Page:Elementary Principles in Statistical Mechanics (1902).djvu/226

From equation (539) we may get an idea of the order of magnitude of the divergences of $\nu _{1}$ from its average value in the ensemble, when that average value is great. The equation may be written

${\frac {(\nu _{1}-{\overline {\nu }}_{1})^{2}}{{\overline {\nu }}_{1}^{2}}}={\frac {\Theta }{{\overline {\nu }}_{1}^{2}}}{\frac {d{\overline {\nu }}_{1}}{d\mu _{1}}},$

(541)

The second member of this equation will in general be small when ${\overline {\nu }}_{1}$ is great. Large values are not necessarily excluded, but they must be confined within very small limits with respect to $\mu$. For if

${\frac {\overline {(\nu _{1}-{\overline {\nu }}_{1})^{2}}}{{\overline {\nu }}_{1}^{2}}}>{\frac {2}{{\overline {\nu }}_{1}^{\frac {1}{2}}}},$

(542)

for all values of $\mu _{1}$ between the limits $\mu _{1}'$ and $\mu _{1}''$, we shall have between the same limits

${\frac {\Theta }{{\overline {\nu }}_{1}^{\frac {3}{2}}}}\,d{\overline {\nu }}_{1}>d\mu _{1},$

(543)

and therefore

${\tfrac {1}{2}}\Theta {\bigg (}{\frac {1}{{\overline {\nu }}_{1}'^{\frac {1}{2}}}}-{\frac {1}{{\overline {\nu }}_{1}''^{\frac {1}{2}}}}{\bigg )}>\mu _{1}''-\mu _{1}'.$

(544)

The difference $\mu _{1}''-\mu _{1}'$ is therefore numerically a very small quantity. To form an idea of the importance of such a difference, we should observe that in formula (498) $\mu _{1}$ is multiplied by $\nu _{1}$ and the product subtracted from the energy. A very small difference in the value of $\mu _{1}$ may therefore be important. But since $\nu \Theta$ is always less than the kinetic energy of the system, our formula shows that $\mu _{1}''-\mu _{1}'$, even when multiplied by ${\overline {\nu }}_{1}'$ or ${\overline {\nu }}_{1}''$, may still be regarded as an insensible quantity. We can now perceive the leading characteristics with respect to properties sensible to human faculties of such an ensemble as we are considering (a grand ensemble canonically distributed), when the average numbers of particles of the various kinds are of the same order of magnitude as the number of molecules in the bodies which are the subject of physical