Page:Elementary Principles in Statistical Mechanics (1902).djvu/98

That is, a diminution of the modulus will diminish the probability of all configurations for which the potential energy exceeds its average value in the ensemble. Again, in the ensemble having modulus $\Theta '$ and average energies ${\overline {\epsilon }}_{p}{}'$ and ${\overline {\epsilon }}_{q}{}'$, values of ${\overline {\epsilon }}_{q}$ sensibly less than ${\overline {\epsilon }}_{q}{}'$ will be so rare as to be practically negligible. They will be still more rare in an ensemble of greater modulus, since by the same equation an increase of the modulus will diminish the probability of configurations for which the potential energy is less than its average value in the ensemble. Therefore, for values of $\Theta$ between $\Theta '$ and $\Theta ''$, and of ${\overline {\epsilon }}_{p}$ between ${\overline {\epsilon }}_{p}{}'$ and ${\overline {\epsilon }}_{p}{}''$, the individual values of ${\overline {\epsilon }}_{q}$ will be practically limited to the interval between ${\overline {\epsilon }}_{q}{}'$ and ${\overline {\epsilon }}_{q}{}''$.

In the cases which remain to be considered, viz., when $d{\overline {\epsilon }}_{q}/d{\overline {\epsilon }}_{p}$ has very large values not confined to narrow limits, and consequently the differences of the mean potential energies in ensembles of different moduli are in general very large compared with the differences of the mean kinetic energies, it appears by (210) that the anomalies of mean square of potential energy, if not small in comparison with the mean kinetic energy, will yet in general be very small in comparison with differences of mean potential energy in ensembles having moderate differences of mean kinetic energy,—the exceptions being of the same character as described for the case when $d{\overline {\epsilon }}_{q}/d{\overline {\epsilon }}_{p}$ is not in general large.

It follows that to human experience and observation with respect to such an ensemble as we are considering, or with respect to systems which may be regarded as taken at random from such an ensemble, when the number of degrees of freedom is of such order of magnitude as the number of molecules in the bodies subject to our observation and experiment, $\epsilon -{\overline {\epsilon }}$, $\epsilon _{p}-{\overline {\epsilon }}_{p}$, $\epsilon _{q}-{\overline {\epsilon }}_{q}$ would be in general vanishing quantities, since such experience would not be wide enough to embrace the more considerable divergencies from the mean values, and such observation not nice enough to distinguish the ordinary divergencies. In other words, such ensembles would appear to human observation as ensembles of systems of uniform energy, and in which the potential and kinetic energies (sup-