Page:Elementary Principles in Statistical Mechanics (1902).djvu/196

where the second and third members of the equation denote average values in an ensemble in which the compound system is microcanonically distributed in phase. Let us suppose the two original systems to be identical in nature. Then

$\epsilon _{1}=\epsilon _{2}={\overline {\epsilon _{1}}}|_{\epsilon _{12}}={\overline {\epsilon _{2}}}|_{\epsilon _{12}}.$

The equation in question would require that

${\frac {d\epsilon _{1}}{d\log V_{1}}}={\overline {\frac {d\epsilon _{1}}{d\log V_{1}}}}{\bigg |}_{\epsilon _{12}},$

i. e., that we get the same result, whether we take the value of $d\epsilon _{1}/d\log V_{1}$ determined for the average value of $\epsilon _{1}$ in the ensemble, or take the average value of $d\epsilon _{1}/d\log V_{1}$. This will be the case where $d\epsilon _{1}/d\log V_{1}$ is a linear function of $\epsilon _{1}$. Evidently this does not constitute the most general case. Therefore the equation in question cannot be true in general. It is true, however, in some very important particular cases, as when the energy is a quadratic function of the $p$'s and $q$'s, or of the $p$'s alone.^[1] When the equation holds, the case is analogous to that of bodies in thermodynamics for which the specific heat for constant volume is constant.

Another quantity which is closely related to temperature is $d\phi /d\epsilon$. It has been shown in Chapter IX that in a canonical ensemble, if $n>2$, the average value of $d\phi /d\epsilon$ is $1/\Theta$, and that the most common value of the energy in the ensemble is that for which $d\phi /d\epsilon =1/\Theta$. The first of these properties may be compared with that of $d\epsilon /d\log V$, which has been seen to have the average value $\Theta$ in a canonical ensemble, without restriction in regard to the number of degrees of freedom.

With respect to microcanonical ensembles also, $d\phi /d\epsilon$ has a property similar to what has been mentioned with respect to $d\epsilon /d\log V$. That is, if a system microcanonically distributed in phase consists of two parts with separate energies, and each

1. This last case is important on account of its relation to the theory of gases, although it must in strictness be regarded as a limit of possible cases, rather than as a case which is itself possible.