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

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

we get by (336)

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

(339)

Substituting this value in

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

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

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

(340)

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

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

(341)

This shows that for a very great number of degrees of freedom the probability of deviations of energy from the most probable value (${\displaystyle \epsilon _{0}}$) 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

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

   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

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