7.5 The Liouville and Boltzmann Equations
393
We shall now restrict ourselves to the binary collision of two atoms. The
Hamiltonian for a system of two atoms in interaction through a central potential
energy function V is given by
H(r 1 , r 2 , p 1 , p 2 ) =
p 2
1
2m 1
+
p 2
2
2m 2
+ V (|r 2 − r 1 |) ,
(7.5.27)
with r 1 , r 2 the positions of the atoms relative to some space-fixed origin O, r ≡
|r 2 − r 1 | is the distance between them (r = r 2 − r 1 is the position of atom 1 relative
to atom 2, and is called the relative position vector), m 1 , m 2 are the atomic masses,
and p 1 , p 2 are the linear momenta of the atoms.
Panel (a) of Fig. 7.3 depicts the overall geometry associated with a binary elastic
collision between two particles, and shows the particle masses, both pre- and postcollisional velocities and relative positions, together with the path taken by their
centre-of-mass (CM) during the interaction. The geometry associated with the two
χ
χ
r
x
χ
Fig. 7.3 Schematics associated with a binary collision
393
We shall now restrict ourselves to the binary collision of two atoms. The
Hamiltonian for a system of two atoms in interaction through a central potential
energy function V is given by
H(r 1 , r 2 , p 1 , p 2 ) =
p 2
1
2m 1
+
p 2
2
2m 2
+ V (|r 2 − r 1 |) ,
(7.5.27)
with r 1 , r 2 the positions of the atoms relative to some space-fixed origin O, r ≡
|r 2 − r 1 | is the distance between them (r = r 2 − r 1 is the position of atom 1 relative
to atom 2, and is called the relative position vector), m 1 , m 2 are the atomic masses,
and p 1 , p 2 are the linear momenta of the atoms.
Panel (a) of Fig. 7.3 depicts the overall geometry associated with a binary elastic
collision between two particles, and shows the particle masses, both pre- and postcollisional velocities and relative positions, together with the path taken by their
centre-of-mass (CM) during the interaction. The geometry associated with the two
χ
χ
r
x
χ
Fig. 7.3 Schematics associated with a binary collision