7.5 The Liouville and Boltzmann Equations
401
Fig. 7.9 Determination of
the volume element dr 2 for
the Boltzmann collision
integral. After figure 3.10 of
[5]
(−) δt =
f
(2) (x 1 , x 2 , t) dp 1 dr 1 dp 2 dr 2 .
All atoms with momentum p 2 that lie within a cylindrical shell of height gδt and
base area bdbdφ, depicted in Fig. 7.9, will undergo collision with an atom having
momentum p 1 in time δt, which means that dr 2 will be given as
dr 2 = gδt bdbdφ = gδtσ de
,
with the consequence that the loss term becomes
(−)
=
f
(2) (x 1 , x 2 , t) dp 2 gσ de
dp 1 dr 1 .
(7.5.43)
For the calculation of (+) , we note that a binary collision that sends a particle
labelled 1 into dp 1 about p 1 in time δt is the inverse of the original collision, a
process which we can write as
(p 1 , p 2 ) −→ (p
1 , p
2 ) ,
or
(p
1 , p
2 ) −→ (p
1 , p 2 ) ,
so that the gain term can be written down as
(+) δt =
f
(2) (x
1 , x
2 , t) dp
2 dr
2 dp
1 dr
1 δt .
401
Fig. 7.9 Determination of
the volume element dr 2 for
the Boltzmann collision
integral. After figure 3.10 of
[5]
(−) δt =
f
(2) (x 1 , x 2 , t) dp 1 dr 1 dp 2 dr 2 .
All atoms with momentum p 2 that lie within a cylindrical shell of height gδt and
base area bdbdφ, depicted in Fig. 7.9, will undergo collision with an atom having
momentum p 1 in time δt, which means that dr 2 will be given as
dr 2 = gδt bdbdφ = gδtσ de
,
with the consequence that the loss term becomes
(−)
=
f
(2) (x 1 , x 2 , t) dp 2 gσ de
dp 1 dr 1 .
(7.5.43)
For the calculation of (+) , we note that a binary collision that sends a particle
labelled 1 into dp 1 about p 1 in time δt is the inverse of the original collision, a
process which we can write as
(p 1 , p 2 ) −→ (p
1 , p
2 ) ,
or
(p
1 , p
2 ) −→ (p
1 , p 2 ) ,
so that the gain term can be written down as
(+) δt =
f
(2) (x
1 , x
2 , t) dp
2 dr
2 dp
1 dr
1 δt .