Page:Elementary Principles in Statistical Mechanics (1902).djvu/102

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78

AVERAGE VALUES IN A CANONICAL

or since by (218)

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

\Theta ^{2}{\frac {d}{d\Theta }}{\overline {(\epsilon -{\overline {\epsilon }})^{h}}}+{\overline {(\epsilon -{\overline {\epsilon }})^{h}}}{\overline {\epsilon }}={\overline {\epsilon (\epsilon -{\overline {\epsilon }})^{h}}}-h{\overline {(\epsilon -{\overline {\epsilon }})^{h-1}}}\Theta ^{2}{\frac {d{\overline {\epsilon }}}{d\Theta }},

{\overline {(\epsilon -{\overline {\epsilon }})^{h+1}}}=\Theta ^{2}{\frac {d}{d\Theta }}{\overline {(\epsilon -{\overline {\epsilon }})^{h}}}+h{\overline {(\epsilon -{\overline {\epsilon }})^{h-1}}}\Theta ^{2}{\frac {d{\overline {\epsilon }}}{d\Theta }}.

(231)

In precisely the same way we may obtain for the potential energy

{\overline {(\epsilon _{q}-{\overline {\epsilon }}_{q})^{h+1}}}=\Theta ^{2}{\frac {d}{d\Theta }}{\overline {(\epsilon _{q}-{\overline {\epsilon }}_{q})^{h}}}+h{\overline {(\epsilon _{q}-{\overline {\epsilon }}_{q})^{h-1}}}\Theta ^{2}{\frac {d{\overline {\epsilon }}_{q}}{d\Theta }}.

(232)

By successive applications of (231) we obtain

{\begin{aligned}{\overline {(\epsilon -{\overline {\epsilon }})^{2}}}&=D{\overline {\epsilon }}\\{\overline {(\epsilon -{\overline {\epsilon }})^{3}}}&=D^{2}{\overline {\epsilon }}\\{\overline {(\epsilon -{\overline {\epsilon }})^{4}}}&=D^{3}{\overline {\epsilon }}+3(D{\overline {\epsilon }})^{2}\\{\overline {(\epsilon -{\overline {\epsilon }})^{5}}}&=D^{4}{\overline {\epsilon }}+10D{\overline {\epsilon }}D^{2}{\overline {\epsilon }}\\{\overline {(\epsilon -{\overline {\epsilon }})^{6}}}&=D^{5}{\overline {\epsilon }}+15D{\overline {\epsilon }}D^{3}{\overline {\epsilon }}+10(D^{2}{\overline {\epsilon }})^{2}+15(D{\overline {\epsilon }})^{3}\mathrm {etc.} \end{aligned}}

where

D

represents the operator

\Theta ^{2}\,d/d\Theta

. Similar expressions relating to the potential energy may be derived from (232).

For the kinetic energy we may write similar equations in which the averages may be taken either for a single configuration or for the whole ensemble. But since

{\frac {d{\overline {\epsilon }}_{p}}{d\Theta }}={\frac {n}{2}}

the general formula reduces to

{\overline {(\epsilon _{p}-{\overline {\epsilon }}_{p})^{h+1}}}=\Theta ^{2}{\frac {d}{d\Theta }}{\overline {(\epsilon _{p}-{\overline {\epsilon }}_{p})^{h}}}+{\tfrac {1}{2}}nh\Theta ^{2}{\overline {(\epsilon _{p}-{\overline {\epsilon }}_{p})^{h-1}}}

(233)

or

{\frac {\overline {(\epsilon _{p}-{\overline {\epsilon }}_{p})^{h+1}}}{{\overline {\epsilon }}_{p}{}^{h+1}}}={\frac {2\Theta }{n}}{\frac {d}{d\Theta }}{\frac {\overline {(\epsilon _{p}-{\overline {\epsilon }}_{p})^{h}}}{{\overline {\epsilon }}_{p}{}^{h}}}+{\frac {2h}{n}}{\frac {\overline {(\epsilon _{p}-{\overline {\epsilon }}_{p})^{h}}}{{\overline {\epsilon }}_{p}{}^{h}}}+{\frac {2h}{n}}{\frac {\overline {(\epsilon _{p}-{\overline {\epsilon }}_{p})^{h-1}}}{{\overline {\epsilon }}_{p}{}^{h-1}}}

(234)