Page:Elementary Principles in Statistical Mechanics (1902).djvu/141

($\epsilon _{q}$) vary in the different systems, subject of course to the condition

$\epsilon _{p}+\epsilon _{q}=\epsilon =\mathrm {constant} .$

(373)

Our first inquiries will relate to the division of energy into these two parts, and to the average values of functions of $\epsilon _{p}$ and $\epsilon _{q}$.

We shall use the notation ${\overline {u}}|_{\epsilon }$ to denote an average value in a microcanonical ensemble of energy $\epsilon$. An average value in a canonical ensemble of modulus $\Theta$, which has hitherto been denoted by ${\overline {u}}$, we shall in this chapter denote by ${\overline {u}}|_{\Theta }$, to distinguish more clearly the two kinds of averages.

The extension-in-phase within any limits which can be given in terms of $\epsilon _{p}$ and $\epsilon _{q}$ may be expressed in the notations of the preceding chapter by the double integral

$\iint dV_{p}\,dV_{q}$

taken within those limits. If an ensemble of systems is distributed within those limits with a uniform density-in-phase, the average value in the ensemble of any function ($u$) of the kinetic and potential energies will be expressed by the quotient of integrals

${\frac {\iint u\,dV_{p}\,dV_{q}}{\iint dV_{p}\,dV_{q}}}$

Since $dV_{p}=e^{\phi _{p}}\,d\epsilon _{p}$, and $d\epsilon _{p}=d\epsilon$ when $\epsilon _{q}$ is constant, the expression may be written

${\frac {\iint u\,e^{\phi _{p}}\,d\epsilon \,dV_{q}}{\iint e^{\phi _{p}}\,d\epsilon \,dV_{q}}}$

To get the average value of $u$ in an ensemble distributed microcanonically with the energy $\epsilon$, we must make the integrations cover the extension-in-phase between the energies $\epsilon$ and $\epsilon +d\epsilon$. This gives