Page:Elementary Principles in Statistical Mechanics (1902).djvu/217

The value of $\epsilon _{p}$ is of course zero in this case. But the value of $\epsilon _{q}$ contains an arbitrary constant, which is generally determined by considerations of convenience, so that $\epsilon _{q}$ and $\epsilon$ do not necessarily vanish with $\nu _{1},\ldots \nu _{h}$.

Unless $-\Omega$ has a finite value, our formulae become illusory. We have already, in considering petit ensembles canonically distributed, found it necessary to exclude cases in which $-\psi$ has not a finite value.^[1] The same exclusion would here make $-\psi$ finite for any finite values of $\nu _{1}\ldots \nu _{h}$. This does not necessarily make a multiple series of the form (506) finite. We may observe, however, that if for all values of $\nu _{1}\ldots \nu _{h}$

$-\psi \leq c_{0}+c_{1}\nu _{1},\ldots +c_{h}\nu _{h},$

(507)

where $c_{0},c_{1},\ldots c_{h}$ are constants or functions of $\Theta$,

$e^{-{\frac {\Omega }{\Theta }}}\leq \Sigma _{\nu _{1}}\ldots \Sigma _{\nu _{h}}{\frac {e^{\frac {c_{0}+(\mu _{1}+c_{1})\nu _{1}\ldots +(\mu _{h}+c_{h})\nu _{h}}{\Theta }}}{{\nu _{1}}!\ldots {\nu _{h}}!}}$

i. e.,

$e^{-{\frac {\Omega }{\Theta }}}\leq e^{\frac {c_{0}}{\Theta }}\Sigma _{\nu _{1}}{\frac {e^{{\frac {\mu _{1}+c_{1}}{\Theta }}\nu _{1}}}{{\nu _{1}}!}}\ldots \Sigma _{\nu _{h}}{\frac {e^{{\frac {\mu _{h}+c_{h}}{\Theta }}\nu _{h}}}{{\nu _{h}}!}}$

i. e.,

$e^{-{\frac {\Omega }{\Theta }}}\leq e^{\frac {c_{0}}{\Theta }}e^{e^{\frac {\mu _{1}+c_{1}}{\Theta }}}\ldots e^{e^{\frac {\mu _{h}+c_{h}}{\Theta }}}$

i. e.,

$-{\frac {\Omega }{\Theta }}\leq {\frac {c_{0}}{\Theta }}+e^{\frac {\mu _{1}+c_{1}}{\Theta }}\ldots +e^{\frac {\mu _{h}+c_{h}}{\Theta }}$

(508)

The value of $-\Omega$ will therefore be finite, when the condition (507) is satisfied. If therefore we assume that $-\Omega$ is finite, we do not appear to exclude any cases which are analogous to those of nature.^[2] The interest of the ensemble which has been described lies in the fact that it may be in statistical equilbrium, both in

1. See Chapter IV, page 35.
2. If the external coördinates determine a certain volume within which the system is confined, the contrary of (507) would imply that we could obtain an infinite amount of work by crowding an infinite quantity of matter into a finite volume.