Plasma Primer Part II
(Boltzmann’s Equation and Conservation Laws)
Boltzmann’s Equation – Continuum Conservation Laws:
Boltzmann’s Equation governs temporal and spatial variations in each species
distribution function. Severl approaches to deriving this central relationship are
possible. The following is probably the easiest to understand since it uses a
generalized conservation argument (See Alternative Argument in Tien &
Leinhard, Statistical Thermodynamics, Hemisphere 1979).
Since fs  fs ( x,w, t )  fs ( x1, x2, x3 ,w1,w 2,w 3 , t ) then we can consider f as being
defined in a G-dimensional space (phase space) in which the independent
variables are x1, x2 , x3 ,w1,w 2 ,w3 . Boltzmann argues that the total time rate of
change of fs must be due to collisions between particles. In other words:
dfs fs

dt
t
(1)
c
fs
denotes the net rate of change in fs due to collisions. We will go into
t c
more detail on this term later (when deriving continuum equations). Now, since
fs = fs ( x1, x2 , x3 ,w1,w 2 ,w 3 , t ) then by chain rule:
Where
3
dfs
f dxi 3 fs dw i fs
 s


dt
t
i 1 xi dt
i 1 w i dt
dxi
dw i
is just the absolute velocity component w i , while
is the
dt
dt
particle acceleration which we can determine using Newton’s Second Law:
dw i
1

Fi
dt
ms
Where Fs i is the net force on s particles (ith component). In a plasma, the
dominant force is the electromagnetic body force, which is given by:
(2)
The term
F  ezs E  wxB 
(3)
(4)
Where E is the electric field, zs is free charge on species s, and B is the
magnetic field. The first form on the right represents the coulomb force while the
second follows from ampere’s law. Thus (3) becomes:
dw i ezs
Ei  (wxB)i 
(5)

dt
ms 
dxi
Inserting (5) into (2) and using
 dw i we obtain Boltzmann’s equation:
dt
3
3
fs
f
ez
f
 df 
  w i s  s Ei  (wxB )i  s   s 
t i 1 xi i 1 ms
w i  dt c
=Boltzmann’s Equation for species s in a plasma
(6)
Notes
 df 
1) Under nominally collisionless conditions, the collision term  s  =0.
 dt c
2) Under equilibrium conditions (i.e. no velocity gradients, no temperature
gradients, and no specie gradients) the distribution functions for each
species does not change with time. In other words collisions do not
 df 
change fs . Thus under equilibrium conditions  s  =0. (Note that
 dt c
collisions do occur under equilibrium conditions they do not affect fs ,
however.)
 df 
3) We will find that the collision term  s  gives risk to a mass source term
 dt c
(due to chemical reactions) when deriving the continuum conservation of
mass. Similarly, this term gives rise to a dynamical friction terms in the
species momentum equation. (Note: Viscous forces will come out of the
stress tensor [ J s ] we discussed earlier.)
4) In flows where collision between particles are elastic (typically true in nonplasma gas flows) it can be shown that the collision term equals zero.
df
5) Under steady-state conditions s  0 .
dt
Derivation of Conservation Laws (Continuum):
First, multiply (6) by a molecular property s and integrate with respect to
dw1, dw 2 , dw 3Fdw :

Where
s
 F

f
f
dw   s (w )fs dw   s 
w  fs dw   s ( )c dw
t
t
 ms

(7)
f
xi
3
(w  )fs   w i
i 1
3
 F

fs

w
f


 s  Fi
w i
i 1
 ms

3

w   ei
w i
i 1
dw  dw1dw 2dw 3
Now:
fs


dw   s fs dw   s fs dw
t
t
t
s

  ns s   ns
t
t
(7a)
  (w )f dw    wf dw   ( w )f dw
   n  w   n   w 
   n  w   n w 
(7b)

s
s
s
s
s
s
s
s
s
s
s
s
s
s
s
 Fs

w  fs dw   w (Fss fs )dw   fs w (Fss )dw
m

  (Fss fs )nˆo ds  ns w (Fss )
 s 
(7c)
S
Note that in the last step we used a generalized form of the divergence theorem
to convert 1st ‘volume’ integral in the 1st line to a ‘surface’ integral in the second
line (i.e. we’ve done integration by parts). nˆ o is a generalized unit normal on
surface S , and S is the surface that n lies at w1  ,w 2  ,w3   since
fs  0 for large velocities then the
 term  0 .
Exercise: Show that
w (Fs )  Fs ws
(7d)
for F  ezs E  (wxB ) . Using (7a)-(7d) in (7) we obtain:

ns 
t

s
n
s

t

  ns  w
s
n
s
s
w 
s


ns
F w
ms

s
 df 
     dw
 dt c
This is called Maxwell’s Transport Equation.
Conservation of Species  s  ms  mass of species s. For s  ms :
(8)

s s
s
t
 ms

s
ns msw
s
ms
0
t
 ns ms w
s

 v   ns msw
s
   v 
s s
ns w   0 Since ms =constant
F w ms  0 Since ms =constant
Thus (8) becomes:

  s      s v s s
t
 df 
     dw
 dt  c
IN words, L.H.S. states that local rate of change of specie
(9)
mass
+ net flux of
volume
mass
with and through an infinitesimal volume of space = R.H.S. Thus,
volume
R.H.S. is interpreted as a mass source form (of species s), which in Boltzmann’s
picture, arises due to collisions. Therefore, R.H.S. is written as s where s is
created by chemical reactions or ionization or recombination.

(9)
 s     sv s   s = Conservation of Species s
t
Overall Mass Conservation
Sum up all species conservation equations:

s ts  s   sv s   s s
or


 s  
t s
s
(10)
 v   0
s s
s
=0 since there can be no net course or sink of mass during chemical
s
reactions, ionizations and/or recombination. Note that (10) is exactly the equation
we derive when doing a mass balance on an infinitesimal volume, i.e., it’s the
same conservation of mass equation we use in fluid mechanics.
Conservation of Species Momentum
Let s  msw s . The transport equation (8) becomes:

ns msw s
t

s
ns w msw
n
s

s
ns
ms
ms
w s
  ns msww
t

F
msw
w

s

(11)
s
 df 
  msw   dw
 dt c


ns msw s s   sv s 
t
t
Since w are independent variables, then are not functions of time, thus:
w
ns ms
0
t


0

 ns ms ww
s
   
s
ww
s

But,
ww
 vv
 (v  c )(v  c )
s
s
 cv
s
 vc
 vv  v sv  vv s 
ns (w  )msw  s (w  )w
F1
 ns
s
 cc
s
P s
s


(w1eˆ1  w 2eˆ2  w 3eˆ3 )  F2
(w1eˆ1  w 2eˆ2  w 3eˆ3 )
w1
w 2
F3

(w1eˆ1  w 2eˆ2  w 3eˆ3 )
w 3
 ns F1eˆ1  F2eˆ2  F3eˆ3
 ns F
1
s
s
s
s
 zes  E  w  B 
s

Thus, collecting all of these into equation (11) we obtain:
 


1
 sv s     s  vv  v sv  vv s  P s    ns ezs F s  zes E  w  B s
t
s
 
 
(12)
 df 
  msw   dw
 dt c
Specie conservation of linear momentum. Note that the term on right side
represents a momentum source term associated with collisions of s with other
species and with itself. We’ll consider this term in detail later.
F
s
Overall Conservation of Linear Momentum
Sum all terms in equation (12) over all species.



s t  sv s   t s  sv s   t  v 
   svv     svv     vv 
s
s
Now vv is a diadic, which can be expressed in indicial notation as:
vv  v i v j eˆi  eˆ j
The divergence of a diadic can be calculated as follows:

 (vv )  eˆe
(v i v j eˆi  eˆ j )
xe

(v i ,ev j  v i v j ,e )(eˆe eˆi )eˆ j
xe
 (v i ,i v j  v i v j ,i )eˆ j
 eˆe
 ( v )v  (v  )v
Thus:

( v i v j eˆi  eˆ j )
xe
 (( v i ),e )v j  v i v j ,e )( ei eˆ j )
 ( vv )  eˆe
 (( v i ),i )v   (v  )v
  ( v )v   (v  )v
 v (  v v )   (  v v )
s s
s s
s
s


  (  sv s )v 
 s



  (  s (v s  v )v 
 s



  (  s (v s )v  vv  s 
s
 s

   vv  vv   0
Similarly,
  (  vv
s
s
s
)0
 (   )    (J   P I )
s
s
s
s
s


     Js   (  Ps ) I  
s
 s

   J   P I 
   J   eˆk

(P ij eˆi  eˆ j )
xk
  J  
P
 ij ik eˆ j
xk
  J  
P
eˆi
xi
   J   P
n
Es  nE ( E is not a function of w )
s
s
e  ns zs w  B  e  zs   w  B fs dw 
s
s
 e   zs  (c  v )  Bfs (c , x, t )dc 
s
ˆ  (v  B )fdc
ˆ 
 e  zs   (c  B )fdc


s
ˆ  (v  B ) fˆ dc 
 e  zs    ijk c j Bk eˆi fdc
s 
s
 e  zs ( ijk Bk eˆi c j
s
s
)   ens (v  B )zs
s
 e  ns zsv s  B  e (v  B )
s
e   ens zs =Free charge density
s
We will call
j   ens zsv s
the conductive current. Physically it’s the electric conduction component due to
the charged particle diffusion. Thus:
 ens zs w  B  j  B  ev  B
s
s
Due to the mobility of electrons with in plasmas, and positive charge is rapidly
neutralized by a swarm of electrons. The result is that free charge density, e ,
tends to be negligible (except for near boundaries, where the rapidly moving
electrons can pass preferentially into or away from the solid; in these regions,
called sheaths, non-neglible e ' s do exist). Finally we interpret
 df 
 m w  dt 
dw
s
c
as a momentum source term in the momentum balance for species s (due to
collisions between particles). Due to conservation of momentum during
collisions, the sum of all s momentum source terms equals 0. Thus the overall
conservation of momentum equation is given by:

( v )  v (v v )   (v )v   J   P
(13)
t
  e(E  v  B )  j  B
=overall conservation of momentum within plasma
Equation (13) can be simplified by use of overall conservation of mass, equation
(10), and by defining:
J   ev  j = current density
(= current due to bulk flow + current due to diffusion)
Thus,
v


  (v  )v  v (
  ( v )    J   P
t
t
 e E  J  B
Or
Dv

   J   P   e E  J  B
(14)
Dt
= Overall conservation of momentum
Note that the only difference between (14) and the Navier-Stokes questions is
the body force term on the R.H.S. In the Navier-Stokes equation, this is  g
(we’ve neglected gravitational force in our derivation; no real need to, gravity is
typically much smaller than the electromagnetic forces).
Note: To actually solve (14) or (13) we need explicit expressions for J  and j
(shear stress tensor and conduction current). We might try to obtain the shear
stress tensor.
By letting Js   s in the transport equation (8). Unfortunately, since
Js   s
c 2c
s1
cc
s
 s I  , then the term s (w )fs in (7) leads to a term
which (in analogy with the c 2c
s
term that arises in the momentum
equations) is unknown. To obtain a closed problem, we return to the Botzmann
f
ez
f
equation (6) ( s  (w )fs  s (E  wxB )   fs  s ) and derive approximate
t
ms
x c
fs
involving only distribution functions (of both s and other
x c
species). This leads to N coupled Boltzmann Equations in N distribution
relationships for
functions (where N=number of species). Once these N equations are obtained,
then the usual procedure is to assume that each distribution functions is nearly
maxwellian (isotropic) (which arises when each specie is in equilibrium). Each
distribution function is the expanded as:
fs  fs 0   fs1   2fs 2 ...
Where fs 0 is the (known) Maxwell distribution,  is a small (perturbation)
parameter, and fs1, fs 2 ... are higher order corrections to fs 0 , this perturbation
approach leads to fs1, fs 2 etc. Once fs is thus obtained, then terms like
1
cc s  (w  v )(w  v ) 
(w  v )(w  v )fs dw can be explicitly calculated.
ns 
Using this approach, it is that:
v v
2
J ij    ( i  j )   ij  v
(15)
x j xi
3
Which is the well-known constitution relation for Newtonian Fluids (i.e. fluids in
v i
which shear strain rate
).
x j
The relationship for j will either be derived or given later.
Species Energy Equation
1
1
s    msw 2   int  msww   int = Total energy of an individual particle
2
2
Referring to equation (8) lets obtain expressions for each term:
1
 s  msw 2   int
2
s
Follow the same procedures as before and express everything in terms of mass
average velocity v and diffusion velocity v s :
ns
1
msw 2   int
2
1
  ms (v  c ) (v  c )fˆsdc   int
2
s
(Where fˆ  fˆ (c, x, t ) )
s
s
ms 2
(v  2v c  c 2 )fˆs dc   int s
2
mn
ˆ  m cfˆ dc   
 s s v 2  v   ms cfdc
 s s  int

2

mn
 s v 2  v (ns ms c )  s s c 2   int s
s
2
2

3
 s v 2  sv v s  Ps   int s
2
2

s
s
1
s c 2 s  Ps ) using Ps  ns kTs (Ideal gas law applied to
3
species s). We obtain:

1
3
(16)
ns
msw 2   int  s v 2  sv v s  ns kTs  ns  int s s
2
2
2
s
(where we have used
Gint,s 1
s
w 2

 ms
00
t s
t
2
t
Since a particles internal energy and velocity aren’t explicitly functions of t.
m w2
ns  w s s   (tint,s  s )wfs dw
2
2
mw
m
m
2
2
 s2 )fsdw   2s (v  c ) (v  c )(v  c )fˆsdc   2s (v  2v c  c )cfˆsdc
nm
nm
nm
 s s v 2v  ns ms (v vˆs )v  v s s c 2  s s v 2vˆs
s
2
2
2
m
v i eˆi (  ms ci eˆi c j eˆ j fˆs dc )   s (c 2 )cfˆs dc
2


3
 s v 2v  s (v vˆs )v  ns kTsv  s v 2vˆs
2
2
2
nm
v i eˆi (  ms ci c j eˆi  eˆ j fˆs dc )  s s c 2c
2
Now
v i eˆi (  ms ci c j eˆi  eˆ j fˆs dc )  v  ms ccfˆs dc
1
 v s Ps 
 sv Ps 

s
v
s
s
1
s
  J   P I  
s
s
Also we define the best flux vector Es as
ms c 2
  int,s )
2
Physically, the term in parentheses is the particle energy associated with its
random motion and its internal energy. Thus qs is a measure of the molecular
energy transport due to random molecular motion (i.e. diffusion of molecular
energy).
Es  ns c (
ˆ .
Finally to complete the evaluation of ns  w s , we have to include  v  int,s fdc
Thus:
ns  w
s

s

3
v 2v  ns kTsv  s v 2v s  v Js   Psv  qs  vns  int
2
2
2
ˆ
ˆ
v I   v k ek  ij ei  eˆ j
s
(17)
 v k  ij  ki eˆ j
(where we have used  v i  ij eˆ j
)
 v j eˆ j
v
ns w    w   int,s 
Since  int,s and
1
msw 2 )fs dw  0
2
1
msw 2 aren’t explicit functions of position
2
ns
m
1

F w  
Fi
( int,s  s w j w j )fs dw

s
ms
ms
w i
2

1 ms
Fi ( ij w j  w j  ij )fs dw
2 ms 

2
Fw
i i fs dw
2
i i fs dw   ezs (E  w  B ) wfs dw
 Fw
  ezs E wfs dw
  ezs E (v  c )fs dw
 ezs ns E v  ezs ns E v s
 E (ezs nsv  j s )
 Fw f dw  E J
i
i s
s
= Joule heating term due to motion of species s
ns
s m
s
Using (16), (17) and (18) in the transport equation (8) we obtain:
  s 2
3

v  sv v s  ns kTs  ns  int s 

t  2
2

 F w
3


  s v 2  sv 2v s  ns kTs  Ps  ns  int s  v
2
2


 f 
 s v 2v s  v  Js   E Js     s  dw
2
s  t c
This is the specie energy equation.
Overall energy equation
(18)
(19)
Let’s use the following definition of the average specie internal energy es :
es 
1
 int
ms
s

 int s 1
3 kTs

 cc
2 ms
ms
2
(20)
And rewrite (19):
  s 2

v  sv v s  s es  

t  2

 

 ( s v 2  sv 2v s  ns kTs  s es )v 
 2

(19a)


 f 
  s v 2v s   v (v  Js )   Es  E Js   s  s  dw
2

 t c
= Species energy equation
Now sum each term in (19a) over all species:

 

 
s ( t 2s v 2 )  t (s 2s v 2 )  t 2s v 2



s ( t sv vs )  t (s sv v s )  t (v s sv s )  0



s ( t s es )  t (s s es )  t ( e)
Where :
e
s
s es

Similar manipulations can be used on the 4th through the 7th terms in (19a):
 

 v ( s v 2  0  P   e) (where we have used  ns kTs   Ps  P  total
 2

s
s
pressure
Or
 

 v ( s v 2  P   e)
 2

s
v2
 svs ) 0
2
2 s
s
  (v Js )  v (Js )  v ( Js )  Rate of work due to shear stresses
 (
s
  qs 
s
s
v 2v s )  (
s
 qs  qs  net conduction heat transfer (into differential volume)
s
 (E J
s
fs
s
) E (  Js ) E Js  Joule heating
s
 (  E ( t ) dw ) 0 (i.e. net energy transfer during collisions between all species
s
s
is zero)
c
Thus, overall energy equation is:
  s 2

v  e   

t  2

 s 2

v ( 2 v  P   e )


 (v  Js )   q  E J
(20)
Heat Flux Vector q :
1) When all species temperatures Ts are equal to a single temperature, T,
and when all fs are Maxwellian (we’ll describe this later), then
E  k T   v s hs
(21)
s
Where k= the ordinary thermal conductivity = fn(T)
P
hs  es  s  specie specific enthalpy
s
This version of k is sometimes called ‘frozen’ thermal conductivity.
2) In the case where all Ts  T , all fs  fMaxwellian , and v s  fn (gradient in
specie density) then:
q  k T
(22)
Since v s typically depends on other properties beside specie density
gradients, (22) only applies under special conditions.
3) In cases where heavy particle temperatures = T  , temperatures = Te , then
we can express k in terms of a heavy particle thermal conductivity, k  ,
and an electron thermal conductivity, k e ; in this case:
q  k T   keTe   v s s hs
(23)
s
This expression reflects the fact that electron collisions with heavy particles
(essentially) have no effect on the heavy specie distribution functions. The
existence of (at least) two temperatures, a heavy particle temperature, T  ,
and an electron temperature, Te , is due to the much greater mobility of
electrons (reflected in a higher c 2 ).
e
Exercise: Show that the overall energy equation (20) can be expressed as
follows:
De

 P ( v )     q  E J  v ( j  B )
(24)
Dt
v
Where:   Jij i = dissipation function
x j
And j   ns ezsv s
s
Use continuity equation (10) and momentum equation (14).