Page:Elementary Principles in Statistical Mechanics (1902).djvu/101

We have therefore

${\displaystyle {\overline {\epsilon _{p}{}^{2}}}={\bigg (}{\frac {n}{2}}+1{\bigg )}{\frac {n}{2}}\Theta ^{2}.}$

(226)

${\displaystyle {\overline {\epsilon _{p}{}^{3}}}={\bigg (}{\frac {n}{2}}+2{\bigg )}{\bigg (}{\frac {n}{2}}+1{\bigg )}{\frac {n}{2}}\Theta ^{2}.}$

(227)

${\displaystyle {\overline {\epsilon _{p}{}^{h}}}={\frac {\Gamma ({\tfrac {1}{2}}n+h)}{\Gamma ({\tfrac {1}{2}}n)}}\Theta ^{h}.}$

^[1](228)

The average values of the powers of the anomalies of the energies are perhaps most easily found as follows. We have identically, since ${\displaystyle {\overline {\epsilon }}}$ is a function of ${\displaystyle \Theta }$, while ${\displaystyle \epsilon }$ is a function of the ${\displaystyle p}$'s and ${\displaystyle q}$'s,

${\displaystyle {\begin{aligned}&\Theta ^{2}{\frac {d}{d\Theta }}\mathop {\int \ldots \int } _{\rm {phases}}^{\rm {all}}(\epsilon -{\overline {\epsilon }})^{h}e^{-{\frac {\epsilon }{\Theta }}}\,dp_{1},\ldots dq_{n}=\\&\mathop {\int \ldots \int } _{\rm {phases}}^{\rm {all}}{\bigg [}\epsilon (\epsilon -{\overline {\epsilon }})^{h}-h(\epsilon -{\overline {\epsilon }})^{h-1}\Theta ^{2}{\frac {d{\overline {\epsilon }}}{d\Theta }}{\bigg ]}e^{-{\frac {\epsilon }{\Theta }}}\,dp_{1},\ldots dq_{n}\end{aligned}}}$

(229)

${\displaystyle \Theta ^{2}{\frac {d}{d\Theta }}{\bigg [}{\overline {(\epsilon -{\overline {\epsilon }})^{h}}}e^{-{\frac {\psi }{\Theta }}}{\bigg ]}={\bigg [}{\overline {\epsilon (\epsilon -{\overline {\epsilon }})^{h}}}-h{\overline {(\epsilon -{\overline {\epsilon }})^{h-1}}}\Theta ^{2}{\frac {d{\overline {\epsilon }}}{d\Theta }}{\bigg ]}e^{-{\frac {\psi }{\Theta }}},}$

(230)

1. ↑ In the case discussed in the note on page 54 we may easily get
   ${\displaystyle {\overline {(\epsilon _{q}-\epsilon _{a})^{h}}}={\bigg (}{\overline {\epsilon }}_{q}-\epsilon _{a}+\Theta ^{2}{\frac {d}{d\Theta }}{\bigg )}{\overline {(\epsilon _{q}-\epsilon _{a})^{h-1}}},}$
   which, with
   ${\displaystyle {\overline {\epsilon }}_{q}-\epsilon _{a}={\frac {n}{2}}\Theta ,}$
   gives
   ${\displaystyle {\overline {(\epsilon _{q}-\epsilon _{a})^{h}}}={\bigg (}{\frac {n}{2}}\Theta +\Theta ^{2}{\frac {d}{d\Theta }}{\bigg )}{\overline {(\epsilon _{q}-\epsilon _{a})^{h-1}}}={\frac {n}{2}}{\bigg (}{\frac {n}{2}}\Theta +\Theta ^{2}{\frac {d}{d\Theta }}{\bigg )}^{h-1}\Theta .}$
   Hence
   ${\displaystyle {\overline {(\epsilon _{q}-\epsilon _{a})^{h}}}={\overline {\epsilon _{p}{}^{h}}}.}$
   Again
   ${\displaystyle {\overline {(\epsilon -\epsilon _{a})^{h}}}={\bigg (}{\overline {\epsilon }}-\epsilon _{a}+\Theta ^{2}{\frac {d}{d\Theta }}{\bigg )}{\overline {(\epsilon -\epsilon _{a})^{h-1}}},}$
   which with
   ${\displaystyle {\overline {\epsilon }}-\epsilon _{a}=n\Theta }$
   gives
   ${\displaystyle {\overline {(\epsilon -\epsilon _{a})^{h}}}={\bigg (}n\Theta +\Theta ^{2}{\frac {d}{d\Theta }}{\bigg )}{\overline {(\epsilon -\epsilon _{a})^{h-1}}}=n{\bigg (}n\Theta +\Theta ^{2}{\frac {d}{d\Theta }}{\bigg )}^{h-1}\Theta ,}$
   hence
   ${\displaystyle {\overline {(\epsilon -\epsilon _{a})^{h}}}={\frac {\Gamma (n+h)}{\Gamma (n)}}\Theta ^{h}.}$