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

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

(516)

or

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

(517)

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

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

(518)

which agrees with (506).

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

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

(519)

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

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

(520)

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

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

(521)

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