Page:Elementary Principles in Statistical Mechanics (1902).djvu/213

Rh of a generic phase is the sum of the probability-coefficients of the specific phases which it represents. When these are equal among themselves, the probability-coefficient of the generic phase is equal to that of the specific phase multiplied by $${\nu_1}! \, {\nu_2}! \ldots {\nu_h}!$$. It is also evident that statistical equilibrium may subsist with respect to generic phases without statistical equilibrium with respect to specific phases, but not vice versa.

Similar questions arise where one particle is capable of several equivalent positions. Does the change from one of these positions to another change the phase? It would be most natural and logical to make it affect the specific phase, but not the generic. The number of specific phases contained in a generic phase would then be $${\nu_1}! \, \kappa_1^{\nu_1} \ldots {\nu_h}! \, \kappa_h^{\nu_h}$$, where $$\kappa_1 \ldots \kappa_h$$ denote the numbers of equivalent positions belonging to the several kinds of particles. The case in which a $$\kappa$$ is infinite would then require especial attention. It does not appear that the resulting complications in the formulae would be compensated by any real advantage. The reason of this is that in problems of real interest equivalent positions of a particle will always be equally probable. In this respect, equivalent positions of the same particle are entirely unlike the $$\nu!$$ different ways in which $$\nu$$ particles may be distributed in $$\nu$$ different positions. Let it therefore be understood that in spite of the physical equivalence of different positions of the same particle they are to be considered as constituting a difference of generic phase as well as of specific. The number of specific phases contained in a generic phase is therefore always given by the product $${\nu_1}! \, {\nu_2}! \ldots {\nu_h}!$$.

Instead of considering, as in the preceding chapters, ensembles of systems differing only in phase, we shall now suppose that the systems constituting an ensemble are composed of particles of various kinds, and that they differ not only in phase but also in the numbers of these particles which they contain. The external coördinates of all the systems in the ensemble are supposed, as heretofore, to have the same value, and when they vary, to vary together. For distinction, we may call such an ensemble a grand ensemble, and one in