Probability Calculations

The principle of equal a priori probabilities is fundamental to all of statistical mechanics, and allows a complete description of the properties of macroscopic systems in equilibrium. In principle, statistical mechanics calculations are very simple. Consider a system in equilibrium that is isolated, so that its total energy is known to have a constant value lying somewhere in the range $E$ to $E+\delta E$ . In order to make statistical predictions, we focus attention on an ensemble of such systems, all of which have their energy in this range. Let ${\mit\Omega}(E)$ be the total number of different states in the ensemble with energies in the specified range. Suppose that, among these states, there are a number ${\mit\Omega}(E; y_k)$ for which some parameter, $y$ , of the system assumes the discrete value $y_k$ . (This discussion can easily be generalized to deal with a parameter that can assume a continuous range of values.) The principle of equal a priori probabilities tells us that all of the ${\mit\Omega}(E)$ accessible states of the system are equally likely to occur in the ensemble. It follows that the probability, $P(y_k)$ , that the parameter $y$ of the system assumes the value $y_k$ is simply

$$P(y_k) = \frac{{\mit\Omega}(E; y_k)}{{\mit\Omega}(E)}.$$ (3.18)

Clearly, the mean value of $y$ for the system is given by

$$\bar{y} = \frac{\sum_k {\mit\Omega}(E; y_k) y_k}{{\mit\Omega}(E)},$$ (3.19)

where the sum is over all possible values that $y$ can assume. In the previous formula, it is tacitly assumed that ${\mit\Omega}(E)\rightarrow \infty$ , which is generally the case in thermodynamic systems.

It can be seen that, using the principle of equal a priori probabilities, all calculations in statistical mechanics reduce to counting states, subject to various constraints. In principle, this is a fairly straightforward task. In practice, problems arise if the constraints become too complicated. These problems can usually be overcome with a little mathematical ingenuity. Nevertheless, there is no doubt that this type of calculation is far easier than trying to solve the classical equations of motion (or Schrödinger's equation) directly for a many-particle system.