So far in our discussion of multidimensional problems we have been focussing on continuum fluids governed by partial differential equations. Despite the fact that treating fluids as continua seems entirely natural, and gives remarkably accurate representation in many cases, we know that fluids in nature are not continuous. They are made up of individual molecules. A continuum representation is expected to work well only when the molecules experience collisions on a time and space scale much shorter than those of interest to our situation. By contrast, when the collision mean-free-path is either an important part of the problem, as it is, for example, when calculating the viscosity of a fluid, or when the collision mean-free-path (or time) is long compared with the typical scales of the problem, as it is for very dilute gases and for many plasmas, a fluid treatment cannot cope. We then need to represent the discrete molecular nature of the substance as well as its collective behavior.

Even so, it is unrealistic in most problems to suppose that we can follow the detailed dynamics of each individual molecule. There are p/kT = 105 (Pa)/ [1.38 × 10−23 (J/K) × 273(K)] = 2.65 × 1025 molecules, for example, in a cubic meter of gas at atmospheric pressure and 0°C temperature (STP). Even computers of the distant future are not going to track every particle in such an assembly. Instead, a statistical description is used. The treatment is common to many different types of particles. The particles under consideration might be neutrons in a fission reactor, neutral molecules of a gas, electrons of a plasma, and so on.

The distribution function

Consider a volume element, small compared with the size of the problem but still large enough to contain very many particles. The element is cuboidal d3x = dx.dy.dz with sides dx, see Fig. 8.1. It is located at the position x. We want a sufficient description of the average properties of the particles in this element.