Page:Elementary Principles in Statistical Mechanics (1902).djvu/129

${\frac {\psi -\epsilon }{\Theta }}={\frac {\psi -\epsilon _{0}}{\Theta }}-{\frac {\epsilon -\epsilon _{0}}{\Theta }},$

we get by (336)

${\frac {\psi -\epsilon }{\Theta }}+\phi ={\frac {\psi -\epsilon _{0}}{\Theta }}+\phi _{0}+{\bigg (}{\frac {d^{2}\phi }{d\epsilon ^{2}}}{\bigg )}{\frac {(\epsilon -\epsilon _{0})^{2}}{2}}+{\bigg (}{\frac {d^{3}\phi }{d\epsilon ^{3}}}{\bigg )}{\frac {(\epsilon -\epsilon _{0})^{3}}{3!}}+\mathrm {etc.}$

(339)

Substituting this value in

$\int \limits _{\epsilon '}^{\epsilon ''}e^{{\frac {\psi -\epsilon }{\Theta }}+\phi }\,d\epsilon ,$

which expresses the probability that the energy of an unspecified system of the ensemble lies between the limits $\epsilon '$ and $\epsilon ''$, we get

$e^{{\frac {\psi -\epsilon _{0}}{\Theta }}+\phi _{0}}\int \limits _{\epsilon '}^{\epsilon ''}e^{{\big (}{\frac {d^{2}\phi }{d\epsilon ^{2}}}{\big )}_{0}{\frac {(\epsilon -\epsilon _{0})^{2}}{2}}+{\big (}{\frac {d^{3}\phi }{d\epsilon ^{3}}}{\big )}_{0}{\frac {(\epsilon -\epsilon _{0})^{3}}{3!}}+\mathrm {etc.} }\,d\epsilon .$

(340)

When the number of degrees of freedom is very great, and $\epsilon -\epsilon _{0}$ in consequence very small, we may neglect the higher powers and write^[1]

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

(341)

This shows that for a very great number of degrees of freedom the probability of deviations of energy from the most probable value ($\epsilon _{0}$) approaches the form expressed by the 'law of errors.' With this approximate law, we get

1. If a higher degree of accuracy is desired than is afforded by this formula, it may be multiplied by the series obtained from

   $e^{{\big (}{\frac {d^{3}\phi }{d\epsilon ^{3}}}{\big )}_{0}{\frac {(\epsilon -\epsilon _{0})^{3}}{3!}}+\mathrm {etc.} }$

   by the ordinary formula for the expansion in series of an exponential function. There would be no especial analytical difficulty in taking account of a moderate number of terms of such a series, which would commence

   $1+{\bigg (}{\frac {d^{3}\phi }{d\epsilon ^{3}}}{\bigg )}_{0}{\frac {(\epsilon -\epsilon _{0})^{3}}{3!}}+{\bigg (}{\frac {d^{4}\phi }{d\epsilon ^{4}}}{\bigg )}_{0}{\frac {(\epsilon -\epsilon _{0})^{4}}{4!}}+\mathrm {etc.}$