Page:Elementary Principles in Statistical Mechanics (1902).djvu/214

which the systems differ only in phase a petit ensemble. A grand ensemble is therefore composed of a multitude of petit ensembles. The ensembles which we have hitherto discussed are petit ensembles.

Let $\nu _{1},\ldots \nu _{h}$, etc., denote the numbers of the different kinds of particles in a system, $\epsilon$ its energy, and $q_{1},\ldots q_{n}$, $p_{1},\ldots p_{n}$ its coördinates and momenta. If the particles are of the nature of material points, the number of coördinates ($n$) of the system will be equal to $3\nu _{1}\ldots +3\nu _{h}$. But if the particles are less simple in their nature, if they are to be treated as rigid solids, the orientation of which must be regarded, or if they consist each of several atoms, so as to have more than three degrees of freedom, the number of coördinates of the system will be equal to the sum of $\nu _{1}$, $\nu _{2}$, etc., multiplied each by the number of degrees of freedom of the kind of particle to which it relates.

Let us consider an ensemble in which the number of systems having $\nu _{1},\ldots \nu _{h}$ particles of the several kinds, and having values of their coördinates and momenta lying between the limits $q_{1}$ and $q_{1}+dq_{1}$, $p_{1}$ and $p_{1}+dp_{1}$, etc., is represented by the expression

${\frac {Ne^{\frac {\Omega +\mu _{1}\nu _{1}\ldots +\mu _{h}\nu _{h}-\epsilon }{\Theta }}}{{\nu _{1}}!\ldots {\nu _{h}}!}}\,dp_{1}\ldots dq_{n},$

(4998)

where $N$, $\Omega$, $\Theta$, $\mu _{1},\ldots \mu _{h}$ are constants, $N$ denoting the total number of systems in the ensemble. The expression

${\frac {Ne^{\frac {\Omega +\mu _{1}\nu _{1}\ldots +\mu _{h}\nu _{h}-\epsilon }{\Theta }}}{{\nu _{1}}!\ldots {\nu _{h}}!}}$

(499)

evidently represents the density-in-phase of the ensemble within the limits described, that is, for a phase specifically defined. The expression

${\frac {e^{\frac {\Omega +\mu _{1}\nu _{1}\ldots +\mu _{h}\nu _{h}-\epsilon }{\Theta }}}{{\nu _{1}}!\ldots {\nu _{h}}!}}$

(500)