### Paper Menu >>

### Journal Menu >>

Journal of Modern Physics, 2013, 4, 555-558 http://dx.doi.org/10.4236/jmp.2013.44077 Published Online April 2013 (http://www.scirp.org/journal/jmp) Causality and Collisionless Damping in Plasma Hee J. Lee1*, Mi Young Song2 1Department of Physics, Hanyang University, Seoul, Korea 2Plasma Technology Research Center, National Fusion Research Institute, Gunsan, Korea Email: *ychjlee@yahoo.com Received January 21, 2013; revised February 22, 2013; accepted March 2, 2013 Copyright © 2013 Hee J. Lee, Mi Young Song. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. ABSTRACT We derive the collisionless Landau damping in a plasma by satisfying the causal requirement that the susceptibility function of the plasma for time t < 0 should be nil. The causality condition should be satisfied by the susceptibility function of a plasma no matter what equations we employ to describe the plasma. Thus we conclude that the fundamen- tal reason of the collisionless damping can be traced to the causality. As an example, we derive the collisionless damp- ing of ion acoustic wave in a plasma by employing fluid equations. Keywords: Causality; Landau Damping; Ion Acoustic Wave 1. Introduction Landau damping discovered by L. Landau in his famous paper [1] is collisionless damping, i.e. damping irrelevant to particle-particle collisions in plasmas. Landau solved the Vlasov-Poisson equations as an initial value problem by means of Laplace transform. In the course of analysis, one has to deal with the singular integral in the expres- sion of the susceptibility function 2 ,d pe k 0 d1 d g v kvkv (1) where p e is the electron plasma frequency, 0 g v i is the zero order equilibrium velocity distribution function. In the velocity integral in Equation (1), ri en is a complex frequency and v is a real variable. If the imaginary frequcy i becomes zero, and thus lands on somewhere on the real v-axis, the velocity inte- gral becomes undefined since the integrand is singular at vk . Landau evaluated this singular integral along his celebrated Landau contour by invoking the mathe- matical argument of analytic continuation in the complex -plane. The Landau contour for the integral of Equation (1) consists of the principal value part along the real v-axis plus the infinitesimal semi-circle around the sin- gular point vk in a concave-down shape. There- fore, the value of the v-integral is the sum of the principal value and the value contributed by the infinitesimal semi- circle around the singular point vk which can be expressed in terms of -function. Thus, Equation (1) yields the real part ,kRe corresponding to the principal value of the integral as well as the imaginary part Im ,k obtained from the -function part. Depending upon the slope of the distribution function, Im ,k can give rise to either damping or growing of the plasma wave. In this work, we show that the Landau damping can be derived from the causal requirement that must be satis- fied for the dielectric permittivity function. The causal requirement is mathematically expressed by Equation (3), and should be satisfied always in electrodynamics no matter what way it is derived. Its physical meaning is that the response of the medium must follow the cause; the cause cannot be precedent to the effect. By enforcing this causality condition, the velocity integral in Equation (1) is shown to be equivalent to the integral along the Lan- dau contour. Recently Lim and Lee [2] derived the Landau damping from the Kramers-Kronig relations, and concluded that the fundamental reason of the collisionless damping of plasma waves can be traced to the causality. In this work, we derive the collisionless damping directly from the causal requirement without the aid of the Kramers- Kronig relations; thus, we show more directly that the collisionless damping is a consequence of the causality. It is emphasized that Im ,k is derived without the mathematical argument of the analytic continuation. As an example, we employ the fluid equations to derive the collisionless damping of an ion acoustic wave; this deri- *Corresponding author. C opyright © 2013 SciRes. JMP H. J. LEE, M. Y. SONG 556 vation clearly shows that the causal requirement is re- sponsible for the collisionless damping. 2. Evaluation of Susceptibility from the Causal Requirement In evaluation of plasma wave dispersion relation, one often encounters with the algebraic expression, the re- ciprocal of the Doppler-shifted frequency, 1 kv kv . This algebraic quantity is well defined if kv , but it should be more defined when because the de- nominator gives a singularity. In this case, the expression should be rephrased so that the singularity can be dealt with without ambiguity. We can use: Pk v 11 kv kv (2) where the symbol P denotes the principal value and is an undetermined constant. Equation (2) is correct be- cause of the two identities: 11, xaPxa 0xa x a In fact, Equation (2) was the mathematical motivation of the Van Kampen modes of plasma wave [3]. Here we use the causal requirement in electrodynamics to deter- mine the constant . Causality in electrodynamics means that the response must always follow the cause; the cause cannot be prece- dent to the effect. Therefore, the future susceptibility should be irrelevant to the present field (displacement). This causal notion leads to the mathematical condition expressed by [4] 0 for 0tt (3) where is the susceptibility of the medium. As an application, we consider the susceptibility of electron plasma (ions are assumed to be immobile) gov- erned by the Vlasov-Poisson equation as given by Equa- tion (1). Using Equation (2) in Equation (1), the sus- ceptibility function takes the form kv kv 2 0 d1 ,d d pe g kvP kv (4) In order to determine the constant , let us invert Equation (4): i 2 i 0 d ,,e 2π d de 2πd t pe kvt kt k g vP kv i e d t (5) The principal part integral is the step function [3,5]: i e diπwhen 0oriπwhen 0 xt Px tt x iπ (6) The causal requirement in Equation (3) is satisfied by , and therefore the plasma susceptibility in Equa- tion (4) becomes 2 0 d1 ,d iπ d pe g kvP kv kvkv ii ekx t (7) which agrees with the susceptibility evaluated along the Landau contour. Therefore we conclude that the colli- sionless damping of plasma waves is fundamentally due to the causal requirement. 3. Ion Landau Damping Derived from Fluid Equations An ion in the electric wave of a Fourier component with phasor is subject to the equation of motion i d, e d kx t i ve Ek tm (8) In Equation (8), the Lagrangian equation of motion, x xt 00 , the particle position at time t, and the sole independent variable is t. As the zero order solution, we use the unperturbed orbit x xvt where 0 x and 0 are the initial position and velocity, respectively, at the initial time . Then Equation (8) with the aforementioned boundary conditions is solved by v 0t 0 0 i i 0 0 e , 1e i kx kv t i e vt vEk mkv v (9) The terms 0 and 1 in the above equation are neces- sary to have the initial condition satisfied. However, even without those terms, the rest of the terms in Equation (9) satisfies Equation (8). Equation (9) is the particular solu- tion of the differential Equation (8). The the first order homogeneous solution is meant by 00 ii 0 e , i kx kvt i e vt Ek mkv ii ekxt (10) which corresponds to the Fourier component with phasor 00 . It is trivial to obtain Eulerian velocity corre- sponding to Equation (10); we simply put x xvt in Equation (10) to get ii 0 e ,, i kx t i e vxt Ek mkv (11) We immediately recognize that Equation (11) solves the Eulerian equation of motion Copyright © 2013 SciRes. JMP H. J. LEE, M. Y. SONG 557 ii ,e kx t e k 0 i vv vE txm v (12) where x and t are now independent. The above Lagran- gian consideration clarifies the meaning of 0 which is the initial velocity, and v is the perturbed velocity. Proceeding from the above preliminary consideration of fluid equation, we derive the collisionless damping of ion acoustic wave via the fluid equations. We need both electron and ion equations. The plasma at hand is con- sidered to be a group of ion beams; each beam is charac- terized by the initial velocity v0 in the background of Boltzmann-distributed electrons. Modeling a plasma as a group of beams with varying beam velocities was earlier adopted by Bohm and Gross [6]. The ion equations read 0 v i vve txm (13) 00 0 nvn v txx Nv Nv (14) where v0 is the zero order initial ion velocity, v is the perturbed ion velocity, 0 is the equilibrium ion number density of the v0-beam, n is the perturbation of the ion number density of the v0-beam, and is the electric potential E. We assume that the electrons are Boltzmann-distrib- uted in the background of the plasma. Thus, the perturb- ed electron number density in each beam can be written as 0 e eNv T e1 e e T e nN (15) The Poisson equation reads 00 e eNv T 0 2=4π v en v (16) Equations (13) and (14) yield, in terms of the Fourier amplitudes, 2 0 2 0 i Nv ek m kv 0 ,,nk v (17) Substituting Equation (17) into the Fourier transform- ed equation of Equation (16) and integrating over the distribution of the initial velocities 0 f v 000 dfvv by putting , we obtain the dispersion relation, Nv 00 2 0 d 1 1 0 pi D kk fv v kkv 2 22 ,1 , (18) where p i is the ion plasma frequency and the Debye length of the plasma is defined by 2 2 00 4πd De e f vv T Integrating by parts in Equation (18), the susceptibility is found to be 2 0 0 22 0 d d 1 ,d pi D f v kv kkvk (19) In order to enforce the causal requirement ,00kt , we take the steps parallel to Equations (3)-(7) by introducing Equation (2) for the quantity 0 in the above integral. We are led to the fol- lowing equation by the causal requirement. 1kv 2 00 22 00 , 1d1 d iπ d pi D k f vP kv kvkvk (20) The first term on the right side of Equation (20) comes from the Boltzmann-distributed electron density which replaces the electron Vlasov equation. Equation (20) agrees with the kinetic theory result. In summary, we derived the the susceptibility of ion acoustic wave via fluid equations and without resorting to the argument of analytic continuation; we used more direct condition of the causal requirement expressed by Equation (3). 4. Discussion Im We have shown that as obtained by the Lan- dau contour follows from the causal requirement express- ed by Equation (3), 00t 00t . This derivation of the collisionless damping makes it transparent that the cau- sality is responsible for the collisionless damping. The collisionless damping of plasma waves appears to be uni- versal because the causality prevails regardless of the way of describing plasmas. The causal requirement, , is mathemati- cally equivalent to the analyticity of in the upper half -plane; this analyticity is basic to the derivation of the Landau damping via the argument of the analytic continuation, as is read in standard text books. It can be also shown that the causality condition directly leads to the Kramers-Kronig relations [2]. We also showed that Landau damping can also be de- rived from the fluid equations by applying the causal requirement. Understanding the Landau damping in terms of causality might provide further insight for plasma electrodynamics. The Cerenkov radiation is emitted by a medium and is known to be the inverse process of Landau damping. The Cerenkov radiation may be also interpreted in the light of causality. Copyright © 2013 SciRes. JMP H. J. LEE, M. Y. SONG Copyright © 2013 SciRes. JMP 558 5. Acknowledgements The authors thank Dr. Y. D. Jung at Hanyang University for many valuable discussions. REFERENCES [1] L. Landau, “On the Vibrations of the Electronic Plasma,” Journal of Physics, Vol. 10, No. 1, 1946, pp. 25-34. [2] Y. K. Lim and H. J. Lee, “Causality, Kramers-Kronig Relations, and Landau Damping,” The Open Plasma Phy- sics Journal, Vol. 5, No. 1, 2012, pp. 36-40. [3] N. G. Van Kampen and B. U. Felderhof, “Theoretical Methods in Plasma Physics,” John Wiley, New York, 1967. [4] J. D. Jackson, “Classical Electrodynamics,” Wiley, New York, 1974. [5] G. Arfken, “Mathematical Methods for Physicists,” Aca- demic Press, Orlando, 1985, p. 415. [6] D. Bohm and E. P. Gross, “Theory of Plasma Oscillations. A. Origin of Medium-Like Behavior,” Physical Review, Vol. 75, No. 12, 1949, pp. 1851-1864. |