ( ) denotes sign of . The resonance broadening disappears and a discontinuity풟 ∝ 푠푖푔푛 appears훾 휋훿 at Ω −=푘0푣. To overcome this problem, one has to turn to Landau’s treatment푠푖푔푛 to get훾 the distribution function.훾 It is difficult to get an explicit function of distribution since an inverse훾 Laplace transform is involved. To circumvent this problem9, Hellinger et al recently extended the linear solution to Vlasov equation for the < 0 case by adding a term involving a complex Dirac delta function. However, these authors didn’t realize that the extended solution is the real solution to Vlasov equation and is related훾 to linear Landau damping, and moreover Dirac delta function is not a well-defined function in complex plane. In the following, we derive a general solution to Vlasov equation for the electrostatic Langmuir waves and from this solution the quasi-linear coefficient is derived and shown to be continuous from growing to damped modes. We also show that the general solution is equivalent to the Landau treatment and leads to the famous landau damping. We consider the one dimentional electrostatic oscillation in a homogeneous unmagnetized plasma, the linearized Vlasov equation is + = (5) 1 1 1 0 The perturbation휕푓 휕푓 of푒퐸 electric휕푓 field and particle distribution is written in the normal mode 휕푡 푣 휕푥 푚 휕푣 form = ( ) (6) = −푖 (휔푡−푘푥 ) (7) 1 �푘 Eq. (6퐸) is then퐸 푒 reduced−푖 휔푡−푘푥 to 푓1 푓̂푘푒 + = (8) �푘 0 The general form of solution푒퐸 휕푓 to Eq. (8) is −푖휔푓̂푘 푖푘푣푓̂푘 푚 휕푣 = + , ( ) ( , ) (9) 푒퐸�푘 휕푓0 1 where ( ) is any function of , and , and ̂푘 , 푚푘 휕푣 푣−휔⁄푘 휔 푘 ⁄ 푓 −푖 � 푆 푣 훥 푣 휔 푘 � ( / ) ( 휔, 푘 ) = ( ) + ( / ) ( ) + ( ) + 푆 푣 휔 푘 푣 ! 2 (10) ′ −푖훾 푘 ′′ Δ 푣 휔⁄푘 훿 푣 − Ω⁄푘 −푖훾 푘 훿 푣 − Ω⁄푘 2 훿 푣 − Ω⁄푘 ⋯ Setting , ( ) = 0, we recover the Vlasov treatment and get the dispersion relation Eq. (1). To prove that Eq. (9) is the solution of Eq. (8), one needs to demonstrate 휔 푘 ( 푆 )푣( , ) = 0 (11) This relation is obvious for = 0. For 0, we have the following integrals for any smooth푣 − 휔function⁄푘 Δ 푣 휔(⁄푘) ( ) ( ) ( ,훾 ) =훾 ≠( / ) ( ) (12a) 퐹 푣 +∞ ∫−∞ 퐹 푣 푣 − 휔⁄푘 δ 푣 Ω⁄푘 푑푣 − 푖훾 푘 퐹 Ω⁄푘 ( ) ( )( / ) ( ) = ( / ) ( ) ( / ) ( ) (12b)+∞ ′ 2 ′ −∞ ∫ 퐹 푣 푣 − 휔⁄푘 −푖훾 푘 훿 푣 − Ω⁄푘 푑푣 푖훾 푘 퐹 Ω⁄푘 − 푖훾 푘 퐹 Ω⁄푘

( ) ( )( / ) ( )( ) = ! +∞ 1 ( / ) ( ) ( / ) 푛( )푛 −∞ ( ) ( ) (12c) ∫( )푛푛! 퐹 푣 푣 − 휔⁄푘 −푖훾! 푛+푘1 훿 푣 − Ω⁄푘 푑푣 푖훾 푘 푛−1 푖훾 푘 푛 푛−1 퐹 Ω⁄푘 − 푛 퐹 Ω⁄푘 Combining (10) (12a-c), we obtain ( ) ( ) ( , ) = 0 (13) Since+ ∞( ) can be any smooth function, then from the basic theorem of function theory, −∞ Eq. (11)∫ is퐹 verified.푣 푣 − 휔 ⁄푘 Δ 푣 휔⁄푘 푑푣 The 퐹form푣 of ( , ) is superficially the Taylor expansion of the complex Dirac delta function ( ). However, it is impossible to define a proper complex Dirac delta function ( Δ) 푣which휔⁄푘 is analytical at = 0 since we know that a complex function analytical at whole훿 푣 complex− 휔⁄푘 plane is only a constant. Now we need훿 푧 to determine , ( ).푧 Without loss of generality, we set , ( ) = 0 for > 0 and derive the value for 0 through the continuity of density perturbation 휔 푘 휔 푘 at = 0. 푆 푣 푆 푣 The 훾density perturbation is given by 훾 ≤ 훾 = + , ( ) ( , ) (14) +∞ 푒퐸�푘 휕푓0 1 The integration route is from to + in -space. In Eq. (14), the principal value of 푛�푘 −푖 ∫−∞ 푚푘 휕푣 �푣−휔⁄푘 푆휔 푘 푣 훥 푣 휔⁄푘 � 푑푣 integral is taken if a singularity lies on the integration route. For > 0, we get from (14)− ∞ ∞ 푣 = + 2 (15) 훾 푒퐸�푘 휕푓0 1 푒퐸�푘 휕푓0 where 푘 denotes푅 the integration route of the semi-circle on the upper half complex 푛� 푖 ∫퐶 푚푘 휕푣 푣−휔⁄푘 푑푣 휋 푚푘 휕푣 �푣=휔⁄푘 plane from + to , as indicated in Fig. 1. For < 0, the density perturbation is given 푅 by 퐶 ∞ −∞ 훾 ( ) = + ( ) + + (16) 2 ! 2 3 푒퐸�푘 휕푓0 1 푒퐸�푘 휕푓0 휕 푓0 푖훾 휕 푓0 2 3 푘 퐶푅 = 2 < 0 휔 휔 휔 ( 0+) = One푛� obtains푖 ∫ 푚푘 휕푣 푣 −for휔⁄ 푘 푑푣 − 푖푆from푚푘 �(15)휕푣 � 푣and=푘 (16)푖훾 by휕 the푣 � 푣requirement=푘 2 휕푣 of� 푣=푘 ∙∙∙� ( 0 ). The same procedure yields = for = 0. In summary, we get the 푘 general solution푆 to 휋푖Eq. (8) given훾 by Eq. (9) and (10) with 푛� 훾 → 푛�푘 훾 → − 0 > 0 푆 휋푖 훾 , ( ) = = 0 (17) 2 <훾0 In the weak non푆휔 푘-linear푣 level� 휋푖 the averaged훾 background particle distribution changes slowly due to the wave-particle interaction,휋푖 훾 which is described through a quasi-linear approach. Keeping to the second order of perturbation in Vlasov equation and taking a space averaging in large space scale, one obtains the distribution evolution = ∗ (18) 0 1 1 where휕푓 ( 푒퐸 휕)푓 denotes the real ( imaginary ), bracket denotes space averaging, and the 휕푡 푅푒 〈 푚 휕푣 〉 superscript star denotes complex conjugate. Inserting푅푒 (6)퐼푚 (7) and (9) into (18), we get the diffusion equation, Eq. (2), for distribution with the diffusion coefficient , ( ) ( , ) 푓 0 = + 푒 2 2 1 푆휔 푘 푣 훥 푣 휔⁄푘 풟 = �푚� �퐸�푘� 퐼푚 �푘푣−휔 + 푘( � ) ( ) + (19) 2 ( ) 2 푒 2 훾 푆 훾 ′′ For simplicity we have푘 kept only2 2 one wave number while the real diffusion is contributed �푚� �퐸� � � Ω−푘푣 +훾 푖푘 �훿 푣 − Ω⁄푘 − �푘� 훿 푣 − Ω⁄푘 ⋯ �� from all the excited modes. It is obvious that the diffusion coefficient given by (19) is continuous as 0 ±, which is ( 0 +) = ( 0 ) ( ) (20) 훾 →

풟 훾 → 풟 훾 → − ∝ 휋훿 푘푣 − Ω The general solution, Eq. (9), can lead to linear Landau damping. Substituting the perturbation of the electric field and the distribution into the one dimensional Poisson equation = (21) 1 we obtain휕퐸 the dispersion+∞ relation 휖0 휕푥 −푒 ∫−∞ 푓1 푑푣 1 + ( ) ( , ) 2 , (22) 휔푝푒 +∞ 휕푓0 1 2 휔 푘 where− 푛0푘 =∫−∞ 휕푣 �푣is− 휔the⁄푘 usual푆 Langmuir푣 훥 푣 휔 frequency⁄푘 � 푑푣 and is the background plasma 2 푛0푒 density.휔 푝푒 � 휖0푚 푛0 Substituting (10) in (22) yields 1 = 0 2 2 (23) 휔푝푒 +∞ 휕푓0 1 휔푝푒 휕푓0 2 2 휔 0 −∞ 0 This is exactly− 푛 푘the∫ plasma휕푣 푣 −dielectric휔⁄푘 푑푣 − function푆 푛 푘 휕푣 one�푣= 푘obtains from Landau’s treatment by adopting a modified Landau integration contour in complex -plane. Hence, the general solution to Vlasov equation is equivalent to the Landau’s solution in the treatment of plasma normal modes and leads to the well-known Landau damping.푣 In a recent work10, Wesson reexamined the problem of Landau damping. He solved Vlasov equation separately for > 0 and < 0 using real variables. The two cases are related through the continuity of density perturbation at 0. Our present treatment is equivalent to Wesson’s since we훾 also solve Vlasov훾 equation separately for > 0 and < 0 and also use the continuity condition to relate the two훾 → cases. To show this, we take the case < 0 as an example. Like Wesson’s treatment, the density perturbation훾 is separated훾 into two parts, and . is the contribution from the basic thermal distribution훾 , which is treated푏 as a Dirac푤 delta푏 function, i. e. = ( ). Hence 1 1 1 푛 푛 푛 1 = 0 0 +∞ 푓 푛 훿 푣 푏 푒퐸�푘 휕푓0 푛 1 =−푖 � 푑푣 −∞ ( ) ( ) +∞푚푘푒퐸�푘 휕휕푣푓0 푣푣−−Ω휔⁄푘⁄+푘푖γ⁄푘 2 2 = −∞ 푚푘 휕푣 푣−Ω⁄푘 + γ+⁄푘 −푖 ∫ 푑푣 ( ) 푒퐸�푘 +∞ 휕푓0 1 +∞ 휕푓0 푖γ⁄푘 2 = 푚푘 −∞ 휕푣 푣−Ω⁄푘 − ∞ 휕푣 푣 − Ω ⁄ 푘(24) −푖 �(∫ ) ( )푑푣 ∫ 푑푣� 푒퐸�푘 푛0 2푖푛0γ⁄푘 where we have neglected the term ( ) in the2 denominate3 of the second line since −푖 푚푘 � Ω⁄푘 − Ω⁄푘 � . is contributed from the particles2 in resonance with the wave. Taking to be γ⁄푘 0 a constant,푤 we have 휕푓 훾 ≪ Ω 푛1 휕푣

1 = + ( , ) +∞ 푤 푒퐸�푘 휕푓0 푛 1 − 푖=� � 푆훥 푣 휔⁄푘+� 푑푣( , ) (25) −∞ ( ) ( ) 푒푚푘퐸�푘 휕푓휕푣0 푣 −+∞휔⁄푘푣−Ω⁄푘+푖γ⁄푘 2 2 Ω −∞ Changing to the coordinate system−푖 푚푘 travelling휕푣 �푣=푘 ∫with �the푣− Ωwave⁄푘 + velocity,γ⁄푘 푆훥one푣 obtains휔⁄푘 � 푑푣 = + ( , ) ( ) 푤 푒퐸�푘 휕푓0 +∞ 푣+푖γ⁄푘 1 휔 2 2 푛 −푖 푚푘 휕푣 �푣= −∞ �푣 + γ⁄푘 푆훥 푣 휔⁄푘 � 푑푣 = 푘 ∫ 2 + 2 (26) 푒퐸�푘 휕푓0 휕 푓0 휔 2 휔 Substitution of (24) (26)− into푖 푚푘 the�푖휋 Poisson휕푣 �푣=푘 equation− 훾 휋 휕 푣(21)�푣= yields푘 ⋯ respectively� from the real and the imaginary parts = 3 (27) 휋Ω 휕푓0 2 휔 0 and 훾 2푛 푘 휕푣 �푣=푘 = = (28) 2 푛0푒 They are exactlyΩ 휔 the푝푒 same� 휖 0as푚 those given by Wesson. In summary, we have presented the general form of solution to Vlasov equation for the electrostatic plasma wave, shown by Eqs. (9) (10) and (17). From this solution, one can get the quasi-linear diffusion coefficients valid for both damping and growing modes. The solution can also lead to the well-known Landau damping and it is equivalent to Wesson’s recent explanation of Landau damping using real variables.

ACKNOWLEDGEMENT

This work is supported by National Natural Science Foundation of China under Grant No. 11175213.

1A. Vlasov, J. Phys. (USSR) 9, 25 (1945). 2L. D. Landau, J. Phys. (USSR) 10, 25 (1946). 3J. Malmberg and C. Wharton, Phys. Rev. Lett. 17, 175 (1966). 4R. Goldston and P. Rutherford, Introduction to Plasma Physics (Institute of Physics Publishing, 1995). 5D. Ryutov, Plasma phys. Contr. Fusion 41, A1 (1999). 6W. Drummond, Phys. Plasmas 11, 552 (2004). 7A. Vedenov, E. Velikhov, and R. Sagdeev, Nucl. Fusion 1, 82 (1961). 8W. Drummond and D. Pines, Nucl. Fusion Suppl. 3, 1049 (1962). 9P. Hellinger and P. Travnicek, Phys. Plasmas 19, 062307 (2012). 10J. Wesson, Phys. Plasmas 22, 022519 (2015).

Fig. 1 The integration route used in calculation of density perturbation.