Page:Elementary Principles in Statistical Mechanics (1902).djvu/130

$e^{{\frac {\psi -\epsilon _{0}}{\Theta }}+\phi _{0}}\left({\frac {-2\pi }{{\Big (}{\frac {d^{2}\phi }{d\epsilon ^{2}}}{\Big )}_{0}}}\right)^{\frac {1}{2}}=1,$

(342)

${\overline {\epsilon }}=\epsilon _{0},\qquad {\overline {(\epsilon -\epsilon _{0})^{2}}}=-{\bigg (}{\frac {d^{2}\phi }{d\epsilon ^{2}}}{\bigg )}^{-},$

(343)

whence

${\frac {\psi -\epsilon _{0}}{\Theta }}+\phi _{0}={\tfrac {1}{2}}\log {\frac {{\Big (}{\frac {d^{2}\phi }{d\epsilon ^{2}}}{\Big )}_{0}}{-2\pi }}=-{\tfrac {1}{2}}\log(2\pi {\overline {(\epsilon -{\overline {\epsilon }})^{2}}}).$

(344)

Now it has been proved in Chapter VII that

${\overline {(\epsilon -{\overline {\epsilon }})^{2}}}={\frac {2}{n}}{\frac {d{\overline {\epsilon }}}{d{\overline {\epsilon }}_{p}}}{\overline {\epsilon }}_{p}{}^{2}.$

We have therefore

${\overline {\eta }}+\phi _{0}={\frac {\psi -\epsilon _{0}}{\Theta }}+\phi _{0}=-{\tfrac {1}{2}}\log(2\pi {\overline {(\epsilon -{\overline {\epsilon }})^{2}}})=-{\tfrac {1}{2}}\log {\bigg (}{\frac {4\pi }{n}}{\frac {d{\overline {\epsilon }}}{d{\overline {\epsilon }}_{p}}}{\overline {\epsilon }}_{p}{}^{2}{\bigg )}$

(345)

approximately. The order of magnitude of ${\overline {\eta }}-\phi _{0}$ is therefore that of $\log n$. This magnitude is mainly constant. The order of magnitude of ${\overline {\eta }}+\phi _{0}-{\tfrac {1}{2}}\log n$ is that of unity. The order of magnitude of $\phi _{0}$, and therefore of $-{\overline {\eta }}$, is that of $n$.^[1]

Equation (338) gives for the first approximation

${\overline {\phi }}-\phi _{0}={\bigg (}{\frac {d^{2}\phi }{d\epsilon ^{2}}}{\bigg )}_{0}{\frac {\overline {(\epsilon -\epsilon _{0})^{2}}}{2}}=-{\tfrac {1}{2}},$

(346)

${\overline {(\phi -\phi _{0})(\epsilon -\epsilon _{0})}}={\frac {\overline {(\epsilon -\epsilon _{0})^{2}}}{\Theta }}={\frac {d{\overline {\epsilon }}}{d{\overline {\epsilon }}_{p}}}{\overline {\epsilon }}_{p},$

(347)

${\overline {(\phi -\phi _{0})^{2}}}={\frac {\overline {(\epsilon -\epsilon _{0})^{2}}}{\Theta ^{2}}}={\frac {n}{2}}{\frac {d{\overline {\epsilon }}}{d{\overline {\epsilon }}_{p}}}$

(348)

The members of the last equation have the order of magnitude of $n$. Equation (338) gives also for the first approximation

${\frac {d\phi }{d\epsilon }}-{\frac {1}{\Theta }}={\bigg (}{\frac {d^{2}\phi }{d\epsilon ^{2}}}{\bigg )}_{0}(\epsilon -\epsilon _{0}),$

1. Compare (289), (314).