402
7 Classical Statistical Mechanics
To proceed further, we note (see Appendix G) that volume elements in phase
space are invariant under a canonical transformation of the phase-space variables. This means for the present case that dp
2 dr
2 dp
1 dr
1 = dp 2 dr 2 dp 1 dr 1 =
dp 2 gδt σ de dp 1 dr 1 , which allows us now to rewrite the gain term as
(+) δt =
f
(2) (x
1 , x
2 , t) dp 2 dr 2 dp 1 dr 1 δt
or
(+)
=
f
(2) (x
1 , x
2 , t) σ de
dp 2 dp 1 dr 1 .
(7.5.44)
With these two results, we can write the binary collision term as
δf
δt
coll
=
(+)
−
(−)
=
f
(2) (x
1 , x
2 , t) − f
(2) (x 1 , x 2 , t)
gbdbdφdr 1 dp 1 dp 2 .
(7.5.45)
Completion of our derivation of the Boltzmann (transport) equation requires
three additional assumptions: the first of these assumptions is called, for want of a
better terminology, ‘molecular chaos’, the second assumption is that of ‘uniformity
in space’, and the third assumption required is that external forces on the system
are not sufficiently strong to influence binary collisions between molecules during
the rather short time that such collisions typically take. Let us summarize them
briefly.
1. Molecular chaos assumption: only binary collisions occur, and molecular linear
momenta before and after a collision are uncorrelated (i.e., independent); it is
based in part upon the observation that for many binary collisions the relative
kinetic energy (‘collision energy’) is sufficiently large that the intermolecular
forces in operation are all of a short-ranged nature, so that for distances larger
than the range, there are no forces active on the molecules. This also causes the
duration of a collision to be very short [typically of order of a few picoseconds
(10 −12 s)]. The end result of this assumption is that we can approximate the twomolecule distribution function as the product of two single-molecule (singlet)
distribution functions, viz.,
f
(2) (x 1 , x 2 , t) f
(1) (x 1 , t)f
(1) (x 2 , t) .
(7.5.46)
2. Uniformity approximation: this is the assumption that the singlet distribution
function is a slowly-changing function of position, so that during a binary
collision we can write
7 Classical Statistical Mechanics
To proceed further, we note (see Appendix G) that volume elements in phase
space are invariant under a canonical transformation of the phase-space variables. This means for the present case that dp
2 dr
2 dp
1 dr
1 = dp 2 dr 2 dp 1 dr 1 =
dp 2 gδt σ de dp 1 dr 1 , which allows us now to rewrite the gain term as
(+) δt =
f
(2) (x
1 , x
2 , t) dp 2 dr 2 dp 1 dr 1 δt
or
(+)
=
f
(2) (x
1 , x
2 , t) σ de
dp 2 dp 1 dr 1 .
(7.5.44)
With these two results, we can write the binary collision term as
δf
δt
coll
=
(+)
−
(−)
=
f
(2) (x
1 , x
2 , t) − f
(2) (x 1 , x 2 , t)
gbdbdφdr 1 dp 1 dp 2 .
(7.5.45)
Completion of our derivation of the Boltzmann (transport) equation requires
three additional assumptions: the first of these assumptions is called, for want of a
better terminology, ‘molecular chaos’, the second assumption is that of ‘uniformity
in space’, and the third assumption required is that external forces on the system
are not sufficiently strong to influence binary collisions between molecules during
the rather short time that such collisions typically take. Let us summarize them
briefly.
1. Molecular chaos assumption: only binary collisions occur, and molecular linear
momenta before and after a collision are uncorrelated (i.e., independent); it is
based in part upon the observation that for many binary collisions the relative
kinetic energy (‘collision energy’) is sufficiently large that the intermolecular
forces in operation are all of a short-ranged nature, so that for distances larger
than the range, there are no forces active on the molecules. This also causes the
duration of a collision to be very short [typically of order of a few picoseconds
(10 −12 s)]. The end result of this assumption is that we can approximate the twomolecule distribution function as the product of two single-molecule (singlet)
distribution functions, viz.,
f
(2) (x 1 , x 2 , t) f
(1) (x 1 , t)f
(1) (x 2 , t) .
(7.5.46)
2. Uniformity approximation: this is the assumption that the singlet distribution
function is a slowly-changing function of position, so that during a binary
collision we can write