Page:Elementary Principles in Statistical Mechanics (1902).djvu/220

bility-coefficient is evidently zero, as they do not occur in the ensemble.

Now this third ensemble is in statistical equilibrium, with respect both to specific and generic phases, since the ensembles from which it is formed are so. This statistical equilibrium is not dependent on the equality of the modulus and the co-efficients $\mu _{1},\ldots \mu _{h}$ in the first and second ensembles. It depends only on the fact that the two original ensembles were separately in statistical equilibrium, and that there is no interaction between them, the combining of the two ensembles to form a third being purely nominal, and involving no physical connection. This independence of the systems, determined physically by forces which prevent particles from passing from one system to the other, or coming within range of each other's action, is represented mathematically by infinite values of the energy for particles in a space dividing the systems. Such a space may be called a diaphragm.

If we now suppose that, when we combine the systems of the two original ensembles, the forces are so modified that the energy is no longer infinite for particles in all the space forming the diaphragm, but is diminished in a part of this space, so that it is possible for particles to pass from one system to the other, this will involve a change in the function $\epsilon '''$ which represents the energy of the combined systems, and the equation $\epsilon '''=\epsilon '+\epsilon ''$ will no longer hold. Now if the co-efficient of probability in the third ensemble were represented by (513) with this new function $\epsilon '''$, we should have statistical equilibrium, with respect to generic phases, although not to specific. But this need involve only a trifling change in the distribution of the third ensemble,^[1] a change represented by the addition of comparatively few systems in which the transference of particles is taking place to the immense number

1. It will be observed that, so far as the distribution is concerned, very large and infinite values of $\epsilon$ (for certain phases) amount to nearly the same thing,—one representing the total and the other the nearly total exclusion of the phases in question. An infinite change, therefore, in the value of $\epsilon$ (for certain phases) may represent a vanishing change in the distribution.