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

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

(342)

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

(343)

whence

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

(344)

Now it has been proved in Chapter VII that

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

We have therefore

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

(345)

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

Equation (338) gives for the first approximation

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

(346)

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

(347)

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

(348)

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

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

1. ↑ Compare (289), (314).