2 Boltzmann's Equation and Vlasov's Equation
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8.2 Boltzmann’s Equation and Vlasov’s Equation
and
165
qi = x i
(8.8)
∂H
dxi
=
= vi ,
dt
∂ pi
(8.9)
d pi
∂H
=
=−
dt
∂xi
k
( pk − q A k ) ∂ A k
∂φ
−q
.
q
m
∂xi
∂xi
(8.10)
Consequently (8.5) becomes
3
∂F
+
∂t
vk
i=k
∂F
+q
∂xk
3
3
vk
i=1
k=1
∂F
=
∂ pi
∂ Ak
∂φ
−
∂xi
∂xi
δF
δt
.
(8.11)
coll
By use of (8.7) and (8.8), independent variables are transformed from (qi , pi , t) to
(x j , v j , t) and
∂v j (xk , pk , t)
1
= δi j ,
∂ pi
m
∂v j (xk , pk , t)
q ∂Aj
=−
,
∂xi
m ∂xi
∂v j (xk , pk , t)
q ∂Aj
=−
.
∂t
m ∂t
We denote F(xi , pi , t) = F(xi , pi (x j , v j , t), t) ≡ f (x j , v j , t)/m 3 . Then we have
m 3 F(xi , pi , t) = f (x j , v j (xi , pi , t), t) and
m3
m3
∂
∂
F(x h , ph , t) =
f (x j , v j (x h , ph , t), t) =
∂ pi
∂ pi
j
∂ f ∂v j
∂f 1
=
,
∂v j ∂ pi
∂vi m
∂
∂
∂f
F(x h , ph , t) =
f (xi , vi (x h , ph , t), t) =
+
∂xk
∂xk
∂xk
∂f
=
+
∂xk
m3
∂f
∂vi
i
−q
m
i
∂ f ∂vi
∂vi ∂xk
∂ Ai
∂xk
∂
∂
∂f
F(x h , ph , t) =
f (xi , vi (x h , ph , t), t) =
+
∂t
∂t
∂t
i
∂f
∂vi
−q
m
Accordingly (8.11) is reduced to
∂f
+
∂t
i
∂f
∂vi
−q
m
∂ Ai
+
∂t
vk
k
∂f
+
∂xk
i
∂f
∂vi
−q
m
∂ Ai
∂xk
∂ Ai
.
∂t
166
8 Boltzmann’s Equation
+
vk
i
∂f
+
∂t
=
k
vk
k
δf
δt
q ∂f
=
m ∂vi
∂ Ak
∂φ
−
∂xi
∂xi
∂f
+
∂xk
−
i
∂ Ai
−
∂t
δf
δt
vk
k
,
coll
∂ Ai
+
∂xk
vk
k
∂ Ak
∂φ
−
∂xi
∂xi
q ∂f
m ∂vi
.
coll
Since the following relation is hold
vk
k
we have
∂ Ak
=
∂xi
∂f
+
∂t
vk
k
vi
i
∂ Ai
+ (v × (∇ × A))i =
∂xk
∂f
+
∂xi
i
vk
k
∂ Ai
+ (v × B)i .
∂xk
q
∂f
(E + v × B)i
=
m
∂vi
δf
δt
.
(8.12)
coll
This equation is called Boltzmann’s equation. The electric charge density ρ and the
electric current j are expressed by
ρ=
q
f dv1 dv2 dv3 ,
(8.13)
q
v f dv1 dv2 dv3 .
(8.14)
i,e
j=
i,e
Accordingly Maxwell equations are given by
1
∇·E=
q
f dv,
(8.15)
0
∂E
+
∂t
q
∇×E=−
∂B
,
∂t
1
∇×B=
μ0
0
∇ · B = 0.
v f dv,
(8.16)
(8.17)
(8.18)
When the plasma is rarefied, the collision term (δ f /δt)coll may be neglected. However, the interactions of the charged particles are still included through the internal
electric and magnetic field which are calculated from the charge and current densities by means of Maxwell equations. The charge and current densities are expressed
by the distribution functions for the electron and the ion. This equation is called
collisionless Boltzmann’s equation or Vlasov’s equation.
8.3 Fokker–Planck Collision Term
167
8.3 Fokker–Planck Collision Term
When Fokker–Planck collision term is adopted as the collision term of Boltzmann’s
equation, this equation is called Fokker Planck equation. In the case of Coulomb
collision, scattering into small angles has a large cross-section and a test particle
interacts with many field particles at the same time, since the Coulomb force is a
long-range interaction. Consequently it is appropriate to treat Coulomb collision
statistically. Assume that the velocity v of a particle is changed to v + Δv after the
time Δt by Coulomb collisions; denote the probability of this process by W (v, Δv).
Then the distribution function f (r, v, t) satisfies
f (r, v, t + Δt) =
f (r, v − Δv, t)W (v − Δv, Δv)d(Δv).
(8.19)
In this process the state at t + Δt depends only on the state at t. Such a process (i.e.,
one independent of the history of the process) is called the Markoff process. The
change of the distribution function by virtue of Coulomb collision is
δf
δt
Δt = f (r, v, t + Δt) − f (r, v, t).
coll
Taylor expansion of the integrand of (8.19) gives
f (r, v − Δv, t)W (v − Δv, Δv)
= f (r, v, t)W (v, Δv) −
∂( f W )
1 ∂2( f W )
Δvr +
Δvr Δvs + · · ·
∂vr
2 ∂vr ∂vs
r
rs
(8.20)
From the definition of W (v, Δv), the integral of W is
W d(Δv) = 1.
Introducing the quantities
W Δvd(Δv) ≡ Δv t Δt,
W Δvr Δvs d(Δv) ≡ Δvr Δvs t Δt,
we find
δf
δt
= −∇v ( Δv t f ) +
coll
1 ∂2
( Δvr Δvs t f ).
2 ∂vr ∂vs
(8.21)
168
8 Boltzmann’s Equation
This term is called the Fokker–Planck collision term and Δv t , Δvr Δvs t are called
Fokker–Planck coefficients. W Δvr Δvs d(Δv) is proportional to Δt. Δvr is the sum
of Δvri , which is, the change of vr due to the ith collisions during Δt, i.e., Δvr =
j
i
i
i Δvr , so that Δvr Δvs =
i
j Δvr Δvs . When the collisions are statistically
j
independent, statistical average of Δvri Δvs
t
W Δvr Δvs d(Δv) =
(i = j) is zero and
Δvri Δvsi d(Δv).
i
This expression is proportional to Δt.
The Fokker–Planck equation can be expressed in the form [1]
F
∂f
+ v · ∇r f + ∇v f + ∇v · J = 0,
∂t
m
where
Ji = Ai f −
Di j
j
Ai = Δvi
t
−
Di j =
1
2
j
(8.22)
∂f
,
∂v j
∂
Δvi Δv j t ,
∂v j
1
Δvi Δv j t .
2
The tensor D is called the diffusion tensor in the velocity space and A is called the
coefficient of dynamic friction. For convenience we consider the components of J
parallel and perpendicular to the velocity v of the test particle. When the distribution
function of field particles is isotropic, we find
J = −D ∇ f + A f,
J ⊥ = −D⊥ ∇⊥ f.
(8.23)
A is parallel to v and the diffusion tensor becomes diagonal. When the distribution
function of field particles is Maxwellian, A and D are given by [1]
mv D = −T ∗ A,
(8.24)
(qq ∗ )2 n ∗ ln Λ Φ1 (b∗ v)
,
b∗2 v 2
8πε20 vm 2
(8.25)
(qq ∗ )2 n ∗ ln Λ
Φ1 (b∗ v)
∗
Φ(b
.
v)
−
2b∗2 v 2
8πε20 vm 2
(8.26)
D =
D⊥ =
8.3 Fokker–Planck Collision Term
169
q ∗ , n ∗ , b∗ , and T ∗ are those of field particles (b∗2 ≡ m ∗ /2T ∗ ) and q, m, and v are
those of test particles. Φ(x) and Φ1 (x) are
Φ(x) =
x
2
π 1/2
exp(−ξ 2 )dξ,
0
Φ1 (x) = Φ(x) −
2x
exp(−x 2 ).
π 1/2
When x > 2, then Φ(x) ≈ Φ1 (x) ≈ 1.
When the distribution function of field particles is generally given by f ∗ (v ∗ ), the
Fokker–Planck coefficients Δvi t , Δvi Δv j t are given by [2]
Δvi
= −L 1 +
t
Δvi Δv j
t
ui ∗ ∗ ∗
f (v )dv
u3
m
m∗
δi j
ui u j
− 3
u
u
=L
where
u = v − v∗,
u ≡ |u|,
L≡
(8.27)
f ∗ (v ∗ )dv ∗ ,
(8.28)
(eZ ∗ e)2 ln Λ
.
4π 20 m 2
(8.29)
v, m are the velocity and the mass of test particles respectively and v ∗ , m ∗ are the
velocity and the mass of field particles respectively. There are following relations
about u;
ui j ≡
ui u j
δi j
∂2u
− 3 ,
=
∂vi ∂v j
u
u
j
∂u i j
=
∂v j
j
∂u i j
=−
∂v ∗j
j
j
∂
∂v j
δi j
ui u j
− 3
u
u
= −2
ui
,
u3
∂u i j
ui
= 2 3.
∂v j
u
The coefficient of dynamic friction A and the diffusion tensor Di j are given by [1]
Ai = Δvi
t
−
= −L 1 +
m
= −L ∗
m
1
2
m
m∗
j
∂
Δvi Δv j
∂v j
t
ui ∗ ∗ ∗ L
f (v )dv −
u3
2
ui ∗ ∗ ∗
f (v )dv ,
u3
j
∂
∂v j
δi j
ui u j
− 3
u
u
f ∗ (v ∗ )dv ∗
(8.30)
170
8 Boltzmann’s Equation
ui u j
δi j
1
L
f ∗ (v ∗ )dv ∗
Δvi Δv j t =
− 3
2
2
u
u
L
L ∂2
=
u f ∗ (v ∗ )dv ∗ ,
u i j f ∗ (v ∗ )dv ∗ =
2
2 ∂vi ∂v j
Di j =
(8.31)
We define E iv (v) and G(v) by
E iv (v) ≡
G(v) ≡
ui ∗ ∗ ∗
f (v )dv ,
u3
(8.32)
u f ∗ (v ∗ )dv ∗ ,
(8.33)
then we have
Ai = −L
Di j =
L
2
m v
E ,
m∗ i
u i j f ∗ (v ∗ )dv ∗ =
⎛
and
Ji (v) = ⎝ Ai f (v) −
j
δf
δt
(8.34)
L ∂2
G(v),
2 ∂v i ∂v j
⎞
∂ f (v) ⎠
,
Di j
∂v j
∗i
= −∇v · J
⎞
∂ ⎝m v
1
E (v) f (v) +
∂vi m ∗ i
2
Since
E iv =
1
2
(8.36)
coll
⎛
=L
(8.35)
u i j f ∗ (v ∗ )dv ∗
j
∂u i j ∗ ∗ ∗
1
∗ f (v )dv = −
∂v j
2
j
ui j
j
∂ f (v) ⎠
.
∂v j
(8.37)
∂ f ∗ (v) ∗
dv ,
∂v ∗j
so that Fokker–Planck collision term is reduced to
δf
δt
=
coll
Lm
2
∗i j
∂
∂vi
−
f ∗ (v ∗ ) ∂ f (v)
f (v) ∂ f ∗ (v ∗ )
+
u i j dv ∗ .
m∗
∂v ∗j
m
∂v j
(8.38)
This equation is called by Landau collision integral [1, 2].
Next the function H (v) is defined by
H (v) =
f ∗ (v ∗ ) ∗
dv .
u
(8.39)
8.3 Fokker–Planck Collision Term
171
Then we have
E iv (v) = −
∂ H (v)
=
∂vi
ui ∗ ∗ ∗
f (v )dv .
u3
(8.40)
The (8.37) and (8.35) are reduced to
δf
δt
=L
coll
∗i
=L
∗i
=L
∗i
From
⎡
m
∂ ⎣
−
∂vi
m∗
⎛
m
∂ ⎝
−
∂vi
m∗
⎛
∂ ⎝
− 1+
∂vi
j ∂u i j /∂v j
∂H
f (v) +
∂vi
∂H
f (v) +
∂vi
m
m∗
j
j
∂H
f (v) +
∂vi
∂2 G
∂vi ∂v j
1
2
⎤
∂ f (v) ⎦
∂v j
⎞
1
∂2 G
∂ ∂2 G
f (v) −
f (v)⎠
∂vi ∂v j
2
∂v j ∂vi ∂v j
j
⎞
∂2 G
1 ∂
f (v) ⎠ ,
(8.41)
2 ∂v j ∂vi ∂v j
1 ∂
2 ∂v j
j
= −2u i /u 3 , we used the following equation to derive (8.41)
−
1
2
j
∂ ∂2 G
∂H
=−
∂v j ∂vi ∂v j
∂vi
(1 + m/m ∗ )H (v), G(v) are called Rosenbluth Potential [3].
8.4 Quasi Linear Theory of Evolution in Distribution
Function
It has been assumed that the perturbation is small and the zeroth-order terms do not
change. Under these assumption, the linearized equations on the perturbations are
analyzed. However if the perturbations grow, then the zeroth-order quantities may
change and the growth rate of the perturbations may change due to the evolution of
the zeroth order quantities. Finally the perturbations saturate (growth rate becomes
zero) and shift to steady state. Let us consider a simple case of B = 0 and one
dimensional electrostatic perturbation (B 1 = 0). Ions are uniformly distributed.
Then the distribution function f (x, v, t) of electrons obeys the following Vlasov
equation;
∂f
e ∂f
∂f
+v
− E
= 0.
(8.42)
∂t
∂x
m ∂v
Let the distribution function f be divided into two parts
f (x, v, t) = f 0 (v, t) + f 1 (x, v, t)
(8.43)
where f 0 is slowly changing zeroth order term and f 1 is the oscillatory 1st order
term. It is assumed that the time derivatives of f 0 is the 2nd order term. When (8.43)
is substituted into (8.42), the 1st and the 2nd terms satisfy following equations;
172
8 Boltzmann’s Equation
∂ f1
e ∂ f0
∂ f1
+v
= E
,
∂t
∂x
m ∂v
(8.44)
∂ f0
e ∂ f1
= E
.
∂t
m ∂v
(8.45)
f 1 and E are expressed by Fourier integrals;
f 1 (x, v, t) =
1
(2π)1/2
f k (v) exp(i(kx − ω(k)t))dk,
(8.46)
E(x, t) =
1
(2π)1/2
E k exp(i(kx − ω(k)t))dk.
(8.47)
Since f 1 and E are real, f −k = f k∗ , E −k = E k∗ , ωr (−k) = −ωr∗ (k), γ(−k) = γ(k),
( ω(k) = ωr (k) + iγ(k)). The substitution of (8.46) and (8.47) into (8.44) yields
f k (v) =
e
m
i
ω(k) − kv
Ek
∂ f0
.
∂v
(8.48)
If (8.47) and (8.48) are substituted into (8.45), we find
∂ f 0 (v, t)
e
=
∂t
m
∂ 1
E k exp(i(k x − ω(k )t))dk
∂v 2π
∂ f 0 (v, t)
i
E k exp(i(kx − ω(k)t))dk
×
ω(k) − kv
∂v
2
means statistical average and
∂
∂ f 0 (v, t)
=
∂t
∂v
e
m
e
=
m
Dv (v) =
2
2
1
2π
1
2πw
Dv (v)
∂ f 0 (v, t)
,
∂v
(8.49)
i Ek Ek
exp i(k + k)x − i(ω(k ) + ω(k))t dk dk
ω(k) − kv
i Ek Ek
exp i(k + k)x − i(ω(k ) + ω(k))t .
dx dkdk
ω(k) − kv
The notation of the statistical average is substituted by the integral of x. w is the
range of x integral.
Since (1/2π) exp i(k + k)xdx = δ(k + k), Dv is reduced to
e
m
e
=
m
Dv (v) =
2
∞
2
−∞
∞
−∞
i(|E k |2 /w) exp(2γ(k)t)
dk
ωr (k) − kv + iγ(k)
γ(k)(|E k |2 /w) exp(2γ(k)t)
dk.
(ωr (k) − kv)2 + γ(k)2
(8.50)
8.4 Quasi Linear Theory of Evolution in Distribution Function
When |γ(k)|
173
|ωr (k)|, the diffusion coefficient in velocity space is
e
m
e
=
m
Dv (v) =
2
2
π
(|E k |2 /w) exp(2γ(k)t) δ(ωr (k) − kv)dk
π
(|E k |2 /w) exp(2γ(k)t)
|v|
ω/k=v
.
(8.51)
Equation (8.49) is the diffusion equation in the velocity space. When the distribution
function of electrons are given by the profile shown in Fig. 10.2b, in which the
positive gradient of v ∂ f /∂v > 0 exists near v1 . Then waves with the phase velocity
of ω/k ≈ v1 grow due to Landau amplification and the amplitude of |E k | increases.
The diffusion coefficient Dv in velocity space becomes large and anomalous diffusion
takes place in velocity space. The positive gradient of ∂ f /∂v near ∼ v1 decreases
and finally the profile of the distribution function becomes flat near v ∼ v1 .
Let us consider the other case. When a wave is externally exited (by antenna) in
a plasma with Maxwellian distribution function as is shown in Fig. 10.2a, diffusion
coefficient Dv at v = ω/k is increased. The gradient of the distribution function near
v = ω/k becomes flat as will be seen in Fig. 11.7 of Chap. 11.
References
1. D.V. Sivukhin, in Reviews of Plasma Physics, ed. by M.A. Leontovich (Consultant Bureau, New
York, 1966), vol. 4, p. 93
2. B.A. Trubnikov, in Reviews of Plasma Physics, ed. by M.A. Leontovich (Consultant Bureau,
New York, 1965), vol. 1, p. 105
3. M.N. Rosenbluth, W.M. MacDonald, D.L. Judd, Phys. Rev. 107, 1 (1957)
Chapter 9
Waves in Cold Plasmas
Abstract Dispersion relation of waves in cold plasma is given by (9.20) in Sect. 9.1.
CMA diagram of a two components plasma (electron and ion) is shown in Fig. 9.5.
Alfven wave, ion cyclotron wave and fast wave, lower hybrid wave, upper hybrid
wave, electron cyclotron wave are described in Sect. 9.4. The condition of electrostatic wave and its dispersion relation (9.92) are described in Sect. 9.5.
A plasma is an ensemble of an enormous number of moving ions and electrons
interacting with each other. In order to describe the behavior of such an ensemble,
the distribution function was introduced in Chap. 7; and Boltzmann’s and Vlasov’s
equations were derived with respect to the distribution function. A plasma viewed as
an ensemble of a large number of particles has a large number of degrees of freedom;
thus the mathematical description of plasma behavior is feasible only for simplified
analytical models.
In Chap. 3, statistical averages in velocity space, such as mass density, flow velocity, pressure, etc., were introduced and the magnetohydrodynamic equations for these
averages were derived. We have thus obtained a mathematical description of the
magnetohydrodynamic fluid model; and we have studied the equilibrium conditions,
stability problems, etc., for this model in Chaps. 3–7. Since the fluid model considers
only average quantities in velocity space, it is not capable of describing instabilities
or damping phenomena, in which the profile of the distribution function plays a significant role. The phenomena which can be handled by means of the fluid model are
of low frequency (less than the ion or electron cyclotron frequency); high-frequency
phenomena are not describable in terms of it.
In this chapter, we will focus on a model which allows us to study wave phenomena while retaining the essential features of plasma dynamics, at the same time
maintaining relative simplicity in its mathematical form. Such a model is given by
a homogeneous plasma of ions and electrons at 0 K in a uniform magnetic field.
In the unperturbed state, both the ions and electrons of this plasma are motionless.
Any small deviation from the unperturbed state induces an electric field and a timedependent component of the magnetic field, and consequently movements of ions
and electrons are excited. The movements of the charged particles induce electric and
magnetic fields which are themselves consistent with the previously induced small
perturbations. This is called the kinetic model of a cold plasma. We will use it in this
© Springer-Verlag Berlin Heidelberg 2016
K. Miyamoto, Plasma Physics for Controlled Fusion, Springer Series on Atomic,
Optical, and Plasma Physics 92, DOI 10.1007/978-3-662-49781-4_9
175
176
9 Waves in Cold Plasmas
chapter to derive the dispersion relation which characterizes wave phenomena in the
cold plasma.
Although this model assumes uniformity of the magnetic field, and the density and
also the zero temperature, this cold plasma model is applicable for an nonuniform,
warm plasma, if the typical length of variation of the magnetic field and the density
is much larger than the wavelength and the phase velocity of wave is much larger
than the thermal velocity of the particles.
It is possible to consider that the plasma as a medium of electromagnetic wave
propagation with a dielectric tensor K . This dielectric tensor K is a function of the
magnetic field and the density which may change with the position. Accordingly
plasmas are in general an nonuniform, anisotropic and dispersive medium.
When the temperature of plasma is finite and the thermal velocity of the particles is
comparable to the phase velocity of propagating wave, the interaction of the particles
and the wave becomes important. A typical interaction is Landau damping, which
will be explained in Chap. 10. The general mathematical analysis of the wave in hotplasma will be discussed. The references [1–3] describe the plasma wave in more
detail.
9.1 Dispersion Equation of Waves in Cold Plasma
In an unperturbed cold plasma, the particle density n and the magnetic field B 0
are both homogeneous in space and constant in time. The ions and electrons are
motionless.
Now assume that the first-order perturbation term exp i(k · r − ωt) is applied.
The ions and electrons are forced to move by the perturbed electric field E and the
induced magnetic field B 1 . Let us denote velocity by v k , where the suffix k indicates
the species of particle (electrons, or ions of various kinds). The current j due to the
particle motion is given by
n k qk v k .
(9.1)
j=
k
n k and qk are the density and charge of the kth species, respectively. The electric
displacement D is
D=
j=
+ P,
(9.2)
∂P
= −iω P
∂t
(9.3)
0E
where E is the electric intensity, P is the electric polarization, and
constant of vacuum. Consequently D is expressed by
D=
0E
+
i
j≡
ω
0K
· E.
0
is the dielectric
(9.4)