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

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

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

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

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

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 (${\displaystyle u}$) of the kinetic and potential energies will be expressed by the quotient of integrals Since ${\displaystyle dV_{p}=e^{\phi _{p}}\,d\epsilon _{p}}$, and ${\displaystyle d\epsilon _{p}=d\epsilon }$ when ${\displaystyle \epsilon _{q}}$ is constant, the expression may be written To get the average value of ${\displaystyle u}$ in an ensemble distributed microcanonically with the energy ${\displaystyle \epsilon }$, we must make the integrations cover the extension-in-phase between the energies ${\displaystyle \epsilon }$ and ${\displaystyle \epsilon +d\epsilon }$. This gives