2. Elements of Kinetic Theory

Ludwig Boltzmann

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 Gases''.

First he defined a distribution density $f$ such that $f\left( \vec{r}, \vec{v} ; t \right) d\vec{r} d\vec{v}$ denotes the number of particles which at time $t$ situated at $\vec{r}$ and have a velocity $\vec{v}$. The $6$-dimensional space of the variables $\{\vec{r},\vec{v} \}$ has later been called ``$\mu$ space''.^2.1

The evolution in time of the function $f\left( \vec{r}, \vec{v}; t \right)$ 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 $t \rightarrow \infty$. It turns out that this equilibrium distribution in $\mu$ 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.