Page:Elementary Principles in Statistical Mechanics (1902).djvu/95

or

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

(197)

Combining this with (191),

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

(198)

In precisely the same way, from the equation

${\overline {\epsilon }}_{q}=\mathop {\int \ldots \int } _{\rm {config.}}^{\rm {all}}\epsilon _{q}e^{\frac {\psi _{q}-\epsilon _{q}}{\Theta }}\Delta _{\dot {q}}{}^{\frac {1}{2}}\,dq_{1}\ldots dq_{n},$

(199)

we may obtain

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

(200)

In the same way also, if we confine ourselves to a particular configuration, from the equation

${\overline {\epsilon }}_{p}=\mathop {\int \ldots \int } _{\rm {veloc.}}^{\rm {all}}\epsilon _{p}e^{\frac {\psi _{p}-\epsilon _{p}}{\Theta }}\Delta _{p}{}^{\frac {1}{2}}\,dp_{1}\ldots dp_{n},$

(201)

we obtain

${\overline {\epsilon _{p}{}^{2}}}={\overline {\epsilon _{p}}}^{2}+\Theta ^{2}{\frac {d{\epsilon _{p}}}{d\Theta }}={\bigg (}\psi _{p}-\Theta {\frac {d\psi _{p}}{d\Theta }}{\bigg )}^{2}-\Theta ^{3}{\frac {d^{2}\psi _{p}}{d\Theta ^{2}}},$

(202)

which by (187) reduces to

${\overline {\epsilon _{p}{}^{2}}}=({\tfrac {1}{4}}n^{2}+{\tfrac {1}{2}}n)\Theta ^{2}.$

(203)

Since this value is independent of the configuration, we see that the average square of the kinetic energy for every configuration is the same, and therefore the same as for the whole ensemble. Hence ${\overline {\epsilon _{p}{}^{2}}}$ may be interpreted as the average either for any particular configuration, or for the whole ensemble. It will be observed that the value of this quantity is determined entirely by the modulus and the number of degrees of freedom of the system, and is in other respects independent of the nature of the system. Of especial importance are the anomalies of the energies, or their deviations from their average values. The average value