Page:Elementary Principles in Statistical Mechanics (1902).djvu/92

CHAPTER VII.

FARTHER DISCUSSION OF AVERAGES IN A CANONICAL ENSEMBLE OF SYSTEMS.

Returning to the case of a canonical distribution, we have for the index of probability of configuration

$\eta _{q}={\frac {\psi _{q}-\epsilon _{q}}{\Theta }}$

(178)

as appears on comparison of formulae (142) and (161). It follows immediately from (142) that the average value in the ensemble of any quantity $u$ which depends on the configuration alone is given by the formula

${\overline {u}}=\mathop {\int \ldots \int } _{\rm {config.}}^{\rm {all}}ue^{\frac {\psi _{q}-\epsilon _{q}}{\Theta }}\Delta _{\dot {q}}{}^{\frac {1}{2}}\,dq_{1}\ldots dq_{n},$

(179)

where the integrations cover all possible configurations. The value of $\psi _{q}$ is evidently determined by the equation

$e^{-{\frac {\psi _{q}}{\Theta }}}=\mathop {\int \ldots \int } _{\rm {config.}}^{\rm {all}}e^{-{\frac {\epsilon _{q}}{\Theta }}}\Delta _{\dot {q}}{}^{\frac {1}{2}}\,dq_{1}\ldots dq_{n}.$

(180)

By differentiating the last equation we may obtain results analogous to those obtained in Chapter IV from the equation

$e^{-{\frac {\psi }{\Theta }}}=\mathop {\int \ldots \int } _{\rm {phases}}^{\rm {all}}e^{-{\frac {\epsilon }{\Theta }}}\,dp_{1}\ldots dq_{n}.$

As the process is identical, it is sufficient to give the results:

$d\psi _{q}={\overline {\eta }}_{q}\,d\Theta -{\overline {A}}_{1}\,da_{1}-{\overline {A}}_{2}\,da_{2}-\mathrm {etc.} ,$

(181)