Page:Elementary Principles in Statistical Mechanics (1902).djvu/227

experiment. Although the ensemble contains systems having the widest possible variations in respect to the numbers of the particles which they contain, these variations are practically contained within such narrow limits as to be insensible, except for particular values of the constants of the ensemble. This exception corresponds precisely to the case of nature, when certain thermodynamic quantities corresponding to $\Theta$, $\mu _{1}$, $\mu _{2}$, etc., which in general determine the separate densities of various components of a body, have certain values which make these densities indeterminate, in other words, when the conditions are such as determine coexistent phases of matter. Except in the case of these particular values, the grand ensemble would not differ to human faculties of perception from a petit ensemble, viz., any one of the petit ensembles which it contains in which ${\overline {\nu }}_{1}$, ${\overline {\nu }}_{2}$, etc., do not sensibly differ from their average values.

Let us now compare the quantities $\mathrm {H}$ and $\eta$, the average values of which (in a grand and a petit ensemble respectively) we have seen to correspond to entropy. Since

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

and

$\eta ={\frac {\psi -\epsilon }{\Theta }},$

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

(545)

A part of this difference is due to the fact that $\mathrm {H}$ relates to generic phases and $\eta$ to specific. If we write $\eta _{\rm {gen}}$ for the index of probability for generic phases in a petit ensemble, we have

$\eta _{\rm {gen}}=\eta +\log {\nu _{1}}!\ldots {\nu _{h}}!,$

(546)

$H-\eta =\mathrm {H} -\eta _{\rm {gen}}+\log {\nu _{1}}!\ldots {\nu _{h}}!,$

(547)

$\mathrm {H} -\eta _{\rm {gen}}={\frac {\Omega +\mu _{1}\nu _{1}\ldots +\mu _{h}\nu _{h}-\psi }{\Theta }}-\log {\nu _{1}}!\ldots {\nu _{h}}!.$

(548)

This is the logarithm of the probability of the petit ensemble ($\nu _{1}\ldots \nu _{h}$).^[1] If we set

1. See formula (517).