7.5 The Liouville and Boltzmann Equations
403
f
(1) (r 1 , p 1 , t)f
(1) (r 2 , p 2 , t) f
(1) (r 1 , t; p 1 )f
(1) (r 1 , t; p 2 ) ,
(7.5.47)
which is sometimes referred to as localization of the collision.
3. Neglect of external forces on the collision process: this simply means that
an applied external field must not be so strong that it contorts the collision
trajectories in the time that it takes for a binary collision to occur.
Upon employing these three assumptions in Eq. (7.5.45), we obtain the final form
for the Boltzmann equation in which the collision integral is frequently written as
δf (1)
δt
collision
=
f
1 f
2 − f 1 f 2
gσ de
dv 2 ,
(7.5.48)
with g being the relative speed of the colliding molecules and σ the collision cross
section. When the collision is between two atoms, the integration over the element of
solid angle de gives rise to a factor 2π (arising from integration over the azimuthal
angle φ shown in Fig. 7.9).
The single-particle distribution functions in Eq. (7.5.48) are now considered as
functions of velocity rather than of linear momentum and are designated by single
subscripts only; moreover, the superscript ‘(1)’ is suppressed, so that f 1 and f
1 are
given as
f 1 ≡ f
(1) (v 1 ) = n 1
m 1
2πk B T
3
2
e
−m 1 v 2
1 /(2k B T ) ,
f
1 ≡ f
(1) (v
1 ) = n 1
m 1
2πk B T
3
2
e
−m 1 v 2
1 /(2k B T ) ,
while f 2 and f
2 are similarly identified as f 2 ≡ f (1) (v 2 ) and f
2 ≡ f (1) (v
2 ).
Upon combining Eq. (7.5.47) with Eq. (7.5.19), we arrive at the Boltzmann
equation for a dilute pure monatomic gas, namely
∂f 1
∂t
+ v 1 · ∇f 1 + X · ∇ v f 1 =
[f
1 f
2 − f 1 f 2 ] gσ de
dv 2 .
(7.5.49)
7.5.5 Nonequilibrium Phenomena and the Boltzmann
Equation
We shall focus upon nonequilibrium phenomena in the dilute gas regime, which in
practice refers to gases at pressures between about 100 Pa and 1 MPa. If we have N
atoms, each with the same mass m 1 , in a container of volume V , then the average
number of atoms at time t in volume element dv 1 dr 1 of the classical single-particle
403
f
(1) (r 1 , p 1 , t)f
(1) (r 2 , p 2 , t) f
(1) (r 1 , t; p 1 )f
(1) (r 1 , t; p 2 ) ,
(7.5.47)
which is sometimes referred to as localization of the collision.
3. Neglect of external forces on the collision process: this simply means that
an applied external field must not be so strong that it contorts the collision
trajectories in the time that it takes for a binary collision to occur.
Upon employing these three assumptions in Eq. (7.5.45), we obtain the final form
for the Boltzmann equation in which the collision integral is frequently written as
δf (1)
δt
collision
=
f
1 f
2 − f 1 f 2
gσ de
dv 2 ,
(7.5.48)
with g being the relative speed of the colliding molecules and σ the collision cross
section. When the collision is between two atoms, the integration over the element of
solid angle de gives rise to a factor 2π (arising from integration over the azimuthal
angle φ shown in Fig. 7.9).
The single-particle distribution functions in Eq. (7.5.48) are now considered as
functions of velocity rather than of linear momentum and are designated by single
subscripts only; moreover, the superscript ‘(1)’ is suppressed, so that f 1 and f
1 are
given as
f 1 ≡ f
(1) (v 1 ) = n 1
m 1
2πk B T
3
2
e
−m 1 v 2
1 /(2k B T ) ,
f
1 ≡ f
(1) (v
1 ) = n 1
m 1
2πk B T
3
2
e
−m 1 v 2
1 /(2k B T ) ,
while f 2 and f
2 are similarly identified as f 2 ≡ f (1) (v 2 ) and f
2 ≡ f (1) (v
2 ).
Upon combining Eq. (7.5.47) with Eq. (7.5.19), we arrive at the Boltzmann
equation for a dilute pure monatomic gas, namely
∂f 1
∂t
+ v 1 · ∇f 1 + X · ∇ v f 1 =
[f
1 f
2 − f 1 f 2 ] gσ de
dv 2 .
(7.5.49)
7.5.5 Nonequilibrium Phenomena and the Boltzmann
Equation
We shall focus upon nonequilibrium phenomena in the dilute gas regime, which in
practice refers to gases at pressures between about 100 Pa and 1 MPa. If we have N
atoms, each with the same mass m 1 , in a container of volume V , then the average
number of atoms at time t in volume element dv 1 dr 1 of the classical single-particle