Page:Elementary Principles in Statistical Mechanics (1902).djvu/222

$\mathop {\int \ldots \int } _{\rm {phases}}^{\rm {all}}{\frac {e^{\frac {\Omega +\mu _{1}\nu _{1}\ldots +\mu _{h}\nu _{h}-\epsilon }{\Theta }}}{{\nu _{1}}!\ldots {\nu _{h}}!}}\,dp_{1}\ldots dq_{n},$

(516)

or

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

(517)

This may be called the probability of the petit ensemble $(\nu _{1},\ldots \nu _{h})$. The sum of all such probabilities is evidently unity. That is,

$\Sigma _{\nu _{1}}\ldots \Sigma _{\nu _{h}}{\frac {e^{\frac {\Omega +\mu _{1}\nu _{1}\ldots +\mu _{h}\nu _{h}-\psi }{\Theta }}}{{\nu _{1}}!\ldots {\nu _{h}}!}}=1,$

(518)

which agrees with (506).

The average value in the grand ensemble of any quantity $u$, is given by the formula

${\overline {u}}=\Sigma _{\nu _{1}}\ldots \Sigma _{\nu _{h}}\mathop {\int \ldots \int } _{\rm {phases}}^{\rm {all}}{\frac {ue^{\frac {\Omega +\mu _{1}\nu _{1}\ldots +\mu _{h}\nu _{h}-\epsilon }{\Theta }}}{{\nu _{1}}!\ldots {\nu _{h}}!}}\,dp_{1}\ldots dq_{n}.$

(519)

If $u$ is a function of $\nu _{1},\ldots \nu _{h}$ alone, i. e., if it has the same value in all systems of any same petit ensemble, the formula reduces to

${\overline {u}}=\Sigma _{\nu _{1}}\ldots \Sigma _{\nu _{h}}{\frac {ue^{\frac {\Omega +\mu _{1}\nu _{1}\ldots +\mu _{h}\nu _{h}-\psi }{\Theta }}}{{\nu _{1}}!\ldots {\nu _{h}}!}}.$

(520)

Again, if we write ${\overline {u}}|_{\rm {grand}}$ and ${\overline {u}}|_{\rm {petit}}$ to distinguish averages in the grand and petit ensembles, we shall have

${\overline {u}}|_{\rm {grand}}=\Sigma _{\nu _{1}}\ldots \Sigma _{\nu _{h}}{\overline {u}}|_{\rm {petit}}{\frac {e^{\frac {\Omega +\mu _{1}\nu _{1}\ldots +\mu _{h}\nu _{h}-\psi }{\Theta }}}{{\nu _{1}}!\ldots {\nu _{h}}!}}.$

(521)

In this chapter, in which we are treating of grand ensembles, ${\overline {u}}$ will always denote the average for a grand ensemble. In the preceding chapters, ${\overline {u}}$ has always denoted the average for a petit ensemble.