Probability calculations

The principle of equal a priori probabilities is fundamental to all 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 which is isolated, so that its total energy is known to have a constant value 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 of the system 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 which can assume a continuous range of values). The principle of equal a priori probabilities tells us that all 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

$$P(y_k) = \frac{{\mit\Omega}(E; y_k)}{{\mit\Omega}(E)}.$$ (102)

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

$$\bar{y} = \frac{\sum_k {\mit\Omega}(E; y_k) \,y_k}{{\mit\Omega}(E)},$$ (103)

where the sum is over all possible values that $y$ can assume. In the above, 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 simply counting states, subject to various constraints. In principle, this is fairly straightforward. 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.