In a dilute gas the molecules are in free, linear flight most of the time;
just occasionally their flight will be interrupted by a collision with a
single other particle. The result of such an event, which in general may
be treated as a classical elastic collision, is a change in the speeds and
directions of both partners. So it should be possible to derive, from a
statistical treatment of such binary collisions, predictions
on the properties of gases. This idea was the starting point from
which Ludwig Boltzmann proceeded to develop his ``Kinetic Theory of
First he defined a distribution density such that
denotes the number of particles which at time
situated at and have a velocity .
The -dimensional space of the variables
has later been called `` space''.2.1
The evolution in time of the function
is then described by
Boltzmann's Transport Equation. We will shortly sketch the derivation
and shape of this important formula.
A particularly important result of Boltzmann's equation is
its stationary solution, i.e. the function that
solves the equation in the long-time limit
It turns out that this equilibrium distribution in space
may be derived without explicitely solving the transport equation itself.
The strategy used to do so, called the
method of the most probable distribution, will be encountered
again in other parts of Statistical Mechanics.