Page:Elementary Principles in Statistical Mechanics (1902).djvu/149

Let us imagine an ensemble of systems distributed in phase according to the index of probability

$c-{\frac {(\epsilon -\epsilon ')^{2}}{\omega ^{2}}},$

where $\epsilon '$ is any constant which is a possible value of the energy, except only the least value which is consistent with the values of the external coördinates, and $c$ and $\omega$ are other constants. We have therefore

$\mathop {\int \ldots \int } _{\rm {phases}}^{\rm {all}}e^{c-{\frac {(\epsilon -\epsilon ')^{2}}{\omega ^{2}}}}\,dp_{1}\ldots dq_{n}=1,$

(403)

or

$e^{-c}=\mathop {\int \ldots \int } _{\rm {phases}}^{\rm {all}}e^{-{\frac {(\epsilon -\epsilon ')^{2}}{\omega ^{2}}}}\,dp_{1}\ldots dq_{n},$

(404)

or again

$e^{-c}=\int \limits _{V=0}^{\epsilon =\infty }e^{-{\frac {(\epsilon -\epsilon ')^{2}}{\omega ^{2}}}+\phi }\,d\epsilon .$

(405)

From (404) we have

${\begin{aligned}{\frac {de^{-c}}{da_{1}}}&=\mathop {\int \ldots \int } _{\rm {phases}}^{\rm {all}}2{\frac {\epsilon -\epsilon '}{\omega ^{2}}}A_{1}\,e^{-{\frac {(\epsilon -\epsilon ')^{2}}{\omega ^{2}}}}\,dp_{1}\ldots dq_{n}\\&=\int \limits _{V=0}^{\epsilon =\infty }2{\frac {\epsilon -\epsilon '}{\omega ^{2}}}{\overline {A_{1}}}|_{\epsilon }\,e^{-{\frac {(\epsilon -\epsilon ')^{2}}{\omega ^{2}}}+\phi }\,d\epsilon ,\end{aligned}}$

(406)

where ${\overline {A_{1}}}|_{\epsilon }$ denotes the average value of $A_{1}$ in those systems of the ensemble which have any same energy $\epsilon$. (This is the same thing as the average value of $A_{1}$ in a microcanonical ensemble of energy $\epsilon$.) The validity of the transformation is evident, if we consider separately the part of each integral which lies between two infinitesimally differing limits of energy. Integrating by parts, we get