Page:Elementary Principles in Statistical Mechanics (1902).djvu/216

The constant $\Omega$ we may regard as determined by the equation

$N=\Sigma _{\nu _{1}}\ldots \Sigma _{\nu _{h}}\mathop {\int \ldots \int } _{\rm {phases}}^{\rm {all}}{\frac {Ne^{\frac {\Omega +\mu _{1}\nu _{1}\ldots +\mu _{h}\nu _{h}-\epsilon }{\Theta }}}{{\nu _{1}}!\ldots {\nu _{h}}!}}\,dp_{1}\ldots dq_{n},$

(504)

or

$e^{-{\frac {\Omega }{\Theta }}}=\Sigma _{\nu _{1}}\ldots \Sigma _{\nu _{h}}{\frac {e^{\frac {\mu _{1}\nu _{1}\ldots +\mu _{h}\nu _{h}}{\Theta }}}{{\nu _{1}}!\ldots {\nu _{h}}!}}\mathop {\int \ldots \int } _{\rm {phases}}^{\rm {all}}e^{-{\frac {\epsilon }{\Theta }}}\,dp_{1}\ldots dq_{n},$

(505)

where the multiple sum indicated by $\Sigma _{\nu _{1}}\ldots \Sigma _{\nu _{h}}$ includes all terms obtained by giving to each of the symbols $\nu _{1}\ldots \nu _{h}$ all integral values from zero upward, and the multiple integral (which is to be evaluated separately for each term of the multiple sum) is to be extended over all the (specific) phases of the system having the specified numbers of particles of the various kinds. The multiple integral in the last equation is what we have represented by $e^{-{\frac {\psi }{\Theta }}}$. See equation (92). We may therefore write

$e^{-{\frac {\Omega }{\Theta }}}=\Sigma _{\nu _{1}}\ldots \Sigma _{\nu _{h}}{\frac {e^{\frac {\mu _{1}\nu _{1}\ldots +\mu _{h}\nu _{h}-\psi }{\Theta }}}{{\nu _{1}}!\ldots {\nu _{h}}!}}.$

(506)

It should be observed that the summation includes a term in which all the symbols $\nu _{1}\ldots \nu _{h}$ have the value zero. We must therefore recognize in a certain sense a system consisting of no particles, which, although a barren subject of study in itself, cannot well be excluded as a particular case of a system of a variable number of particles. In this case $\epsilon$ is constant, and there are no integrations to be performed. We have therefore^[1]

$e^{-{\frac {\psi }{\Theta }}}=e^{-{\frac {\epsilon }{\Theta }}},\quad {\mathit {i.\,e.}},\quad \psi =\epsilon .$

1. This conclusion may appear a little strained. The original definition of $\psi$ may not be regarded as fairly applying to systems of no degrees of freedom. We may therefore prefer to regard these equations as defining $\psi$ in this case.