Page:Elementary Principles in Statistical Mechanics (1902).djvu/94

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70

AVERAGE VALUES IN A CANONICAL

are also identical with those given by Clausius for the corresponding quantities.

Equations (112) and (181) show that if $\psi$ or $\psi _{q}$ is known as function of $\Theta$ and $a_{1}$ , $a_{2}$ , etc., we can obtain by differentiation ${\overline {\epsilon }}$ or ${\overline {\epsilon }}_{q}$ , and ${\overline {A}}_{1}$ , ${\overline {A}}_{2}$ etc. as functions of the same variables. We have in fact

{\overline {\epsilon }}=\psi -\Theta {\overline {\eta }}=\psi -\Theta {\frac {d\psi }{d\Theta }}.

(191)

{\overline {\epsilon }}_{q}=\psi _{q}-\Theta {\overline {\eta }}_{q}=\psi _{q}-\Theta {\frac {d\psi _{q}}{d\Theta }}.

(192)

The corresponding equation relating to kinetic energy,

{\overline {\epsilon }}_{p}=\psi _{p}-\Theta \eta _{p}=\psi _{p}-\Theta {\frac {d\psi _{p}}{d\Theta }}.

(193)

which may be obtained in the same way, may be verified by the known relations (186), (187), and (188) between the variables. We have also

{\overline {A}}_{1}=-{\frac {d\psi }{da_{1}}}=-{\frac {d\psi _{q}}{da_{1}}},

(194)

etc., so that the average values of the external forces may be derived alike from

\psi

or from

\psi _{q}

.

The average values of the squares or higher powers of the energies (total, potential, or kinetic) may easily be obtained by repeated differentiations of $\psi$ , $\psi _{q}$ , $\psi _{p}$ , or ${\overline {\epsilon }}$ , ${\overline {\epsilon }}_{q}$ , ${\overline {\epsilon }}_{p}$ , with respect to $\Theta$ . By equation (108) we have

{\overline {\epsilon }}=\mathop {\int \ldots \int } _{\rm {phases}}^{\rm {all}}\epsilon e^{\frac {\psi -\epsilon }{\Theta }}\,dp_{1}\ldots dq_{n},

(195)

and differentiating with respect to

\Theta

,

{\frac {d{\overline {\epsilon }}}{d\Theta }}=\mathop {\int \ldots \int } _{\rm {phases}}^{\rm {all}}{\bigg (}{\frac {\epsilon ^{2}-\psi \epsilon }{\Theta ^{2}}}+{\frac {\epsilon }{\Theta }}{\frac {d\psi }{d\Theta }}{\bigg )}e^{\frac {\psi -\epsilon }{\Theta }}\,dp_{1}\ldots dq_{n},

(196)

whence, again by (108),

{\frac {d{\overline {\epsilon }}}{d\Theta }}={\frac {{\overline {\epsilon ^{2}}}-\psi {\overline {\epsilon }}}{\Theta ^{2}}}+{\frac {\overline {\epsilon }}{\Theta }}{\frac {d\psi }{d\Theta }},