Page:Elementary Principles in Statistical Mechanics (1902).djvu/197

with more than two degrees of freedom, the average values in the ensemble of $d\phi /d\epsilon$ for the two parts are equal to one another and to the value of same expression for the whole. In our usual notations

${\overline {\frac {d\phi _{1}}{d\epsilon _{1}}}}{\bigg |}_{\epsilon _{12}}={\overline {\frac {d\phi _{2}}{d\epsilon _{2}}}}{\bigg |}_{\epsilon _{12}}={\frac {d\phi _{12}}{d\epsilon _{12}}}$

if $n_{1}>2$, and $n_{2}>2$.

This analogy with temperature has the same incompleteness which was noticed with respect to $d\epsilon /d\log V$, viz., if two systems have such energies ($\epsilon _{1}$ and $\epsilon _{2}$) that

${\frac {d\phi _{1}}{d\epsilon _{1}}}={\frac {d\phi _{2}}{d\epsilon _{2}}},$

and they are combined to form a third system with energy

$\epsilon _{12}=\epsilon _{1}+\epsilon _{2},$

we shall not have in general

${\frac {d\phi _{12}}{d\epsilon _{12}}}={\frac {d\phi _{1}}{d\epsilon _{1}}}={\frac {d\phi _{2}}{d\epsilon _{2}}}.$

Thus, if the energy is a quadratic function of the $p$'s and $q$'s, we have^[1]

${\frac {d\phi _{1}}{d\epsilon _{1}}}={\frac {n_{1}-1}{\epsilon _{1}}},\qquad {\frac {d\phi _{2}}{d\epsilon _{2}}}={\frac {n_{2}-1}{\epsilon _{2}}},$

${\frac {d\phi _{12}}{d\epsilon _{12}}}={\frac {n_{12}-1}{\epsilon _{12}}}={\frac {n_{1}+n_{2}-1}{\epsilon _{1}+\epsilon _{2}}},$

where $n_{1}$, $n_{2}$, $n_{12}$, are the numbers of degrees of freedom of the separate and combined systems. But

${\frac {d\phi _{1}}{d\epsilon _{1}}}={\frac {d\phi _{2}}{d\epsilon _{2}}}={\frac {n_{1}+n_{2}-2}{\epsilon _{1}+\epsilon _{2}}}.$

If the energy is a quadratic function of the $p$'s alone, the case would be the same except that we should have ${\tfrac {1}{2}}n_{1}$, ${\tfrac {1}{2}}n_{2}$, ${\tfrac {1}{2}}n_{12}$, instead of $n_{1}$, $n_{2}$, $n_{12}$. In these particular cases, the analogy

1. See foot-note on page 93. We have here made the least value of the energy consistent with the values of the external coördinates zero instead of $\epsilon _{a}$, as is evidently allowable when the external coördinates are supposed invariable.