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69in b3496-ch01 page 1

Chapter 1
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Boltzmann equation
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1.1. Heuristic derivation of Boltzmann equation

The Boltzmann equation is of fundamental importance in the kinetic
theory of gases and plasmas. It expresses a mathematical description of the
distribution function f (r, v, t) of the particles at (r, v) in the phase space and
at time t. It takes the form of (1.14) below, and gives the rate of change of
f along the characteristics, the particle trajectory. The Boltzmann equation
becomes the Vlasov equation when we regard the change of f due to particle
collisions is zero in (1.14). The Vlasov fluid behaves like an incompressible
fluid in the 6-dimensional phase space, just as the Liouville ensemble behaves
likewise in the 6N -dimensional Γ-space. This Chapter may be read with
a quick consulting of Chapter 12. Boltzmann invented the collision term
in Section (1.2) below, which is his great accomplishment. The following
assumptions are made as a preliminary to the development of Boltzmann’s
equation:

(a) We assume that the state of the gas is described by a one-body

distribution function f (r, v, t).
(b) The density of particles is low enough that only two-body interactions
need to be considered, i.e., the range of inter-particle forces (r0 ) is much
smaller than the mean inter-particle distance.
(c) The duration of an encounter between two particles (interaction time)
is much smaller than the duration of free motion of the particles, i.e.,
r0 /vav ≡ tint  λ/vav where λ is the mean free path and vav is the mean
speed of the particles.
(d) The particles are assumed to be point centers of spherically symmetric
fields, so that the one-body distribution function depends only on the
position r, velocity v of the particles and time t. In case of exceptional
models for the particles, other variables, such as the angular velocity,
may be introduced.

1
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2 Fundamentals of Theoretical Plasma Physics

Let f (r, v, t) d3 rd3 v be the expected number of particles to be found in a

volume element d3 rd3 v of phase space about r and v at time t. The volume
element Δμ ≡ d3 rd3 v must be large enough to contain a sufficient number
of particles in order that probability concepts can be applied at all. Thus

1
f (r, v, t) = F (r, v, t)d3 r1 . . . d3 rN d3 v1 . . . d3 vN (1.1)
Δμ Δμ
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N
F (r, v, t) = δ(r − ri (t))δ(v − vi (t)) (1.2)
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i=1

where i is a particle index. Moreover, the changes in f will be observed over

a time Δt which is much larger than the interaction time tint = ro /vav .
We are concerned with developing an equation which determines the
temporal evolution of f given its value at some initial time t0 for all r and
v. By definition the total number of particles

N = f (r, v, t)d3 rd3 v (1.3)

where the integration is carried over the volume V in configuration space

occupied by the particles and over the accessible region of velocity space.
Furthermore, we define a density n

1
n(r, t) = f (r, v, t)d3 v (1.4)
V all v
with the configuration volume V vanishingly small but large enough to
contain many enough particles.
If we assume that those particles contained in d3 rd3 v do not interact
with each other, then at a time t + dt (dt  tint ) we expect these particles
to be in the volume element d3 r  d3 v  about r and v where
r = r + vdt + O(dt)2 (1.5)
v = v + adt + O(dt)2 (1.6)
where a is the acceleration suffered by the particles due to the fields that
may be applied by external means or those generated by the collective action
of of all the particles excluding those whose trajectories (internal) are under
examination. Thus
a = ae + ai (1.7)
where the subscripts e and i correspond to the cause of the acceleration. The
new volume element d3 r  d3 v  is related to the old volume element d3 rd3 v by
February 19, 2019 14:33 Fundamentals of Theoretical Plasma Physics 9.61in x 6.69in b3496-ch01 page 3

Boltzmann equation 3

the relation
 
3  3  r , v
d rd v =J d3 rd3 v (1.8)
r, v
where J is the Jacobian determinant of the transformation written out in
full as:
  
 ∂r1 ∂r2 ∂r3 
 0 
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0 0
 ∂r1 ∂r1 ∂r1 
 
  
 ∂r1 ∂r2  
 ∂r3

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 ∂r2 ∂r2 ∂r 2
0 0 0 
 
  
 ∂r1 ∂r2 ∂r3
  
 0 0 0 
 ∂r3 ∂r3 ∂r3
 
J = 
 ∂v1 ∂v2 ∂v3 
  
 0
 0 0
∂v1 ∂v1 ∂v1 

 
    
 0 ∂v 1 ∂v 2 ∂v 3 
 0 0 
 ∂v2 ∂v2 ∂v2 
 
    
 ∂v 1 ∂v 2 ∂v 3 
 0 0 0 
∂v3 ∂v3 ∂v3
This expression for the Jacobian assumes that the after-collision position of
a molecule is uncorrelated with the before-collision velocity of the particle,
which is the assumption of molecular chaos. Owing to this assumption, the
chain of hierarchy of distribution functions can be terminated at the one-
particle distribution function, and the Boltzmann equation becomes self-
closed with the collision term that is here derived and is expressed by the
Boltzmann distribution function itself. Making use of (1.5) and (1.6),
∂ai
J =1+ dt + O(dt)2 (1.9)
∂vi
where we note that the determinant J is equal to the product of the two
3 by 3 sub-determinants located at the upper-left and lower-right corners.
The upper-left determinant is unity. The lower-right determinant is diagonal
with elements
∂a1
1+ dt,
∂v1
and the like. Here the repeated indexes are summed over.
 
∂ai
d3 r  d3 v  = 1 + dt + O(dt)2 d3 rd3 v (1.10)
∂vi
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4 Fundamentals of Theoretical Plasma Physics

and further we have

 
δf
f (r , v , t + dt)d3 r  d3 v  = f (r, v, t)d3 rd3 v + d3 rd3 vdt (1.11)
δt c

which means that the same number of particles are in the new volume
element as in the old element except for those gained or lost by interaction
among the particles themselves denoted by the last term of (1.11). Expanding
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the left hand side (LHS) of (1.11) in a Taylor series about (r, v, t), we have
   
∂f ∂f dri ∂f dvi
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LHS(1.11) = f (r, v, t) + + + dt
∂t ∂ri dt ∂vi dt
 
3 3 ∂ai
× d rd vdt 1 + dt + O(dt)2 . (1.12)
∂vi
Thus
 
∂f ∂f dri ∂f dvi ∂ai δf
+ + +f dt = . (1.13)
∂t ∂ri dt ∂vi dt ∂vi δt c

As we restrict ourselves to forces that are either independent of particle

velocity or the Lorentz force e v × B, if they depend on velocity, ∂ai
∂vi = 0
always, and we finally have
 
∂f ∂f ∂f δf
+ vi + ai = . (1.14)
∂t ∂ri ∂vi δt c

1.2. Collision term ( δf )

δt c
Consider a two-body elastic collision. Denote the particle velocities before
and after collision as (v, v1 ) and (v , v1 ), respectively, where the primed
are after-collision velocities, which can be expressed in terms of the pre-
collision velocities if the force law governing the interaction is known. The
collision event is more conveniently described in the center-of-mass frame
of the two particles, in which the collision is equivalent to a deflection of
a fictitious particle of reduced mass (moving with the relative velocity)
from the scattering center. Figure 1.1 shows the geometry of the scattering
process. The incident particle approaches the scattering center with an
impact parameter b and is deflected by a scattering angle θ. The orbit of
the particle involves two angles (ϕ, θ) where ϕ is the azimuthal angle and
the polar angle θ corresponds to the scattering angle. The flux of particles
passing through an elemental area b db dϕ is
b db dϕ|v − v1 | f (r, v1 , t)d3 v1 (sec−1 ). (1.15)
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Boltzmann equation 5
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Fig. 1.1. Collision in the center-of-mass frame.

Then the number of collisions (per unit time) of the particles in the above
flux with the particles of velocity v is (1.15) × f (v, r, t)d3 v:
b db dϕ|v − v1 | f (r, v1 , t)d3 v1 f (v, r, t)d3 v.
Then integrating the above equation over all v1 and dividing by d3 v gives
the total rate of change of f (v, r, t) due to collisions which scatter v-particles
out of the range (v, v + dv):
    ∞  2π
δf 3
= d v1 db b dϕ|v − v1 | f (v, r, t)f (r, v1 , t). (1.16)
δt out 0 0

Note that |v − v1 | ≡ vR , the relative speed, is a function of the scattering

angle θ, which in turn is determined by b and ϕ if the force law is known. It
is customary to introduce the differential scattering cross section
dσ(vR , θ) = b db dϕ; θ = θ(vR , b).
For more about the scattering angle θ, §1.7 and Fig. 1.3 should be consulted.
If we go into the center of mass frame, θ is the angle between the pre and
post relative velocities, and is a function of b and the initial relative velocity.
Then (1.16) is written as
   
δf 3 dσ
= d v1 dΩ |v − v1 | f (v, r, t)f (r, v1 , t),
δt out dΩ
dΩ = sinθ dθ dϕ. (1.17)
The rate of scattering into the range (v, v + dv) can be calculated in the
same manner, giving
     
∂f 3  dσ  
= d v1 dΩ |v − v1 |f (r, v , t)f (r, v1 , t). (1.18)
∂t in dΩ
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6 Fundamentals of Theoretical Plasma Physics

In the above equation, v , v1 are functions of v, v1 because these are

the velocities of the interacting particles before and after the collision,
respectively. Since we have assumed an elastic collision the following
conservation relations hold,

mv + mv1 = mv + mv1 , (1.19)

m 2 m 2 m 2 m 2
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v + v1 = v + v1 . (1.20)
2 2 2 2
The collision may be thought of as a linear orthogonal transformation from
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(v, v1 ) to (v , v1 ), and we have

|v − v1 | = |v − v1 |.

Furthermore, for this transformation, the Jacobian is unity and therefore

d3 v1 d3 v  = d3 v1 d3 v.

Because of the symmetry of the interaction, we also have

   
dσ dσ
dΩ = dΩ .
dΩ dΩ
Collecting the above equations for the collision integral, the Boltzmann
equation becomes
 
∂f ∂f ∂f δf
+ vi + ai =
∂t ∂vi ∂vi δt c
where
„ « Z Z » –
δf dσ
= d3 v1 dΩ |v − v1 | f (r, v , t)f (r, v1 , t) − f (r, v, t)f (r, v1 , t)
δt c dΩ
(1.21)

The actual evaluation of this integral for small angle scattering will be
performed in §1.7. If the fluid contains several kinds of particles, then each
kind of particle has its own distribution function and we readily generalize
Boltzmann’s equation to
∂fα ∂fα ∂fα 
+ vi + ai = Cαβ (fα fβ ) (1.22)
∂t ∂ri ∂vi
β

where α and β refer to the particle species, Cαβ (fα fβ ) denotes the collision
operator and the summation includes α.
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Boltzmann equation 7

1.3. Chapman–Enskog solution of Boltzmann equation

In this section, we shall discuss the solution of the Boltzmann equation
for a simple gas (consisting of one kind of spherical molecules having only
the translational energy) which is not subject to any external forces. The
Chapman and Enskog treatment of the Boltzmann equation is known as the
method of moments.
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Summation invariant. Let Q be any particle property such that the sum
of the Q’s of two particles involved in collision remains unchanged in the
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collision, i.e., a summation invariant. Important examples of Q are 1, mv

(momentum), and mv 2 /2 (kinetic energy) (see Eqs. (1.19) and (1.20)).

Q(v) + Q(v1 ) = Q(v ) + Q(v1 ) (1.23)

Then ΔQ defined below vanishes. This is the property of a summation

invariant which is found useful in later discussion.
  
δf
ΔQ = Q(v) d3 v
δt c

dσ
= Q(v) f  f1 − f f1 g dΩ d3 v1 d3 v = 0. (1.24)
dΩ
In the above equation, we abbreviated

g = v − v1 , (1.25)
f = f (v, r, t), f1 = f (v1 , r, t), f  = f (v , r, t), f1 = f (v1 , r, t).

The proof of ΔQ = 0 goes as follows. Owing to the symmetry of the

integrand, we have

dσ
ΔQ = Q(v1 ) f  f1 − f f1 g dΩd3 v1 d3 v. (1.26)
dΩ
Adding Eqs. (1,24) and (1.26) gives

1 dσ
ΔQ = Q(v) + Q(v1 ) (f  f1 − f f1 ) g dΩd3 v1 d3 v. (1.27)
2 dΩ
Due to the symmetry of the integrand again, (1.27) can be written as

1 dσ
ΔQ = Q(v ) + Q(v1 ) − Q(v) − Q(v1 ) f f1 g dΩd3 v1 d3 v.
2 dΩ
(1.28)

Hence, ΔQ = 0.
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8 Fundamentals of Theoretical Plasma Physics

Exercise. In writing (1.28), we used specifically


dσ
Φ(va )fa (va )fb (vb )g dΩd3 va d3 vb
dΩ

dσ
= Φ(va )fa (va )fb (vb )g dΩd3 va d3 vb (1.29)
dΩ
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where Φ is a summation invariant. Prove this by using (1.19) and (1.20).

Let us consider a gas in equilibrium (steady state in time and uniform
in space). Then LHS of (1.14) vanishes and the distribution function should
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satisfy
  
δf
= d3 v1 dσg f  f1 − f f1 = 0 (1.30)
δt c
which is satisfied if and only if f  f1 = f f1 . Or
logf (v ) + logf (v1 ) = logf (v) + logf (v1 ).
Hence the quantity logf (v) is a summation invariant and it should be
expressible as a linear combination of the three summation invariants, 1,
mv, and mv 2 /2:
m
logf (v) = ma1 + a2 · mv − a3 v 2
2
where a1 , a2 , a3 are arbitrary constants. By completing the squares, one
obtains
   
a22 m a2 2
logf (v) = m a1 + − a3 v − .
2a3 2 a3
By a simple transformation, f (v) can be put into the form
m 32 m(v − V)2
f (v) = N exp − (1.31)
2πT 2T
where N, V, T are related with a1 , a2 , a3 and defined by
  
3 1 3 3 1 m 2 3 m
N = f d v, V − vf d v, T = v f d v − V 2.
N 2 N 2 2
(1.32)
Equation (1.31) is a moving Maxwell–Boltzmann distribution function. If the
gas has no directional preference (V = 0) and is distributed isotropically,
m 3
2 mv2
f (v) = N exp − . (1.33)
2πT 2T
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Boltzmann equation 9

[In the above and following, temperature T should be read as kT where

k is the Boltzmann constant]. Thus the particles in equilibrium state are
Maxwell–Boltzmann distributed. Particles in non-equilibrium state approach
the Maxwell–Boltzmann distribution in the course of time to establish
equilibrium state (relaxation).
Let us consider a special case where an electrostatic force derivable from
a potential,
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F(r) = −e∇φ(r)
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acts on a plasma consisting of immobile ions and warm electrons. We

also assume that a steady state is maintained ( ∂f∂t = 0). In this case the
distribution
m 32 − mv2
f (r, v, t) = N0 e−eφ(r)/T e 2T (1.34)
2πT
solves the Boltzmann equation
 
∂f Fi ∂f δf
vi + = . (1.35)
∂ri m ∂vi δt c
It is readily seen that LHS of (1.35) vanishes per (1.34), and thus ( δf
δt )c = 0
since the Boltzmann factor e −eφ(r)/T can be taken out of the velocity integral
in (1.30). Now the particle number density of the plasma which maintains
an equilibrium under the action of an electrostatic force is calculated as

N (r) = (1.34) d3 v = N0 e−eφ(r)/T . (1.36)

The inhomogeneous distribution (1.36) is the Boltzmann distribution when

the particles are in an electrostatic force field.
Boltzmann’s H-theorem. Boltzmann introduced into his collision integral
the famous quantity called H:

H(t) = d3 vf (v, t) logf (v, t).

This is the kinetic theory version of the entropy, which is in fact the negative
entropy, −S(t). We have

dH ∂f
= d3 v 1 + log f (v, t) .
dt ∂t
For a spatially homogeneous gas, using (1.21) in the above equation gives
   
dH
= d v1 d v2 bdb dϕ|v1 − v2 | f2 f1 − f2 f1 (1 + logf1 ).
3 3
dt
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10 Fundamentals of Theoretical Plasma Physics

The above integral is unchanged upon the interchange of the subscripts 1

and 2, which is the property of the collision integral that we used to derive
(1.30). Also the integral is symmetric with respect to the interchange of the
prime and no prime. Also note that bdb dϕ is invariant in the reverse
event. Hence, we can deduce
   
dH 1
= d v1 d v2 bdb dϕ|v1 − v2 | f2 f1 − f2 f1
3 3
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dt 4
 
× log (f1 f2 ) − log(fi f2 )
where we note that (f2 f1 − f2 f1 ) [log (f1 f2 ) − log(fi f2 )] ≤ 0 always. There-
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fore, we conclude that dH dt ≤ 0 (the equality holds for equilibrium state)

always, which is the Boltzmann’s H-theorem.

1.4. Simple relaxation model of the collision term

In a weakly ionized plasma whose major population is neutrals, the following
model equation called the BGK model (after Bhatnagar, Gross, and Krook,
in Phys. Rev. 94 1954 p. 511) is often made use of for the Boltzmann collision
term:  
δf
= −ν [f (r, v, t) − f0 (v)]. (1.37)
δt c
Here f0 (v) is a suitable Maxwell–Boltzmann distribution, and 1/ν ≡ τ has
the dimension of time and is termed as the relaxation time. A Maxwellian
f0 (v) makes the collision integral vanish, and the negative sign signifies that
the collisions reduce the deviation of the distribution function from the
equilibrium Maxwell–Boltzmann distribution. Assuming that the external
force is absent from the uniform plasma under consideration, f (v, t) is
governed by
∂
f (v, t) = −ν[f (r, v, t) − f0 (v)].
∂t
Integrating this equation gives
f (v, t) − f0 (v) = f (r, v, t) − f0 (v) e−νt .
As time t increases, f (v, t) approaches to the Maxwell–Boltzmann distribu-
tion f0 (v) as the model equation (1.37) intends to.

1.5. Moment equations

Reference: Chapman, S. and T. G. Cowling, The mathematical theory of
non-uniform gases 2nd Ed. Cambridge University Press (1952).
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Boltzmann equation 11

Instead of solving the Boltzmann equation as an initial-value problem,

Chapman and Enskog reduced the description of f (r, v, t) in the six-
dimensional phase space to the description in terms of the following
moments: n(r, t), V(r, t), T (r, t) in the ordinary coordinate space. In this
reduction, there involves a time-wise coarse graining of the Boltzmann
equation by distinguishing two time scales. The relaxation time t0 is the time
for f (r, v, t) to approach the local Maxwellian f (0) in (1.51). That time scale
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is assumed to be much shorter than the time scale for the moments, i.e.,
n(r, t), V(r, t), T (r, t) to change appreciably. Therefore the fluid equations
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describe slow process by suppressing the fast time involved in Boltzmann’s

f . In terms of the three moments, the Boltzmann equation is transformed
to a series of inhomogeneous integral equations, which we shall not step into
for detailed discussion. But we will summarize the salient features of the
Chapman and Enskog solution.
Let us first define the following macroscopic variables which are the first
three moments of the distribution function:

n(r, t) = f (r, t, v)d3 v (density), (1.38)

1
V(r, t) = v f (r, t, v)d3 v (f luid velocity), (1.39)
n
 2
m
T (r, t) = v − V(r, t) f (r, t, v)d3 v (temperature). (1.40)
3n

In order to obtain the transport equations for these quantities of a simple

gas ( a gas consisting of one species), we multiply (1.21) by 1, mv, mv 2 /2,
respectively, and integrate over velocity. The integrals of RHS of (1.21) vanish
since these quantities are summation invariants. Carrying out the integrals
for LHS, we obtain

∂n
+ ∇ · (nV) = 0 (continuity equation) (1.41)
∂t
∂ ∂
(mnVi ) + (mnvi vj ) − nFi = 0 (Fi = mai )
∂t ∂xj
(momentum equation) (1.42)
∂ mn 2 ∂ mn 2
v  + v vj  − nFi Vi = 0
∂t 2 ∂xj 2
(energy equation) (1.43)
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12 Fundamentals of Theoretical Plasma Physics

where the bracket · · · denotes the average value:


1
· · · = (· · ·)f d3 v.
n
It is convenient to transform (1.42) and (1.43) as follows. Let w be the
random velocity (the ‘peculiar velocity’ in Chapman and Cowling’s book)
defined by
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w = v − V(r, t).
Then evidently, w = 0 and vi vj  = Vi Vj + wi wj . Using (1.41) in (1.42)
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yields
dVi ∂Pij
mn =− + nFi (1.44)
dt ∂xj
where
d ∂ ∂
= + Vj
dt ∂t ∂xj

Pij = mnwi wj  = m wi wj f d3 v. (1.45)

The pressure tensor Pij is usually split into two parts:

Pij = pδij + πij (1.46)
where p is the scalar pressure, and πij is the viscous stress tensor with the
following definitions
nm 2 1
p= w  = nT (r, t) = Pii , (1.47)
3 3
 
w2 
πij = nm wi wj  − δij (trace, πii = 0). (1.48)
3
If the velocity distribution function is isotropic, then wx2  = wy2  = wz2  =
1 2
3 w , wx wy  = wx wz  = wy wz  = 0, so Pij = p δij . Also note that πij
is traceless (πii = 0). The stress tensor πij represents the part of Pij that
arises as a result of the deviation of the distribution function from spherical
symmetry. Carrying out similar transformations for (1.43), we can reduce it
to the form
 
d 3 3 ∂Vi
nT + nT ∇ · V + ∇ · q + Pij = 0, (1.49)
dt 2 2 ∂xj
 2 
w
q = nm w (heat f lux vector). (1.50)
2
In obtaining (1.49), we used (1.41) and (1.44).
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Boltzmann equation 13

So far we have obtained the first three moment equations, (1.41), (1.44),
and (1.49). Our purpose is to obtain hydrodynamic equations to govern the
hydrodynamic variables n, V, T . The three moment equations contain πij
and q, which must be expressed in terms of n, V, T . Since the quantities πij
and q involve the integrations of the distribution function f , an approximate
solution for f must be found. If f is obtained approximately in terms of
(n, V, T ), πij and q can be expressed in terms of (n, V, T ), and a closed
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set of equations for (n, V, T ) is obtained. The Chapman-Enskog solution for

the Boltzmann equation is a successive approximation for f . At each stage
Fundamentals of Theoretical Plasma Physics Downloaded from www.worldscientific.com

of approximation f is found in terms of the three hydrodynamic variables,

thereby the three moment equations form a closed set of equations for the
three hydrodynamic variables. The solutions for the three moment equations
then in turn serve to determine the next approximation for f , and so on.
Here we shall not delve into this successive steps, but will write down the
results of primary importance. The lowest order solution of the distribution
function is the local Maxwellian
 3
(0) m 2 m(v − V(r, t))2
f (v, r, t) = n(r, t) exp − . (1.51)
2πT (r, t) 2T (r, t)
The moment equations for the three hydrodynamic variables are as follows.
∂n
+ ∇ · (nV) = 0, (1.52)
∂t
 
∂V
mn + (V · ∇)V = −∇p + nF, (1.53)
∂t
p = nT, (1.54)
 
∂ 2p d
+V·∇ T =− ∇·V or (nT −3/2 ) = 0. (1.55)
∂t 3n dt
Note that the energy equation (1.55) implies the adiabatic law of a perfect
gas, pn−γ = const. with γ = 5/3. Recall that the specific entropy s =
3k −γ
2m ln(pn ), and (1.55) gives ds/dt = 0. A quick way of getting (1.55) is to
put q = 0 and Pij = pδij in (1.49).
The next higher order solutions are as follows.
dVi Fi 1 ∂Pij
= − (1.56)
dt m mn ∂xj
where
 
∂Vi ∂Vj 2
Pij = pδij + πij = pδij − μ + − δij ∇ · V (1.57)
∂xj ∂xi 3
February 19, 2019 14:33 Fundamentals of Theoretical Plasma Physics 9.61in x 6.69in b3496-ch01 page 14

14 Fundamentals of Theoretical Plasma Physics

and μ is the coefficient of shear viscosity. Equation (1.56) together with

(1.57) is the Navier–Stokes equation for a viscous fluid. The quantity in the
parenthesis in (1.57) is called the rate of strain tensor. The viscous stress
tensor πij which represents the internal friction due to shear gives rise to
the irreversible viscous transfer of momentum between the particles in the
fluid, i.e. the exchange of momentum between the particles due to collisions.
When such momentum transfer is absent from a fluid, the fluid is called ideal
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(or perfect).
Using (1.57) in (1.56) yields
Fundamentals of Theoretical Plasma Physics Downloaded from www.worldscientific.com

 
dVi Fi 1 ∂p 2 μ ∂
= − − μ∇ Vi − ∇·V . (1.58)
dt m mn ∂xi 3 ∂xi
The energy equation takes the form
 
3 dT 1 ∂vi ∂vj
n = −λ∇2 T − p∇ · V − πij + (1.59)
2 dt 2 ∂xj ∂xi
where λ is the coefficient of thermal conduction: q = −λ∇T .

1.6. Ideal two-fluid equations

The lowest order moment equations (1.52)–(1.55) are obtained when the
collisional interactions between the constituent particles of a plasma are
ignored. They should be compatible with the collisionless Boltzmann
equation or Vlasov equation. In this context, we often use the ideal two-
fluid equations which hold independently for each species, electron fluid and
ion fluid, in the following form.
∂nα
+ ∇ · (nα Vα ) = 0 (1.60)
∂t
 
∂Vα
mα n α + Vα · ∇Vα = −∇pα + nα F (1.61)
∂t
pα = nα Tα (1.62)
 
∂
+ Vα · ∇ (nα Tα−3/2 ) = 0 (1.63)
∂t
where the subscript α takes e(i) for electron (ion) fluid.
Phenomenological derivation of πij . The pressure tensor Pij is a surface
force: Pij dsj (no sum on j) is the force directed along i-direction, acted upon
the surface whose normal is j-direction. If i = j, it is normal stress (p,
pressure); if i = j, it represents the shear stress (here πij ). For a fluid at rest
February 19, 2019 14:33 Fundamentals of Theoretical Plasma Physics 9.61in x 6.69in b3496-ch01 page 15

Boltzmann equation 15

or in uniform motion, the three normal stresses have the same value, i.e. the
fluid is isotropic:

P11 = P22 = P33 = p. (1.64)

Thus the stress tensor (pressure tensor) can be written

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Pij = pδij + πij .

The shear stress πij should be traceless, because, by definition, it makes zero
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contribution to the mean normal stresses. Furthermore, the conservation of

angular momentum (see Eq. (2.49)) demands

πij = πji . (1.65)

We also assume that in general flow the shear stress components are propor-
tional to the corresponding time rates of angular deformation. Consulting
Fig. 1.2, we have the relations
 
dφ1 dφ1
dx1 = dV2 , π21 ∼
dt dt
 
dφ2 dφ2
dx2 = dV1 . π12 ∼
dt dt

So we can write
   
1 dφ1 dφ2 dV2 dV1
π12 = (π12 + π21 ) = μ + =μ + .
2 dt dt dx1 dx2

Fig. 1.2. Deformation under shear force πij .

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16 Fundamentals of Theoretical Plasma Physics

Generalizing this to all directions, we have

 
dVi dVj
πij = μ + .
dxj dxi

The complete form of πij which is traceless can be easily written down as
 
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dVi dVj 2
πij = μ + − ∇ · Vδij . (1.66)
dxj dxi 3
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1.7. Fokker–Planck equation, Landau collision integral

In a fully ionized plasma consisting of electrons and ions, the meaningful
collisional event occurs by accumulation of small angle scatterings, and
the collision term can be appropriately approximated by Fokker–Planck
equation, which will be derived from the Boltzmann collision integral. The
collision term Cαβ (fα fβ ) in (1.22) can be considerably simplified for small
angle scatterings.

Kinematics of elastic binary collision. We rewrite

 
3 dσ(g, θ)
Cab (fa fb ) = d vb dΩ |va − vb |
dΩ
 
× fa (va )fb (vb ) − fa (va )fb (vb ) . (1.67)

Introduce the following notations for the relative velocities,

g = va − vb , g = va − vb (1.68)

where unprimed (primed) quantities are pre-(post-)collision velocities.

Remember that the scattering angle θ is the angle between g and g . Let G
be the center-of-mass velocity of two particles a and b. By the law of elastic
collisions, we have

(ma + mb )G = ma va + mb vb = ma va + mb vb . (1.69)

The above two equations yield

mb −ma
va = g + G, vb = g + G, (1.70)
ma + mb ma + mb
mb −ma
va = g + G, vb = g + G. (1.71)
ma + mb ma + mb
February 19, 2019 14:33 Fundamentals of Theoretical Plasma Physics 9.61in x 6.69in b3496-ch01 page 17

Boltzmann equation 17

Hence the pre-collision velocities are determined by G and g; the post-

collision velocities by G and g . It is readily shown that
1 1
(ma va2 + mb vb2 ) = [(ma + mb )2 G2 + μg2 ]
2 2
1 1
(ma va2 + mb vb2 ) = [(ma + mb )2 G2 + μg2 ]
2 2
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where μ = mmaa+mmb
b
is the reduced mass. Since the kinetic energy is conserved,
the above relations give g = g . The relative velocity g is deflected through
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the scattering angle θ by collision. The scattering angle θ depends in general

on the magnitude of the initial relative velocity and the impact parameter
b: θ = θ(g, b). Introducing a unit vector k̂ defined by
g − g
k̂ = (see Fig. 1.3)
|g − g|
Equations (1.70) and (1.71) can be written in the form
2mb 2ma
va = va − (g · k̂)k̂, vb = vb + (g · k̂)k̂, (1.72)
ma + mb ma + mb
2mb 2ma
va = va − (g · k̂)k̂, vb = vb + (g · k̂)k̂. (1.73)
ma + mb ma + mb

Fig. 1.3. The unit vector k̂ = (g − g)/|g − g|.

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18 Fundamentals of Theoretical Plasma Physics

Thus when va , vb , k̂ are given, the post-collision velocities are determined

and k̂ is determined by the impact parameter b and the azimuthal angle
ϕ. Equations (1.72) and (1.73) can be used to derive a theorem concerning
the Jacobian determinant needed in transformation of the velocity integral
from the pre-collision variables to post-collision variables. The Jacobian
determinant is defined by
 
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 ∂v       
 ax ∂vay ∂vaz ∂vbx ∂vby ∂vbz 
 
 ∂vax ∂vax ∂vax ∂vax ∂vax ∂vax 
 
Fundamentals of Theoretical Plasma Physics Downloaded from www.worldscientific.com

   
 ∂vax ∂vay
∂v  ∂v  ∂v ∂v  
 az bx by bz 
 ∂v 
 ay ∂vay ∂vay ∂vay ∂vay ∂vay 
 
       
 ∂vax ∂vay ∂vaz ∂vbx ∂vby ∂vbz
 
∂(va , vb )  ∂vaz ∂vaz ∂vaz ∂vaz ∂vaz ∂vaz 
J= = 
∂(va , vb )  ∂v       
 ax ∂vay ∂v ∂v ∂v by ∂vbz 
 az bx

 ∂vbx ∂vbx ∂vbx ∂vbx ∂vbx ∂vbx 
 
 
 ∂v     ∂v 
∂vbz 

 ax ∂vay ∂vaz ∂vbx by
 
 ∂vby ∂vby ∂vby ∂vby ∂vby ∂vby 
 
   
 ∂vax ∂vay ∂vaz ∂v
   ∂vby ∂vbz  
 bx 
 ∂v ∂vbz ∂vbz ∂vbz ∂vbz ∂vbz 
bz

The partial differentiations involved in J are performed regarding k̂ as a

constant. Since (1.72) which gives va , vb in terms of va , vb is linear, J depends
only on k̂, ma , mb . But va , vb are given in terms of va , vb by (1.73) which
can be obtained from (1.72) by interchanging primed and unprimed letters.
Thus the Jacobian J  defined by
∂(va , vb )
J =
∂(va , vb )

is equal to the same function of (k̂, ma , mb ) as J, and therefore J = J  . Since

JJ  = 1, we have J = ±1, where (−1) is irrelevant.
For small angle scattering, the change of velocity
−2mb
va − va ≡ u = (g · k̂)k̂ (1.74)
ma + mb
will be small for most collisions, so that we may expand (1.67) in powers of
u and keep only the first and second powers. This will be carried out. We
February 19, 2019 14:33 Fundamentals of Theoretical Plasma Physics 9.61in x 6.69in b3496-ch01 page 19

Boltzmann equation 19

multiply both sides of (1.67) by an arbitrary function Φ(va ) and integrate

over va . In the final result, Φ can be eliminated owing to the fact that it is
an entirely arbitrary function. We have

Φ(va ) Cab (va )d3 va
  
Φ(va ) fa (va )fb (vb ) − fa (va )fb (vb ) g b db dϕd3 vb d3 va .
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= (1.75)

In the first integral in the above equation, we change the integration variables
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from (va , vb ) → (va , vb ). In doing this, va in Φ should be expressed as a

function of va and vb :

Φ(va )fa (va )fb (vb )g b db dϕd3 vb d3 va

= Φ(va − u )fa (va )fb (vb )g b db dϕd3 vb d3 va

where u = m2m b
a +mb
(g · k̂)k̂ (see (1.73)) and we used d3 va d3 vb = d3 va d3 vb .
In the RHS integral, we rename the integration variables by dropping the
primes (then g goes to g) to obtain

Φ(va )fa (va )fb (vb )g b db dϕd3 vb d3 va

= Φ(va )fa (va )fb (vb )g b db dϕd3 vb d3 va . (1.76)

Using (1.76) in (1.75) gives


Φ(va ) Cab (va )d3 va
  
= Φ(va ) − Φ(va ) fa (va )fb (vb )g b db dϕd3 vb d3 va . (1.77)

Expanding Φ(va ) in the powers of u (see (1.69)), RHS of (1.77) becomes

  
∂Φ(va ) 1 ∂ 2 Φ(va )
[[uj ]] + [[ui uj ]] fa (va )fb (vb ) g d3 va d3 vb
∂vaj 2 ∂vai ∂vaj
(1.78)

where

[[· · ·]] = (· · ·)b db dϕ.
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20 Fundamentals of Theoretical Plasma Physics

By integration by parts, (1.78) can be cast into the form

 
∂
Φ(va ) − ([[ui ]]fa (va )fb (vb )g)
∂vai

1 ∂2
+ ([[ui uj ]]fa (va )fb (vb )g) d3 va d3 vb .
2 ∂vai ∂vaj
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This is equal to LHS of (1.77) for arbitrary choice of the function Φ, and
thus we conclude
  
∂ 3
Cab (fa fb ) = −
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fa (va ) [[ui ]]fb (vb )gd vb

∂vai
  
1 ∂2
+ fa (va ) [[ui uj ]]fb (vb )gd3 vb . (1.79)
2 ∂vai ∂vaj
This is the Fokker–Planck collision term. Now the Boltzmann equation with
the Fokker–Planck collision term, or the Fokker–Planck equation, is not an
integro-differential equation but a differential equation.
To evaluate [[ui ]] and [[ui uj ]], we introduce the spherical coordinates as
shown in Fig. 1.3, in which the ẑ direction is taken to be parallel to g,
i.e. ẑ = g/|g|. Then k̂ in (1.74) is resolved as (note that k̂ = −sin θ2 and
k̂⊥ = cos θ2 , where and ⊥ are referred to the direction of ĝ)
θ θ θ
k̂ = x̂cos cosϕ + ŷcos sinϕ − ẑsin .
2 2 2
Therefore, we have
 
2mb θ θ θ θ
u= g sin x̂cos cosϕ + ŷcos sinϕ − ẑsin . (1.80)
ma + mb 2 2 2 2
Performing the ϕ-integration gives

mb θ
[[ui ]] = −4π gi sin2 b db. (1.81)
ma + mb 2
The tensor [[ui uj ]] will be diagonal because terms like ux uy vanish upon
performing the ϕ-integration. We have
 2
2mb
[[uu]] = 2π
ma + mb
  
θ g2 θ θ
× ẑẑ g2 sin4 + (x̂x̂ + ŷŷ) sin2 cos2 bdb. (1.82)
2 2 2 2
To change the dyad above into tensor notation, note that
gi gj
x̂x̂ + ŷŷ + ẑẑ = δij , x̂x̂ + ŷŷ = δij − 2 .
g
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Boltzmann equation 21

Thus we obtain
 2
2 mb
[[ui uj ]] = 4πg
ma + mb
    
gi gj 4θ gi gj 2θ 2θ
× b db 2 2 sin + δij − 2 sin cos . (1.83)
g 2 g 2 2
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In the above equation, we neglect the terms of sin4 2θ as compared to the

term of sin2 2θ , which is permissible for small angle scattering. Then we finally
obtain
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 2 
2 mb gi gj θ
[[ui uj ]] = 4πg (δij − 2 ) sin2 b db. (1.84)
ma + mb g 2
Using the Rutherford scattering formula for Coulomb collisions,
 
θ ea eb ma mb
tan = , e : charge, μ =
2 b μ g2 ma + mb
the integral in (1.81) is obtained
 ∞
θ ea eb
sin2 b db = J (1.85)
0 2 μg2
 ∞  
λdλ b μg2
J= λ= . (1.86)
0 1 + λ2 ea eb
Using the above equations in (1.81) and (1.83) gives
 
2 2 gi 1 1
[[ui ]] = −4πea eb J + (1.87)
ma g 4 mb ma
4πe2a e2b
[[ui uj ]] = J (g2 δij − gi gj ). (1.88)
g4 m2a
Now the collision term (1.79) becomes, upon using the above equations,
    
2 2 J 1 1 ∂ gi 3
cab (fa fb ) = 4πea eb + fa fb d vb
ma ma mb ∂vai g3
  2 
e2 e2 ∂2 g δij − gi gj 3
+ 2π a 2b J fa f b d vb . (1.89)
ma ∂vai ∂vaj g3
Use the following identity
 2   2 
gi 1 ∂ g δij − gi gj 1 ∂ g δij − gi gj
= =−
g3 2 ∂vbj g3 2 ∂vaj g3
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22 Fundamentals of Theoretical Plasma Physics

in (1.89) and integrate by parts the result to finally obtain the Landau
collision integral
  
e2 e2 ∂ fb ∂fa fa ∂fb
cab (fa fb ) = 2π a b J d3 vb Uij −
ma ∂vai ma ∂vaj mb ∂vbj
(1.90)
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where
g2 δij − gi gj
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Uij = . (1.91)
g3

The integral J diverges as λ → ∞ (λ should run from 0 to ∞ as b does so).

This is a consequence of the long range of the Coulomb potential. However,
the actual interaction does not extend far beyond a Debye length since the
Coulomb potential is screened off. It is therefore reasonable to cut off the
integral at bmax , equal to the Debye length, corresponding to

μg2 T
λmax = .
ea eb 4πn0 e2

With this cut-off, J in (1.86) yields J = log 1 + λ2max . As will become clear
shortly, 1 in the square root can be neglected as compared to λ2max . Since the
logarithmic term changes slowly, we take the mean value of g2 by replacing
it with 3T /μ, We then obtain
 3/2
e2 T
J  logΛ, Λ = 12πn0 . (1.92)
ea eb 4πn0 e2

Apart from a numerical factor of order 10, Λ is the number of particles in a

Debye sphere, and hence much greater than 1.
Landau [Reference: Landau, L. D. The transport equation in the case of
Coulomb interactions in Collected papers of L. D. Landau, Ed. D. Ter Haar
Pergamon Oxford (1965) p. 163] did not make use of the exact formula for
the Rutherford scattering, but used the impulse approximation. One replaces
the hyperbolic paths of the particles by rectilinear motion with constant
velocity, and computes the force which they exert on each other during their
rectilinear motion. The time integral of this force is then taken as the total
momentum transfer:
  
1 1 ea eb ∞ b dx 2ea eb
sinθ = F⊥ dt = 2 F⊥ dx = 2
=  θ.
g g μg −∞ (x2 + b2 )3/2 bμg2
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Boltzmann equation 23

Using the above relation, the integral in (1.85) becomes

 ∞  2   2  
2θ ea eb db ea eb bmax
sin b db  = log
0 2 μg2 b μg2 bmin

where we have taken a second cut-off: bmin = e3T a eb

, which corresponds
to a deflection through an angle of θ = π/2 (in Rutherford scatter-
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ing formula, g2 is replaced by 3T /μ). Taking bmax again to be a Debye

length, one arrives at exactly the same Landau equation in the impulse
approximation.
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Fokker–Planck equation. One can easily verify the following relations by

direct calculation.
∂ 1 gi
=− 3
∂vi g g
 
g2 δij − gi gj ∂ gj ∂2g
= = .
g3 ∂vi g ∂vi ∂vj

Then (1.89) can be put into the Fokker–Planck form of the collision integral:
  
∂ ∂ 1 3
cab (fa fb ) = A −fa fb d vb
∂vai ∂vai g
  
∂2 ∂2 3
+B fa g fb d vb
∂vai ∂vaj ∂vai ∂vaj

where A and B are constants. The integrals in the above equation,

( 1g or g)fb d3 vb , are called Rosenbluth potential. The Landau collision
integral in (1.90) is a variation of the above form obtained after integration
by parts. The above expression applies to the event when two particles a and
b collide. Let us change the notaions: va → v, vb → v . Then g = v − v ,
and the above equation can be written in the form
   
δf (v, t)  ∂ ∂ f (v ) 3 
= −A f (v) d v
δt c ∂vi ∂vi |v − v |
  
∂2 ∂2
+B f (v) |v − v | f (v ) d3 v  .
∂vi ∂vj ∂vi ∂vj

This equation gives the change of the distribution function due to the
collisions between the composing particles. The effect of prime and no prime
particle collisions are summed over the distribution. The first term is termed
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24 Fundamentals of Theoretical Plasma Physics

the dynamic friction and the second the diffusion term. This equation,
although derived from the Boltzmann collision integral here, is valid for
Markovian process of weak scattering and can be derived independently.
For this, see References: Krall, N. A. and Trivelpiece, A. W., Principles of
Plasma Physics, San Francisco Press, 1986; Sturrock, P. A. Plasma Physics
Cambridge University Press (1994). Also the Fokker–Planck equation above
can be derived from Lenard–Balescu equation (Chapter 12) by taking an
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appropriate approximation [see Reference: Schram, P. P. J. M., Kinetic

Theory of Gases and Plasmas, Kluer Academic, Dordrecht (1991)].
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1.8. Braginskii’s two-fluid equations

Reference: S. I. Braginskii, Transport process in a plasma in Reviews of
plasma physics Vol. 1: Consultant’s Bureau, New York (1965).
Braginskii derived two-fluid equations for a plasma consisting of electrons
and ions from the set of Boltzmann equations with the Landau collision terms
(1.90):
 
∂fe ∂fe e 1 ∂fe
+v· − E+ v×B · = Cee + Cei , (1.93)
∂t ∂r me c ∂v
 
∂fi ∂fi e 1 ∂fi
+v· + E+ v×B · = Cii + Cie . (1.94)
∂t ∂r mi c ∂v

(1.93) and (1.94) are two coupled equations through the cross-collision terms
Cei and Cie . Since the Landau collision terms are appropriate for plasma of
frequent Coulomb collisions, Brakinskii’s two-fluid equation is widely used
for collisional plasma investigation.
In contrast with a simple gas, there are no summation invariants. But
we have the following relations:

Cab d3 v = 0 (a = e, i; b = e, i) (1.95)

which implies that the total number of particles of each species is not changed
due to collisions between like particles as well as between unlike particles.
We have for like particles

ma vCaa d3 v = 0, (1.96)

ma v 2 Caa d3 v = 0. (1.97)
February 19, 2019 14:33 Fundamentals of Theoretical Plasma Physics 9.61in x 6.69in b3496-ch01 page 25

Boltzmann equation 25

The above two equations denote, respectively, the conservation of momentum

and energy in collisions between like particles. For unlike particles, we have

(mi Cie + me Cei )vd3 v = 0, (1.98)

(mi Cie + me Cei )v 2 d3 v = 0. (1.99)
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The above two equations express the conservation of momentum and energy
for the total pair of collisions involving unlike particles.
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In order to derive the moment equations, multiply both sides of (1.93)

with 1, me v, m2e v 2 , respectively, and integrate over the velocity. Doing the
same operation with the ion equation (1.94), we obtain the three moment
equations:
∂na
+ ∇ · (na Va ) = 0 (1.100)
  ∂t  
∂ Va
ma n a + Va · ∇ Va = −∇pa − ∇ · πaij + ea na E+ × B + Ra
∂t c
(1.101)
3 ∂ 3 ∂Vai
(na Ta ) + ∇ · (na Ta Va ) + na Ta ∇ · Va + πaij + ∇ · qa = Qa
2 ∂t 2 ∂xj
(1.102)
where a = e, i, and the hydrodynamic variables na , Va , Ta are defined for
each species of particles similarly to (1.38)–(1.40):

na (r, t) = fa (r, v, t)d3 v, (1.103)

1
Va (r, t) = vfa d3 v, (1.104)
na

Paij = ma (vi − Vai )(vj − Vaj )fa d3 v = pa δij + πaij (pressure tensor)
(1.105)

ma
Ta = (v − Va )2 fa d3 v. (1.106)
3na
Exercise. Show that (1.106) can be arranged into the form of (1.59) with
Q = 0.
The appearance of the quantities Ra and Qa in (1.101) and (1.102) is due
to the cross-collisional terms in (1.93) and (1.94); the corresponding terms
in a simple gas do not exist. The quantity Ra represents the mean change
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26 Fundamentals of Theoretical Plasma Physics

of the momentum of particles of species ‘a’ due to collisions with particles

of other species:

Re = Cei me (v − Ve )d3 v, (1.107)

Ri = Cie mi (v − Vi )d3 v. (1.108)
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Due to the relation (1.98) we have

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Re = −Ri . (1.109)

The quantity Qa in (1.102) is the heat generated in the particles of species

‘a’ as a consequence of collisions with particles of other species.

me
Qe = Cei (v − Ve )2 d3 v, (1.110)
2

mi
Qi = Cie (v − Vi )2 d3 v. (1.111)
2
Adding the above two equations give

Qe + Qi = −Re · (Ve − Vi ) (1.112)

where we used (1.97)–(1.99). The foregoing formulas can be obtained

without actually solving the coupled equations (1.93) and (1.94). Braginskii
decoupled these equations by making use of the smallness of the mass ratio of
electron and ion. Then the problem is reduced to that of solving the kinetic
equation of single species, for which the successive approximation method of
Chapman and Enskog can be applied. The key step to separate the electron
and ion kinetic equations is to expand the relative velocity g in powers of
ion velocity. We shall not go into the details of the Braginskii solution, but
will summarize the important results.

Example. Expand Uij = g2 δij − gi gj /g3 in powers of ion velocity.

This is a problem of Taylor expanding a three variable function in the form

∞
1
ψ(r + a) = (a · ∇)n ψ(r)
n=0
n!

∂ 1 ∂ ∂
= ψ(r) + ai ψ(r) + ai aj ψ(r) + · · · .
∂xi 2 ∂xi ∂xj
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Boltzmann equation 27

Note gα = vα − vα where α is the Cartesian index, no prime denotes electron

velocity, prime denotes ion velocity. In the above formula, put aα = −vα .
Then we obtain

∂Vαβ  1 ∂ 2 Vαβ  
Uαβ = Vαβ − vγ + v v + ···
∂vγ 2 ∂vγ ∂vκ γ κ
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1 2
where Vαβ = v δαβ − vα vβ .
v3
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In the lowest order, the electron and ion distributions are the local
Maxwellians
 3
ma 2 ma (v − Va (r, t))2
fa (v, r, t) = na (r, t) exp − (a = e, i)
2πTa (r, t) 2Ta (r, t)
(1.113)

which yield the following two-fluid equations:

∂na
+ ∇ · (na Va ) = 0 (1.114)
∂t
   
∂ Va
ma n a + Va · ∇ Va = −∇pa + ea na E + × B + R(0)
a
∂t c
(1.115)
 
3 ∂
na + Va · ∇ Ta + na Ta ∇ · Va = 0. (1.116)
2 ∂t

In the momentum equation (1.115),

me n e (0)
R(0)
e =− u = −Ri : f rictional f orce, (1.117)
τe
√ 3/2
3 me Te
u = Ve − Vi , τe = √ : collision time,
4 2π ni J Z 2 e4

where Z is ionic charge number and J is given by (1.86). The energy equation
(1.116) is the adiabaticity: the entropy change, dsa /dt = 0.
As the higher order solutions, the momentum and energy equations take
the forms of (1.101) and (1.102), respectively. But the frictional force is
reduced by a factor of 0.51 and, in addition, a thermal force RT , which
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28 Fundamentals of Theoretical Plasma Physics

arises owing to temperature gradients, appears. So we write in (1.101)

Re = Ru + RT (1.118)
me n e
Ru = − (0.51u + u⊥ ) (1.119)
τe
3 ne B0
RT = −0.71ne ∇ Te − × ∇Te . (1.120)
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2 ωce τe B0
The symbols and ⊥ are referred to the direction of the ambient magnetic
field (B0 ). If the plasma is unmagnetized, the ⊥-components are discarded.
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The thermal forces are given rise to, due to the temperature gradients. The
heat Qa is a plasma-proper term which represents the heat generated by
collisions between electrons and ions:
3me ne
Qi = (Te − Ti ) ≡ QΔ . (1.121)
mi τ e
The heat acquired by the electrons is obtained by using (1.121) in (1.112).
Another point worthy of attention in Braginskii’s solution is that a heat
flux (qeu ) is given rise to by the relative motion between electrons and ions,
in addition to the usual heat flux(qeT ) generated by temperature gradients:
qe = qeu + qeT .
3 ne Te B0
qeu = 0.71ne Te u + × u. (1.122)
2 ωce τe B0
The ion heat flux qi takes the familiar form ∝ ∇Ti . Viscosity πij , heat flux q,
heat Q, and frictional and thermal force R all contribute to the production
of entropy in the plasma. Define the entropy per electron as
3
se = lnTe − ln ne + const. (1.123)
2
The electron entropy equation is written in the form
 
∂ qe QΔ
(ne se ) + ∇ · se ne Ve + + = ϑe (1.124)
∂t Te Te
where ϑe is the entropy production per unit volume:
 
1 e ∂Vei ∂Vej 2
Te ϑe = −qe · ∇lnTe − R · u − πij + − δij ∇ · Ve .
2 ∂xj ∂xi 3
(1.125)

The electron entropy production ϑe is shown to be a positive definite

quantity.
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Boltzmann equation 29

Exercise. Show that (1.124) is equivalent to (1.102). Derive the correspond-

ing ion entropy equation.
The entropy equation for the entire plasma can be written in the form
(see Braginskii 1965)
 
∂ qe qi
(se ne + si ni ) + ∇ · se ne Ve + si ni Vi + +
∂t Te Ti
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= Θ : positive def inite. (1.126)

Example. In the Boltzmann equation of electron distribution function, we
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use the BGK type collision integral: ( δf

δt )c = −ν(fe −fi ). This model implies
e

that the electron and ion distributions become the same after infinite time.
Derive the momentum equation for the fluid velocity Ve (r, t). Use (1.93)
for the left hand side. This example demonstrates the necessary algebra.
Multiply (1.93) by v and integrate over d3 v.
 
3 ∂fe ∂ ∂
me d vv = me vfe d3 v = (me ne Ve (r, t))
∂t ∂t ∂t
  
3 ∂fe 3 ∂fe ∂
me d vv v · = me d vvi vj = me d3 v vi vj fe
∂r ∂xj ∂xj
∂
= me (ne vi vj ) (1.127)
∂xj
where v = vi (in the following ‘i’ is the free index indicating the vector
component). We define the random velocity
wi = vi − Vi (r, t). (1.128)
Then we have
∂ ∂
= (me ne Vi Vj + me ne wi wj ) = (me ne Vi Vj + Pij )
∂xj ∂xj
  
3 1 ∂fe 3
−e d vv E + v × B · d v
c ∂v
  
1 ∂fe
= −e d3 vvi Ej + ejlm vl Bm .
c ∂vj
Integrating by parts and using ∂vi /∂vj = δij give
  
3 Bm
= e d v Ej δij + ejlm (vi δlj + vl δij ) (ejlm δlj = 0)
c
 
1
= ene E + Ve × B . (1.129)
c
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30 Fundamentals of Theoretical Plasma Physics

The collision term gives

−νme ne (Ve − Vi ).

Collecting the above results, we write

 
∂ ∂ Ve
(me ne Ve ) + (me ne Vi Vj + Pij ) + ene E + ×B
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∂t ∂xj c
= −νme ne (Ve − Vi ).
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Using the continuity equation in the above yields [Krall and Trivelpiece 1986
p. 89]
   
∂ Ve
me n e + Ve · ∇ Ve + ∇ · Pij + ene E + ×B
∂t c
= −νme ne (Ve − Vi ). (1.130)
∂Pij ∂Pji
Note that ∇ · Pij = ∂xj = ∂xj .

Example. Derive the energy equation by taking the moments of (1.93) and
(1,94) with 12 mα vα2 . The first two terms give
   
∂ 1 ∂ 1
mα nα v 2 α + 2
mα nα vj v α .
∂t 2 ∂xj 2

The electromagnetic force term yields

  
eα 3 2 1 ∂fα
d vv Ej + ejlm vl Bm
2 c ∂vj
    
eα 3 ∂v 1 v2
=− d v 2v Ej + ejlm vl Bm + ejlm δjl Bm
2 ∂vj c c
= −eα nα E · Vα .

Collecting the above results gives for the energy equation in the form
   
∂ 1 2 ∂ 1 2
mα nα v α + mα nα vj v α − eα nα E · Vα = Qα .
∂t 2 ∂xj 2
(1.131)

Exercise. Proceed further with (1.131) to derive (1.102).

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Boltzmann equation 31

References
Braginskii, S. I. Transport process in a plasma, in Reviews of Plasma Physics Vol. 1:
Consultant’s Bureau, New York, 1965.
Braginskii, S. I. Transport phenomena in a completely ionized Two-temperature plasma,
Soviet Phys. JETP, 6 (1958) p. 358.
Bhatnagar, P. L., Gross, E. P. and Krook, M. Phys. Rev. 94 (1954) p. 511
Chapman, S. and Cowling, T. G. The Mathematical Theory of Nonuniform Gases,
Cambridge University Press, 1952.
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Krall, N. A. and Trivelpiece, A. W. Principles of Plasma Physics, San Francisco Press,

1986.
Landau, L. D. The Transport Equation in the Case of Coulomb Interactions, in Collected
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papers of L. D. Landau, Ed. D. Ter Haar Pergamon Oxford, 1965, p. 163.

Reif, F. Fundamentals of Statistical and Thermal Physics, McGraw-Hill International,
1985, Chapter 13.
Sturrock, P. A. Plasma Physics, Cambridge University Press, 1994.
Van Kampen, N. G. and Felderhof, B. U. Theoretical Methods in Plasma Physics, John
Wiley, 1967, Chapter XV.
Wu, T.-Y. Kinetic Equations of Gases and Plasmas, Addison-Wesley, 1966, Chapter 2.