Page:Elementary Principles in Statistical Mechanics (1902).djvu/132

falls within any given limits of energy ($\epsilon '$ and $\epsilon ''$) is represented by

$\int \limits _{\epsilon '}^{\epsilon ''}e^{\eta +\phi }\,d\epsilon .$

If we expand $\eta$ and $\phi$ in ascending powers of $\epsilon -\epsilon _{0}$, without going beyond the squares, the probability that the energy falls within the given limits takes the form of the 'law of errors'—

$e^{\eta _{0}+\phi _{0}}\int \limits _{\epsilon '}^{\epsilon ''}e^{{\big [}{\big (}{\frac {d^{2}\eta }{d\epsilon ^{2}}}{\big )}_{0}+{\big (}{\frac {d^{2}\phi }{d\epsilon ^{2}}}{\big )}_{0}{\big ]}{\frac {(\epsilon -\epsilon _{0})^{2}}{2}}}\,d\epsilon .$

(353)

This gives

$\eta _{0}+\phi _{0}={\tfrac {1}{2}}\log {\bigg [}{\frac {-1}{2\pi }}{\bigg (}{\frac {d^{2}\eta }{d\epsilon ^{2}}}{\bigg )}_{0}-{\frac {1}{2\pi }}{\bigg (}{\frac {d^{2}\phi }{d\epsilon ^{2}}}{\bigg )}_{0}{\bigg ]},$

(354)

and

${\overline {(\epsilon -\epsilon _{0})^{2}}}={\bigg [}-{\bigg (}{\frac {d^{2}\eta }{d\epsilon ^{2}}}{\bigg )}_{0}-{\bigg (}{\frac {d^{2}\phi }{d\epsilon ^{2}}}{\bigg )}_{0}{\bigg ]}^{-1}.$

(355)

We shall have a close approximation in general when the quantities equated in (355) are very small, i. e., when

$-{\bigg (}{\frac {d^{2}\eta }{d\epsilon ^{2}}}{\bigg )}_{0}-{\bigg (}{\frac {d^{2}\phi }{d\epsilon ^{2}}}{\bigg )}_{0}$

(356)

is very great. Now when $n$ is very great, $-d^{2}\phi /d\epsilon ^{2}$ is of the same order of magnitude, and the condition that (356) shall be very great does not restrict very much the nature of the function $\eta$.

We may obtain other properties pertaining to average values in a canonical ensemble by the method used for the average of $d\phi /d\epsilon$. Let $u$ be any function of the energy, either alone or with $\Theta$ and the external coördinates. The average value of $u$ in the ensemble is determined by the equation

${\overline {u}}=\int \limits _{V=0}^{\epsilon =\infty }u\,e^{{\frac {\psi -\epsilon }{\Theta }}+\phi }\,d\epsilon .$

(357)