Page:Elementary Principles in Statistical Mechanics (1902).djvu/146

average value of $u$ in an ensemble in which the whole system is microcanonically distributed in phase, viz.,

${\overline {u}}|_{\epsilon }=e^{-\phi }\int \limits _{V_{2}=0}^{\epsilon _{2}=\epsilon }u\,e^{\phi _{1}}\,dV_{2},$

(387)

where $\phi _{1}$ and $V_{2}$ are connected by the equation

$\epsilon _{1}+\epsilon _{2}=\mathrm {constant} =\epsilon ,$

(388)

and $u$, if given as function of $\epsilon _{1}$, or of $\epsilon _{1}$ and $\epsilon _{2}$, becomes in virtue of the same equation a function of $\epsilon _{2}$ alone.^[1] Thus

${\overline {e^{-\phi _{1}}V_{1}}}|_{\epsilon }=e^{-\phi }\int \limits _{V_{2}=0}^{\epsilon _{2}=\epsilon }V_{1}\,dV_{2},$

(389)

$e^{-\phi }V={\overline {e^{-\phi _{1}}V_{1}}}|_{\epsilon }={\overline {e^{-\phi _{2}}V_{2}}}|_{\epsilon }.$

(390)

This requires a similar relation for canonical averages

$\Theta ={\overline {e^{-\phi }V}}|_{\Theta }={\overline {e^{-\phi _{1}}V_{1}}}|_{\Theta }={\overline {e^{-\phi _{2}}V_{2}}}|_{\Theta }.$

(391)

Again

${\overline {\frac {d\phi _{1}}{d\epsilon _{1}}}}{\bigg |}_{\epsilon }=e^{-\phi }\int \limits _{V_{2}=0}^{\epsilon _{2}=\epsilon }{\frac {d\phi _{1}}{d\epsilon _{1}}}e^{\phi _{1}}\,dV_{2}.$

(392)

But if $n_{1}>2$, $e^{\phi _{1}}$ vanishes for $V_{1}=0$,^[2] and

${\frac {d}{d\epsilon }}e^{\phi }={\frac {d}{d\epsilon }}\int \limits _{V_{2}=0}^{\epsilon _{2}=\epsilon }e^{\phi _{1}}\,dV_{2}=\int \limits _{V_{2}=0}^{\epsilon _{2}=\epsilon }{\frac {d\phi _{1}}{d\epsilon _{1}}}\,e^{\phi _{1}}\,dV_{2}.$

(393)

Hence, if $n_{1}>2$, and $n_{2}>2$,

${\frac {d\phi }{d\epsilon }}={\overline {\frac {d\phi _{1}}{d\epsilon _{1}}}}{\bigg |}_{\epsilon }={\overline {\frac {d\phi _{2}}{d\epsilon _{2}}}}{\bigg |}_{\epsilon },$

(394)

1. In the applications of the equation (387), we cannot obtain all the results corresponding to those which we have obtained from equation (374), because $\phi _{p}$ is a known function of $\epsilon _{p}$, while $\phi _{1}$ must be treated as an arbitrary function of $\epsilon _{1}$, or nearly so.
2. See Chapter VIII, equations (306) and (316).