A kinetic equation, also called transport equation, is an
equation of motion for the space-time behaviour of the
distribution function. The derivation and justification of such
equations for various systems and circumstances is one of the
tasks of statistical mechanics. However, we do not wish to discuss
this in any detail. Instead, we shall rely on consistency and
plausibility arguments.
Our starting point will be the conservation laws. Take, for
example, the energy conservation law (3.7). After substitution of
(2.6) and (3.8), this equation may be written as
1
This implies that the kinetic equation must be of the form
2
with a source term such that
3
The physical significance of may not be clear at this
point, but it seems intuitively obvious that will involve
the interaction between the particles. The constraint (4.3)
expresses then the fact that in any particle interaction energy is
strictly conserved.
By a similar reasoning we arrive at the conclusion that we also
must have
4
5
These relations reflect the conservation laws of momentum and
charge on the microscopic level.
Collision invariants
The quantities , and are often
called the collision invariants, or also summational
invariants, because in an arbitrary -particle collision they
are additively conserved. For example, in a typical two-particle
event we have
6
7
Figure 4.1 Two-particle
collision
If the particles before and after a collision are the same,
one calls the collision elastic, and inelastic
otherwise.
In the simplest case of a one-component system, there are five
collisional invariants: particle number (mass), momentum,
and energy. This number five appears everywhere in
kinetic theory as some kind of "magic number".
