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
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

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.

H. J. LEE, M. Y. SONG 558
5. Acknowledgements
The authors thank Dr. Y. D. Jung at Hanyang University
for many valuable discussions.
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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
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[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,
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