Page:Elementary Principles in Statistical Mechanics (1902).djvu/118

falls within given limits is independent also of the value of the modulus, being determined entirely by the number of degrees of freedom of the system and the limiting values of the ratio.

The average value of any function of the kinetic energy, either for the whole ensemble, or for any particular configuration, is given by

${\overline {u}}={\frac {1}{\Theta ^{\frac {n}{2}}\Gamma ({\tfrac {1}{2}}n)}}\int _{0}^{\infty }u\,e^{-{\frac {\epsilon _{p}}{\Theta }}}{\epsilon _{p}}^{{\tfrac {n}{2}}-1}\,d\epsilon _{p}$

[1](291)

Thus:

${\overline {{\epsilon _{p}}^{m}}}={\frac {\Gamma (m+{\tfrac {1}{2}}n)}{\Gamma ({\tfrac {1}{2}}n)}}\Theta ^{m},\quad \mathrm {if} \quad m+{\tfrac {1}{2}}n>0;$

[2](292)

${\overline {V_{p}}}={\frac {\Gamma (n)}{\Gamma ({\tfrac {1}{2}}n+1)\Gamma ({\tfrac {1}{2}}n)}}(2\pi \Theta )^{\tfrac {n}{2}};$

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1. The corresponding equation for the average value of any function of the potential energy, when this is a quadratic function of the $q$'s, and $\Delta _{\dot {q}}$ is independent of the $q$'s, is

   ${\overline {u}}={\frac {1}{\Theta ^{\frac {n}{2}}\Gamma ({\tfrac {1}{2}}n)}}\int _{\epsilon _{a}}^{\infty }u\,e^{-{\frac {\epsilon _{q}-\epsilon _{a}}{\Theta }}}(\epsilon _{q}-\epsilon _{a})^{{\tfrac {n}{2}}-1}\,d\epsilon _{q}.$

   In the same case, the average value of any function of the (total) energy is given by the equation

   ${\overline {u}}={\frac {1}{\Theta ^{n}\Gamma (n)}}\int _{0}^{\infty }u\,e^{-{\frac {\epsilon -\epsilon _{a}}{\Theta }}}(\epsilon -\epsilon _{a})^{{\tfrac {n}{2}}-1}\,d\epsilon .$

   Hence in this case

   ${\overline {(\epsilon _{q}-\epsilon _{a})^{m}}}={\frac {\Gamma (m+{\tfrac {1}{2}}n)}{\Gamma ({\tfrac {1}{2}}n)}}\Theta ^{m},\quad \mathrm {if} \quad m+{\tfrac {1}{2}}n>0.$

   ${\overline {(\epsilon -\epsilon _{a})^{m}}}={\frac {\Gamma (m+n)}{\Gamma (n)}}\Theta ^{m},\quad \mathrm {if} \quad m+n>0.$

   ${\overline {e^{-\phi _{q}}V_{q}}}={\overline {e^{-\phi }V}}=\Theta ,$

   ${\overline {\frac {d\phi _{q}}{d\epsilon _{q}}}}={\frac {1}{\Theta }},\quad \mathrm {if} \quad n>2,$

   and

   ${\overline {\frac {d\phi }{d\epsilon }}}={\frac {1}{\Theta }},\quad \mathrm {if} \quad n>1.$

   If $n=1$, $e^{\phi }=2\pi$ and $d\phi /d\epsilon =0$ for any value of $\epsilon$. If $n=2$, the case is the same with respect to $\phi _{q}$.
2. This equation has already been proved for positive integral powers of the kinetic energy. See page 77.