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. Consider a system in equilibrium that is isolated, so that its total internal energy is known to have a constant value lying somewhere in the range $U$ to $U+\delta U$. In order to make statistical predictions, we focus attention on an ensemble of such systems, all of which have their internal energy in this range. Let ${\mit\Omega}(U)$ be the total number of different states in the ensemble with internal energies in the specified range. Suppose that, among these states, there are a number ${\mit\Omega}(U; x_k)$ for which some parameter, $x$, of the system assumes the discrete value $x_k$. (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 ${\mit\Omega}(U)$ accessible states of the system are equally likely to occur in the ensemble. It follows that the probability, $P(x_k)$, that the parameter $x$ of the system assumes the value $x_k$ is simply

$$P(x_k) = \frac{{\mit\Omega}(U; x_k)}{{\mit\Omega}(U)}.$$ (5.291)

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

$$\langle x\rangle = \frac{\sum_k {\mit\Omega}(U; x_k) \,x_k}{{\mit\Omega}(U)},$$ (5.292)

where the sum is over all possible values that $x$ can assume.