11.C.3. Boltzmann's H Theorem
The Boltzmann's H function is defined as
H  t    d 3q1  d 3 p1 f  p1, q1, t  ln f  p1, q1, t 
(11.75)
  d 3q1  d 3 p1 f1 ln f1
We shall show that if f satisfies the Boltzmann equation (11.74), then H always
decreases due to collision effects. Now,
H
f
  d 3q1  d 3 p1  ln f1  1 1
t
t

(11.75) 
t
(11.76)
Writing (11.74) as
f1
f
f
 q1  1  p1  1   d 3 p2  d  g  cm  f1' f 2'  f1 f 2 
t
q1
p1
where f j  f  p j , q1 , t  , eq (11.76) can be written as

H
f
f 
  d 3q1  d 3 p1  ln f1  1  q1  1  p1  1 
t
q1
p1 

  d 3q1  d 3 p1  d 3 p2  d  g  cm  f1' f 2'  f1 f 2  ln f1  1
(11.77)
Now,

 ln f1  1  q1 


 ln f1  1  p1 



f1 

 f1 ln f1      q1 f1 ln f1   f1 ln f1  q1
  q1 
q1
q1
q1 
q1


f1 

 f1 ln f1      p1 f1 ln f1   f1 ln f1  p1
  p1 
p1
p1
p1 
p1
which, with the help of


 H
 H
 q1 
 p1 



0
q1
p1
q1 p1 p1 q1
gives
d q d
3
1
3

f
f 
p1  ln f1  1  q1  1  p1  1 
q1
p1 

 


   d 3q1  d 3 p1 
  q1 f1 ln f1  
  p1 f1 ln f1   0
p1
 q1

Hence, (11.77) simplifies to
H
  d 3q1  d 3 p1  d 3 p2  d  g  cm  f1' f 2'  f1 f 2  ln f1  1
t
Setting p1  p2 gives
(11.78)
H
 d 3q1  d 3 p1  d 3 p2  d  g  cm  f1' f 2'  f1 f 2  ln f 2  1
t 
(11.79)
Hence,
H 1 3
  d q1  d 3 p1  d 3 p2  d  g  cm  f1' f 2'  f1 f 2  ln f1 f 2  2 
(11.80)
t
2
Setting p1  p1 , p2  p2 inverts the scattering process but leaves  cm unchanged.
Furthermore,
d
3
p1  d 3 p2   d 3 p1  d 3 p2 so that (11.80) becomes
H
1
   d 3q1  d 3 p1  d 3 p2  d  g  cm  f1' f 2'  f1 f 2  ln f1' f 2'  2 
t
2
(11.80a)
Combining (11.80,a) gives
H 1 3
f f
  d q1  d 3 p1  d 3 p2  d  g  cm  f1' f 2'  f1 f 2  ln 1 2
t
4
f1' f 2'
(11.81)
Now, since ln is a monotonic function,
 a  b ln a  ln b  0

 f1' f 2'  f1 f 2  ln
 a, b  0
f1 f 2
0
f1' f 2'
Hence, we obtain the Boltzmann's H theorem:
H
0
(11.81a)
t
where the equal signs holds only if f1' f 2'  f1 f 2 for all collisions.
This is the
condition of detailed balance and is the equilibrium condition for the gas.
also be written as
ln f1'  ln f 2'  ln f1  ln f 2
(11.82)
It can
Now, at equilibrium, all f must be independent of q1. Therefore, (11.82) can be
satisfied only if ln f is an additive function of only quantities that are conserved in a
collision, i.e.,
ln f1  A  B  p1  C

p12
2m
(11.83)

p2 
f1  exp  A  B  p1  C 1 
2m 

where A, B and C are constants.
Note that this is simply a Maxwell-Boltzmann
distribution with non-zero average momentum.
Since H always decreases with t, we may use it to define an entropy for systems in
non-equilibrium, i.e.,
S  t   kB H  t   k B  d 3q1  d 3 p1 f1 ln f1
(11.84)