Page:Elementary Principles in Statistical Mechanics (1902).djvu/96

of these anomalies is of course zero. The natural measure of such anomalies is the square root of their average square. Now

${\overline {(\epsilon -{\overline {\epsilon }})^{2}}}={\overline {\epsilon ^{2}}}-{\overline {\epsilon }}^{2},$

(204)

identically. Accordingly

${\overline {(\epsilon -{\overline {\epsilon }})^{2}}}=\Theta ^{2}{\frac {d{\overline {\epsilon }}}{d\Theta }}=-\Theta ^{3}{\frac {d^{2}\psi }{d\Theta ^{2}}}.$

(205)

In like manner,

${\overline {(\epsilon _{q}-{\overline {\epsilon }}_{q})^{2}}}=\Theta ^{2}{\frac {d{\overline {\epsilon }}_{q}}{d\Theta }}=-\Theta ^{3}{\frac {d^{2}\psi _{q}}{d\Theta ^{2}}}.$

(206)

${\overline {(\epsilon _{p}-{\overline {\epsilon }}_{p})^{2}}}=\Theta ^{2}{\frac {d{\overline {\epsilon }}_{p}}{d\Theta }}=-\Theta ^{3}{\frac {d^{2}\psi _{p}}{d\Theta ^{2}}}={\tfrac {1}{2}}n\Theta ^{2}.$

(207)

Hence

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

(208)

Equation (206) shows that the value of $d{\overline {\epsilon }}_{q}/d\Theta$ can never be negative, and that the value of $d^{2}\psi _{q}/d\Theta ^{2}$ or $d{\overline {\eta }}_{q}/d\Theta$ can never be positive.^[1] To get an idea of the order of magnitude of these quantities, we may use the average kinetic energy as a term of comparison, this quantity being independent of the arbitrary constant involved in the definition of the potential energy. Since

1. In the case discussed in the note on page 54, in which the potential energy is a quadratic function of the $q$'s, and $\Delta _{\dot {q}}$ independent of the $q$'s, we should get for the potential energy

   ${\overline {(\epsilon _{q}-{\overline {\epsilon }}_{q})^{2}}}={\frac {1}{2}}n\Theta ^{2},$

   and for the total energy

   ${\overline {(\epsilon -{\overline {\epsilon }})^{2}}}=n\Theta ^{2}.$

   We may also write in this case,

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

   ${\frac {\overline {(\epsilon -{\overline {\epsilon }})^{2}}}{({\overline {\epsilon }}-\epsilon _{a})^{2}}}={\frac {1}{n}}.$