Page:Elementary Principles in Statistical Mechanics (1902).djvu/215

is therefore the probability-coefficient for a phase specifically defined. This has evidently the same value for all the ${\nu _{1}}!\ldots {\nu _{h}}!$ phases obtained by interchanging the phases of particles of the same kind. The probability-coefficient for a generic phase will be ${\nu _{1}}!\ldots {\nu _{h}}!$ times as great, viz.,

$e^{\frac {\Omega +\mu _{1}\nu _{1}\ldots +\mu _{h}\nu _{h}-\epsilon }{\Theta }}.$

(501)

We shall say that such an ensemble as has been described is canonically distributed, and shall call the constant $\Theta$ its modulus. It is evidently what we have called a grand ensemble. The petit ensembles of which it is composed are canonically distributed, according to the definitions of Chapter IV, since the expression

${\frac {e^{\frac {\Omega +\mu _{1}\nu _{1}\ldots +\mu _{h}\nu _{h}}{\Theta }}}{{\nu _{1}}!\ldots {\nu _{h}}!}}$

(502)

is constant for each petit ensemble. The grand ensemble, therefore, is in statistical equilibrium with respect to specific phases.

If an ensemble, whether grand or petit, is identical so far as generic phases are concerned with one canonically distributed, we shall say that its distribution is canonical with respect to generic phases. Such an ensemble is evidently in statistical equilibrium with respect to generic phases, although it may not be so with respect to specific phases.

If we write $\mathrm {H}$ for the index of probability of a generic phase in a grand ensemble, we have for the case of canonical distribution

$\mathrm {H} ={\frac {\Omega +\mu _{1}\nu _{1}\ldots +\mu _{h}\nu _{h}-\epsilon }{\Theta }}$

(503)

It will be observed that the $\mathrm {H}$ is a linear function of $\epsilon$ and $\nu _{1},\ldots \nu _{h}$; also that whenever the index of probability of generic phases in a grand ensemble is a linear function of $\epsilon$, $\nu _{1},\ldots \nu _{h}$, the ensemble is canonically distributed with respect to generic phases.