Introduction to statistical ensembles

The concept of a random or stochastic process or function represents a generalization of the idea of a set of random variables x1, x2, …, when the set is no longer countable and the variables form a continuum. We therefore introduce a continuous parameter t, such as time, that labels the variates. We call x(t) a random process or a random function of t if x does not depend on t in a deterministic way. Random processes are encountered in many fields of science, whenever fluctuations are present. Examples of a real random process x(t) are the fluctuating voltage across an electrical resistor, and the coordinates of a particle under-going Brownian motion. We shall see shortly that the optical field generated by any realistic light source must also be treated as a random function of position and time. Of course the parameter t may also stand for some quantity other than time, but for simplicity we shall take it to represent time. In our applications x(t) will frequently represent a Cartesian component of the electric or magnetic field vector in a light beam. To begin with we shall take x(t) to be real, but complex random processes will also be encountered.

The ensemble average

As x does not depend on t deterministically, we can only describe its values statistically, by some probability distribution or probability density.