Page:Elementary Principles in Statistical Mechanics (1902).djvu/97

${\overline {\epsilon }}_{p}={\tfrac {1}{2}}n\Theta ,$

${\frac {\overline {(\epsilon _{p}-{\overline {\epsilon _{p}}})^{2}}}{{\overline {\epsilon _{p}}}^{2}}}={\frac {2}{n}},$

(209)

${\frac {\overline {(\epsilon _{q}-{\overline {\epsilon _{q}}})^{2}}}{{\overline {\epsilon _{p}}}^{2}}}={\frac {2}{n}}{\frac {d{\overline {\epsilon _{q}}}}{d{\overline {\epsilon _{p}}}}},$

(210)

${\frac {\overline {(\epsilon -{\overline {\epsilon }})^{2}}}{{\overline {\epsilon _{p}}}^{2}}}={\frac {2}{n}}{\frac {d{\overline {\epsilon }}}{d{\overline {\epsilon _{p}}}}}={\frac {2}{n}}+{\frac {2}{n}}{\frac {d{\overline {\epsilon _{q}}}}{d{\overline {\epsilon _{p}}}}}.$

(211)

These equations show that when the number of degrees of freedom of the systems is very great, the mean squares of the anomalies of the energies (total, potential, and kinetic) are very small in comparison with the mean square of the kinetic energy, unless indeed the differential coefficient $d{\overline {\epsilon }}_{q}/d{\overline {\epsilon }}_{p}$ is of the same order of magnitude as $n$. Such values of $d{\overline {\epsilon }}_{q}/d{\overline {\epsilon }}_{p}$ can only occur within intervals (${\overline {\epsilon }}_{p}{}''-{\overline {\epsilon }}_{p}{}'$) which are of the order of magnitude of $n^{-1}$ unless it be in cases in which ${\overline {\epsilon }}_{q}$ is in general of an order of magnitude higher than ${\overline {\epsilon }}_{p}$. Postponing for the moment the consideration of such cases, it will be interesting to examine more closely the case of large values of $d{\overline {\epsilon }}_{q}/d{\overline {\epsilon }}_{p}$ within narrow limits. Let us suppose that for ${\overline {\epsilon }}_{p}{}'$ and ${\overline {\epsilon }}_{p}{}''$ the value of $d{\overline {\epsilon }}_{q}/d{\overline {\epsilon }}_{p}$ is of the order of magnitude of unity, but between these values of ${\overline {\epsilon }}_{p}$ very great values of the differential coefficient occur. Then in the ensemble having modulus $\Theta ''$ and average energies ${\overline {\epsilon }}_{p}{}''$ and ${\overline {\epsilon }}_{q}{}''$, values of $\epsilon _{q}$ sensibly greater than ${\overline {\epsilon }}_{q}{}''$ will be so rare that we may call them practically negligible. They will be still more rare in an ensemble of less modulus. For if we differentiate the equation

$\eta _{q}={\frac {\psi _{q}-\epsilon _{q}}{\Theta }}$

regarding $\epsilon _{q}$ as constant, but $\Theta$ and therefore $\psi _{q}$ as variable, we get

${\bigg (}{\frac {d\eta _{q}}{d\Theta }}{\bigg )}_{\epsilon _{q}}={\frac {1}{\Theta }}{\frac {d\psi _{q}}{d\Theta }}-{\frac {\psi _{q}-\epsilon _{q}}{\Theta }},$

(212)

whence by (192)

${\bigg (}{\frac {d\eta _{q}}{d\Theta }}{\bigg )}_{\epsilon _{q}}={\frac {\epsilon _{q}-{\overline {\epsilon }}_{q}}{\Theta ^{2}}}.$

(213)