Page:Elementary Principles in Statistical Mechanics (1902).djvu/103

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ENSEMBLE OF SYSTEMS.

79

But since identically

{\frac {\overline {(\epsilon _{p}-{\overline {\epsilon }}_{p})^{0}}}{{\overline {\epsilon }}_{p}{}^{0}}}=1,\quad {\frac {\overline {(\epsilon _{p}-{\overline {\epsilon }}_{p})^{1}}}{{\overline {\epsilon }}_{p}{}^{1}}}=0

the value of the corresponding expression for any index will be independent of

\Theta

and the formula reduces to

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

(235)

we have therefore

{\overline {{\bigg (}{\frac {\epsilon _{p}-{\overline {\epsilon }}_{p}}{{\overline {\epsilon }}_{p}}}{\bigg )}^{0}}}=1,\quad {\overline {{\bigg (}{\frac {\epsilon _{p}-{\overline {\epsilon }}_{p}}{{\overline {\epsilon }}_{p}}}{\bigg )}^{3}}}={\frac {8}{n^{2}}},

{\overline {{\bigg (}{\frac {\epsilon _{p}-{\overline {\epsilon }}_{p}}{{\overline {\epsilon }}_{p}}}{\bigg )}^{1}}}=0,\quad {\overline {{\bigg (}{\frac {\epsilon _{p}-{\overline {\epsilon }}_{p}}{{\overline {\epsilon }}_{p}}}{\bigg )}^{4}}}={\frac {48}{n^{3}}}+{\frac {12}{n^{2}}},

{\overline {{\bigg (}{\frac {\epsilon _{p}-{\overline {\epsilon }}_{p}}{{\overline {\epsilon }}_{p}}}{\bigg )}^{2}}}={\frac {2}{n}},\quad \mathrm {etc.}

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It will be observed that when $\psi$ or ${\overline {\epsilon }}$ is given as function of $\Theta$ , all averages of the form ${\overline {\epsilon ^{h}}}$ or ${\overline {(\epsilon -{\overline {\epsilon }})^{h}}}$ are thereby deter-

↑ In the case discussed in the preceding foot-notes we get easily
${\overline {(\epsilon _{q}-{\overline {\epsilon }}_{q})^{h}}}={\overline {(\epsilon _{p}-{\overline {\epsilon }}_{p})^{h}}},$
and
${\overline {{\bigg (}{\frac {\epsilon _{q}-{\overline {\epsilon }}_{q}}{{\overline {\epsilon }}_{q}-\epsilon _{a}}}{\bigg )}^{h}}}={\overline {{\bigg (}{\frac {\epsilon _{p}-{\overline {\epsilon }}_{p}}{{\overline {\epsilon }}_{p}}}{\bigg )}^{h}}}.$
For the total energy we have in this case
${\overline {{\bigg (}{\frac {\epsilon -{\overline {\epsilon }}}{{\overline {\epsilon }}-\epsilon _{a}}}{\bigg )}^{h+1}}}={\frac {h}{n}}{\overline {{\bigg (}{\frac {\epsilon -{\overline {\epsilon }}}{{\overline {\epsilon }}-\epsilon _{a}}}{\bigg )}^{h}}}+{\overline {{\frac {h}{n}}{\bigg (}{\frac {\epsilon -{\overline {\epsilon }}}{{\overline {\epsilon }}-\epsilon _{a}}}{\bigg )}^{h-1}}}.$

${\overline {{\bigg (}{\frac {\epsilon -{\overline {\epsilon }}}{{\overline {\epsilon }}-\epsilon _{a}}}{\bigg )}^{2}}}={\frac {1}{n}}.\quad {\overline {{\bigg (}{\frac {\epsilon -{\overline {\epsilon }}}{{\overline {\epsilon }}-\epsilon _{a}}}{\bigg )}^{4}}}={\frac {3}{n^{2}}}+{\frac {6}{n^{3}}},$

${\overline {{\bigg (}{\frac {\epsilon -{\overline {\epsilon }}}{{\overline {\epsilon }}-\epsilon _{a}}}{\bigg )}^{3}}}={\frac {2}{n^{2}}},\quad \mathrm {etc.}$

[p79-1] In the case discussed in the preceding foot-notes we get easily
${\overline {(\epsilon _{q}-{\overline {\epsilon }}_{q})^{h}}}={\overline {(\epsilon _{p}-{\overline {\epsilon }}_{p})^{h}}},$
and
${\overline {{\bigg (}{\frac {\epsilon _{q}-{\overline {\epsilon }}_{q}}{{\overline {\epsilon }}_{q}-\epsilon _{a}}}{\bigg )}^{h}}}={\overline {{\bigg (}{\frac {\epsilon _{p}-{\overline {\epsilon }}_{p}}{{\overline {\epsilon }}_{p}}}{\bigg )}^{h}}}.$
For the total energy we have in this case
${\overline {{\bigg (}{\frac {\epsilon -{\overline {\epsilon }}}{{\overline {\epsilon }}-\epsilon _{a}}}{\bigg )}^{h+1}}}={\frac {h}{n}}{\overline {{\bigg (}{\frac {\epsilon -{\overline {\epsilon }}}{{\overline {\epsilon }}-\epsilon _{a}}}{\bigg )}^{h}}}+{\overline {{\frac {h}{n}}{\bigg (}{\frac {\epsilon -{\overline {\epsilon }}}{{\overline {\epsilon }}-\epsilon _{a}}}{\bigg )}^{h-1}}}.$

${\overline {{\bigg (}{\frac {\epsilon -{\overline {\epsilon }}}{{\overline {\epsilon }}-\epsilon _{a}}}{\bigg )}^{2}}}={\frac {1}{n}}.\quad {\overline {{\bigg (}{\frac {\epsilon -{\overline {\epsilon }}}{{\overline {\epsilon }}-\epsilon _{a}}}{\bigg )}^{4}}}={\frac {3}{n^{2}}}+{\frac {6}{n^{3}}},$

${\overline {{\bigg (}{\frac {\epsilon -{\overline {\epsilon }}}{{\overline {\epsilon }}-\epsilon _{a}}}{\bigg )}^{3}}}={\frac {2}{n^{2}}},\quad \mathrm {etc.}$

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