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# NIFS PROC 88

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```refractive
index N = \ufffd1/2 is larger inside the beam than outside. Then, the plasma acts as an optical \ufb01ber,
focusing the beam to a small diameter. By the ponderomotive force, intense laser beam with
Peta Watt (1015W) can bore a hole and reach to the core of high density fuel pellet in inertial
con\ufb01nement and heat electrons by the oscillating components in (3.62). This concept is called fast
ignition (refer to section 18.3).
References
[1] R. G. Littlejohn: Hamiltonian Formulation of Guiding Center Motion. Phys. Fluids 24, 1730 (1981).
R. G. Littlejohn: Variational Principles of Guiding Center Motion. J. Plasma Phys. 29, 111 (1983).
[2] A. I. Morozov and L. S. Solovev: Rev. of Plasma Physics 2, 201, (edited by M. A. Leontovich),
Consultant Bureau, New York (1966).
38
39
Ch.4 Velocity Space Distribution Function and
Boltzmann\u2019s Equation
A plasma consists of many ions and electrons, but the individual behavior of each particle can
hardly be observed. What can be observed instead are statistical averages. In order to describe the
properties of a plasma, it is necessary to de\ufb01ne a distribution function that indicates particle number
density in the phase space whose ordinates are the particle positions and velocities. The distribution
function is not necessarily stationary with respect to time. In sec.4.1, the equation governing the
distribution function f(qi, pi, t) is derived by means of Liouville\u2019s theorem. Boltzmann\u2019s equation
for the distribution function f(x,v, t) is formulated in sec.4.2. When the collision term is neglected,
Boltzmann\u2019s equation is called Vlasov\u2019s equation.
4.1 Phase Space and Distribution Function
A particle can be speci\ufb01ed by its coordinates (x, y, z), velocity (vx, vy, vz), and time t. More
generally, the particle can be speci\ufb01ed by canonical variables q1, q2, q3, p1, p2, p3 and t in phase
space. The motion of a particle in phase space is described by Hamilton\u2019s equations
dqi
dt
=
\u2202H(qj , pj, t)
\u2202pi
,
dpi
dt
= \u2212\u2202H(qj , pj, t)
\u2202qi
. (4.1)
When canonical variables are used, an in\ufb01nitesimal volume in phase space
\u394 = \u3b4q1\u3b4q2\u3b4q3\u3b4p1\u3b4p2\u3b4p3 is conserved according to Liouville\u2019s theorem, that is,
\u394 = \u3b4q1\u3b4q2\u3b4q3\u3b4p1\u3b4p2\u3b4p3 = const. (4.2)
Let the number of particles in a small volume of phase space be \u3b4N
\u3b4N = F (qi, pi, t)\u3b4q\u3b4p (4.3)
where \u3b4q = \u3b4q1\u3b4q2\u3b4q3, \u3b4p = \u3b4p1\u3b4p2\u3b4p3, and F (qi, pi, t) is the distribution function in phase space.
If the particles move according to the equation of motion and are not scattered by collisions, the
small volume in phase space is conserved. As the particle number \u3b4N within the small phase space
is conserved, the distribution function (F = \u3b4N/\u394) is also constant, i.e.,
dF
dt
=
\u2202F
\u2202t
+
3\u2211
i=1
(
\u2202F
\u2202qi
dqi
dt
+
\u2202F
\u2202pi
dpi
dt
)
=
\u2202F
\u2202t
+
3\u2211
i=1
(
\u2202H
\u2202pi
\u2202F
\u2202qi
\u2212 \u2202H
\u2202qi
\u2202F
\u2202pi
)
= 0. (4.4)
In the foregoing discussion we did not take collisions into account. If we denote the variation of F
due to the collisions by (\u3b4F/\u3b4t)coll, (4.4) becomes
\u2202F
\u2202t
+
3\u2211
i=1
(
\u2202H
\u2202pi
\u2202F
\u2202qi
\u2212 \u2202H
\u2202qi
\u2202F
\u2202pi
)
=
(
\u3b4F
\u3b4t
)
coll
. (4.5)
4.2 Boltzmann\u2019s Equation and Vlasov\u2019s Equation
Let us use the space and velocity-space coordinates x1, x2, x3, v1, v2, v3 instead of the canonical
coordinates. The Hamiltonian is
H =
1
2m
(p\u2212 qA)2 + q\u3c6, (4.6)
pi = mvi + qAi, (4.7)
39
40 4 Velocity Space Distribution Function · · ·
Fig.4.1 Movement of particles in phase space.
qi = xi (4.8)
and
dxi
dt
=
\u2202H
\u2202pi
= vi, (4.9)
dpi
dt
= \u2212\u2202H
\u2202xi
=
\u2211
k
(pk \u2212 qAk)
m
q
\u2202Ak
\u2202xi
\u2212 q \u2202\u3c6
\u2202xi
. (4.10)
Consequently (4.5) becomes
\u2202F
\u2202t
+
3\u2211
i=k
vk
\u2202F
\u2202xk
+ q
3\u2211
i=1
(
3\u2211
k=1
vk
\u2202Ak
\u2202xi
\u2212 \u2202\u3c6
\u2202xi
)
\u2202F
\u2202pi
=
(
\u3b4F
\u3b4t
)
coll
. (4.11)
By use of (4.7) (4.8), independent variables are transformed from (qi, pi, t) to (xj , vj , t) and
\u2202vj(xk, pk, t)
\u2202pi
=
1
m
\u3b4ij ,
\u2202vj(xk, pk, t)
\u2202xi
= \u2212 q
m
\u2202Aj
\u2202xi
,
\u2202vj(xk, pk, t)
\u2202t
= \u2212 q
m
\u2202Aj
\u2202t
.
We denote F (xi, pi, t) = F (xi, pi(xj, vj , t), t) \u2261 f(xj, vj , t)/m3. Then we have m3F (xi, pi, t) =
f(xj, vj(xi, pi, t), t) and
m3
\u2202
\u2202pi
F (xh, ph, t) =
\u2202
\u2202pi
f(xj, vj(xh, ph, t), t) =
\u2211
j
\u2202f
\u2202vj
\u2202vj
\u2202pi
=
\u2202f
\u2202vi
1
m
,
m3
\u2202
\u2202xk
F (xh, ph, t) =
\u2202
\u2202xk
f(xi, vi(xh, ph, t), t) =
\u2202f
\u2202xk
+
\u2211
i
\u2202f
\u2202vi
\u2202vi
\u2202xk
=
\u2202f
\u2202xk
+
\u2211
i
\u2202f
\u2202vi
(\u2212q
m
)
\u2202Ai
\u2202xk
m3
\u2202
\u2202t
F (xh, ph, t) =
\u2202
\u2202t
f(xi, vi(xh, ph, t), t) =
\u2202f
\u2202t
+
\u2211
i
\u2202f
\u2202vi
(\u2212q
m
)
\u2202Ai
\u2202t
.
40
4.3 Fokker-Planck Collision Term 41
Accordingly (4.11) is reduced to
\u2202f
\u2202t
+
\u2211
i
\u2202f
\u2202vi
(\u2212q
m
)
\u2202Ai
\u2202t
+
\u2211
k
vk
(
\u2202f
\u2202xk
+
\u2211
i
\u2202f
\u2202vi
(\u2212q
m
)
\u2202Ai
\u2202xk
)
+
\u2211
i
(\u2211
k
vk
\u2202Ak
\u2202xi
\u2212 \u2202\u3c6
\u2202xi
)
q
m
\u2202f
\u2202vi
=
(
\u3b4f
\u3b4t
)
coll
,
\u2202f
\u2202t
+
\u2211
k
vk
\u2202f
\u2202xk
+
\u2211
i
(
\u2212\u2202Ai
\u2202t
\u2212
\u2211
k
vk
\u2202Ai
\u2202xk
+
\u2211
k
vk
\u2202Ak
\u2202xi
\u2212 \u2202\u3c6
\u2202xi
)
q
m
\u2202f
\u2202vi
=
(
\u3b4f
\u3b4t
)
coll
.
Since the following relation is hold\u2211
k
vk
\u2202Ak
\u2202xi
=
\u2211
k
vk
\u2202Ai
\u2202xk
+ (v × (\u2207×A))i =
\u2211
k
vk
\u2202Ai
\u2202xk
+ (v ×B)i.
we have
\u2202f
\u2202t
+
\u2211
i
vi
\u2202f
\u2202xi
+
\u2211
i
q
m
(E + v ×B)i \u2202f
\u2202vi
=
(
\u3b4f
\u3b4t
)
coll
. (4.12)
This equation is called Boltzmann\u2019s equation. The electric charge density \u3c1 and the electric current
j are expressed by
\u3c1 =
\u2211
i,e
q
\u222b
fdv1dv2dv3, (4.13)
j =
\u2211
i,e
q
\u222b
vfdv1dv2dv3. (4.14)
Accordingly Maxwell equations are given by
\u2207 ·E = 1
\ufffd0
\u2211
q
\u222b
fdv, (4.15)
1
\u3bc0
\u2207×B = \ufffd0\u2202E
\u2202t
+
\u2211
q
\u222b
vfdv, (4.16)
\u2207×E = \u2212\u2202B
\u2202t
, (4.17)
\u2207 ·B = 0. (4.18)
When the plasma is rare\ufb01ed, the collision term (\u3b4f/\u3b4t)coll may be neglected. However, the
interactions of the charged particles are still included through the internal electric and magnetic
\ufb01eld 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\u2019s equation or Vlasov\u2019s equation.
4.3 Fokker-Planck Collision Term
When Fokker-Planck collision term is adopted as the collision term of Boltzmann\u2019s 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 \ufb01eld 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+\u394v
41
42 4 Velocity Space Distribution Function · · ·
after the time \u394t by Coulomb collisions; denote the probability of this process by W (v,\u394v). Then
the distribution function f(r,v, t) satis\ufb01es
f(r,v, t+\u394t) =
\u222b
f(r,v \u2212\u394v, t)W (v \u2212\u394v,\u394v)d(\u394v). (4.19)
In this process the state at t+\u394t depends only on the state at t. Such a process (i.e., one independent
of the history of the process) is called the Marko\ufb00 process. The change of the distribution function
by virtue of Coulomb collision is(
\u3b4f
\u3b4t
)
coll
\u394t = f(r,v, t+\u394t)\u2212 f(r,v, t).
Taylor expansion of the integrand of (4.19) gives
f(r,v \u2212\u394v, t)W (v \u2212\u394v,\u394v)
= f(r,v, t)W (v,\u394v)\u2212
\u2211
r
\u2202(fW )
\u2202vr
\u394vr +
\u2211
r s
1
2
\u22022(fW )
\u2202vr\u2202vs
\u394vr\u394vs + · · · .(4.20)
From the de\ufb01nition of W (v,\u394v), the intgral of W is\u222b
Wd(\u394v) = 1.
Introducing the quantities\u222b
W\u394vd(\u394v) = \u3008\u394v\u3009t\u394t,
\u222b
W\u394vr\u394vsd(\u394v) = \u3008\u394vr\u394vs\u3009t\u394t,
we \ufb01nd (
\u3b4f
\u3b4t
)
coll
= \u2212\u2207v(\u3008\u394v\u3009tf) +
\u2211 1
2
\u22022
\u2202vr\u2202vs
(\u3008\u394vr\u394vs\u3009tf). (4.21)
This term is called the Fokker-Planck collision term```