Page:Elementary Principles in Statistical Mechanics (1902).djvu/104

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mined. So also if ${\displaystyle \psi _{q}}$ or ${\displaystyle {\overline {\epsilon }}_{q}}$ is given as a function of ${\displaystyle \Theta }$, all averages of the form ${\displaystyle {\overline {\epsilon _{q}{}^{h}}}}$ or ${\displaystyle {\overline {(\epsilon _{q}-{\overline {\epsilon }}_{q})^{h})}}}$ are determined. But

${\displaystyle {\overline {\epsilon }}_{q}={\overline {\epsilon }}-{\tfrac {1}{2}}n\Theta .}$

Therefore if any one of the quantities ${\displaystyle \psi }$, ${\displaystyle \psi _{q}}$, ${\displaystyle {\overline {\epsilon }}}$, ${\displaystyle {\overline {\epsilon }}_{q}}$ is known as a function of ${\displaystyle \Theta }$, and ${\displaystyle n}$ is also known, all averages of any of the forms mentioned are thereby determined as functions of the same variable. In any case all averages of the form

${\displaystyle {\overline {{\bigg (}{\frac {\epsilon _{p}-{\overline {\epsilon }}_{p}}{{\overline {\epsilon }}_{p}}}{\bigg )}^{h}}}}$

are known in terms of ${\displaystyle n}$ alone, and have the same value whether taken for the whole ensemble or limited to any particular configuration.

If we differentiate the equation

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

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with respect to ${\displaystyle a_{1}}$, and multiply by ${\displaystyle \Theta }$, we have

${\displaystyle \int \ldots \int {\bigg [}{\frac {d\psi }{da_{1}}}-{\frac {d\epsilon }{da_{1}}}{\bigg ]}e^{\frac {\psi -\epsilon }{\Theta }}\,dp_{1}\ldots dq_{n}=0.}$

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Differentiating again, with respect to ${\displaystyle a_{1}}$, with respect to ${\displaystyle a_{2}}$, and with respect to ${\displaystyle \Theta }$, we have

${\displaystyle \int \ldots \int {\bigg [}{\frac {d^{2}\psi }{da_{1}{}^{2}}}-{\frac {d^{2}\epsilon }{da_{1}{}^{2}}}+{\frac {1}{\Theta }}{\bigg (}{\frac {d\psi }{da_{1}}}-{\frac {d\epsilon }{da_{1}}}{\bigg )}^{2}{\bigg ]}e^{\frac {\psi -\epsilon }{\Theta }}\,dp_{1}\ldots dq_{n}=0,}$

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${\displaystyle {\begin{aligned}\int \ldots \int {\bigg [}{\frac {d^{2}\psi }{da_{1}\,da_{2}}}-{\frac {d^{2}\epsilon }{da_{1}\,da_{2}}}&+{\frac {1}{\Theta }}{\bigg (}{\frac {d\psi }{da_{1}}}-{\frac {d\epsilon }{da_{1}}}{\bigg )}{\bigg (}{\frac {d\psi }{da_{2}}}-{\frac {d\epsilon }{da_{2}}}{\bigg )}{\bigg ]}\\&\qquad \qquad \qquad e^{\frac {\psi -\epsilon }{\Theta }}\,dp_{1}\ldots dq_{n}=0,\end{aligned}}}$

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${\displaystyle {\begin{aligned}\int \ldots \int {\bigg [}{\frac {d^{2}\psi }{da_{1}\,d\Theta }}+{\bigg (}{\frac {d\psi }{da_{1}}}-{\frac {d\epsilon }{da_{1}}}{\bigg )}&{\bigg (}{\frac {1}{\Theta }}{\frac {d\psi }{d\Theta }}-{\frac {\psi -\epsilon }{\Theta ^{2}}}{\bigg )}{\bigg ]}\\&\qquad e^{\frac {\psi -\epsilon }{\Theta }}\,dp_{1}\ldots dq_{n}=0.\end{aligned}}}$

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