Page:Elementary Principles in Statistical Mechanics (1902).djvu/225

If we compare the statistical equations (529) and (532) with (114) and (112), which are given in Chapter IV, and discussed in Chapter XIV, as analogues of thermodynamic equations, we find considerable difference. Beside the terms corresponding to the additional terms in the thermodynamic equations of this chapter, and beside the fact that the averages are taken in a grand ensemble in one case and in a petit in the other, the analogues of entropy, $\mathrm {H}$ and $\eta$, are quite different in definition and value. We shall return to this point after we have determined the order of magnitude of the usual anomalies of $\nu _{1},\ldots \nu _{h}$.

If we differentiate equation (518) with respect to $\mu _{1}$, and multiply by $\Theta$, we get

$\Sigma _{\nu _{1}}\ldots \Sigma _{\nu _{h}}{\bigg (}{\frac {d\Omega }{d\mu _{1}}}+\nu _{1}{\bigg )}{\frac {e^{\frac {\Omega +\mu _{1}\nu _{1}\ldots +\mu _{h}\nu _{h}-\psi }{\Theta }}}{{\nu _{1}}!\ldots {\nu _{h}}!}}=0,$

(536)

whence $d\Omega /d\mu _{1}=-{\overline {\nu }}_{1}$, which agrees with (527). Differentiating again with respect to $\mu _{1}$, and to $\mu _{2}$, and setting

${\frac {d\Omega }{d\mu _{1}}}=-{\overline {\nu }}_{1},\quad {\frac {d\Omega }{d\mu _{2}}}=-{\overline {\nu }}_{2},$

we get

$\Sigma _{\nu _{1}}\ldots \Sigma _{\nu _{h}}{\bigg (}{\frac {d^{2}\Omega }{d\mu _{1}^{2}}}+{\frac {(\nu _{1}-{\overline {\nu }}_{1})^{2}}{\Theta }}{\bigg )}{\frac {e^{\frac {\Omega +\mu _{1}\nu _{1}\ldots +\mu _{h}\nu _{h}-\psi }{\Theta }}}{{\nu _{1}}!\ldots {\nu _{h}}!}}=0,$

(537)

$\Sigma _{\nu _{1}}\ldots \Sigma _{\nu _{h}}{\bigg (}{\frac {d^{2}\Omega }{d\mu _{1}\,d\mu _{2}}}+{\frac {(\nu _{1}-{\overline {\nu }}_{1})(\nu _{2}-{\overline {\nu }}_{2})}{\Theta }}{\bigg )}{\frac {e^{\frac {\Omega +\mu _{1}\nu _{1}\ldots +\mu _{h}\nu _{h}-\psi }{\Theta }}}{{\nu _{1}}!\ldots {\nu _{h}}!}}=0.$

(538)

The first members of these equations represent the average values of the quantities in the principal parentheses. We have therefore

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

(539)

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

(540)