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.

So far we have been focussing on how particle codes work once the particles are launched. We've talked about how they are moved, and how self-consistent forces on them are calculated. What we have not addressed is how they are launched in an appropriate way in the first place, and how particles are reinjected into a simulation. We've also not explained how one decides statistically whether a collision has taken place to any particle and how one would then decide what scattering angle the collision corresponds to. All of this must be determined in computational physics and engineering by the use of random numbers and statistical distributions. Techniques based on random numbers are called by the name of the famous casino at Monte Carlo.

Probability and statistics

11.1.1 Probability and probability distribution

Probability, in the mathematically precise sense, is an idealization of the repetition of a measurement, or a sample, or some other test. The result in each individual case is supposed to be unpredictable to some extent, but the repeated tests show some average trends that it is the job of probability to represent. So, for example, the single toss of a coin gives an unpredictable result: heads or tails; but the repeated toss of a (fair) coin gives on average equal numbers of heads and tails. Probability theory describes that regularity by saying the probability of heads and tails is equal. Generally, the probability of a particular class of outcomes (e.g. heads) is defined as the fraction of the outcomes, in a very large number of tests, that are in the particular class. For a fair coin toss, the probability of heads is the fraction of outcomes of a large number of tosses that is heads, 0.5. For a six-sided die, the probability of getting any particular value, say 1, is the fraction of rolls that come up 1, in a very large number of tests.