Page:Elementary Principles in Statistical Mechanics (1902).djvu/125

its most common or most probable value, and which is determined by the equation

${\frac {d\phi }{d\epsilon }}={\frac {1}{\Theta }}.$

(319)

This value of $d\phi /d\epsilon$ is also, when $n>2$, its average value in the ensemble. For we have identically, by integration by parts,

$\int \limits _{V=0}^{\epsilon =\infty }{\frac {d\phi }{d\epsilon }}e^{{\frac {\psi -\epsilon }{\Theta }}+\phi }\,d\epsilon ={\bigg [}e^{{\frac {\psi -\epsilon }{\Theta }}+\phi }{\bigg ]}_{V=0}^{\epsilon =\infty }+{\frac {1}{\Theta }}\int \limits _{V=0}^{\epsilon =\infty }e^{{\frac {\psi -\epsilon }{\Theta }}+\phi }\,d\epsilon .$

(320)

If $n>2$, the expression in the brackets, which multiplied by $N$ would represent the density-in-energy, vanishes at the limits, and we have by (269) and (318)

${\overline {\frac {d\phi }{d\epsilon }}}={\frac {1}{\Theta }}.$

(321)

It appears, therefore, that for systems of more than two degrees of freedom, the average value of $d\phi /d\epsilon$ in an ensemble canonically distributed is identical with the value of the same differential coefficient as calculated for the most common energy in the ensemble, both values being reciprocals of the modulus.

Hitherto, in our consideration of the quantities $V$, $V_{q}$, $V_{p}$, $\phi$, $\phi _{q}$, $\phi _{p}$, we have regarded the external coördinates as constant. It is evident, however, from their definitions that $V$ and $\phi$ are in general functions of the external coördinates and the energy ($\epsilon$), that $V_{q}$ and $\phi _{q}$ are in general functions of the external coördinates and the potential energy ($\epsilon _{q}$). $V_{p}$ and $\phi _{p}$ we have found to be functions of the kinetic energy ($\epsilon _{p}$) alone. In the equation

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

(322)

by which $\psi$ may be determined, $\Theta$ and the external coördinates (contained implicitly in $\phi$) are constant in the integration. The equation shows that $\psi$ is a function of these constants.