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

This page has been proofread, but needs to be validated.

122

A PERMANENT DISTRIBUTION IN WHICH

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)

↑ 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.
↑ See Chapter VIII, equations (306) and (316).