Page:Elementary Principles in Statistical Mechanics (1902).djvu/203

Rh to temperature is a notion but vaguely defined. Now if the state of a system is given by its energy and the external coördinates, it is incompletely defined, although its partial definition is perfectly clear as far as it goes. The ensemble of phases microcanonically distributed, with the given values of the energy and the external coördinates, will represent the imperfectly defined system better than any other ensemble or single phase. When we approach the subject from this side, our theorems will naturally relate to average values, or most probable values, in such ensembles.

In this case, the choice between the variables of (485) or of (489) will be determined partly by the relative importance which is attached to average and probable values. It would seem that in general average values are the most important, and that they lend themselves better to analytical transformations. This consideration would give the preference to the system of variables in which $$\log V$$ is the analogue of entropy. Moreover, if we make $$\phi$$ the analogue of entropy, we are embarrassed by the necessity of making numerous exceptions for systems of one or two degrees of freedom.

On the other hand, the definition of $$\phi$$ may be regarded as a little more simple than that of $$\log V$$, and if our choice is determined by the simplicity of the definitions of the analogues of entropy and temperature, it would seem that the $$\phi$$ system should have the preference. In our definition of these quantities, $$V$$ was defined first, and $$e^\phi$$ derived from $$V$$ by differentiation. This gives the relation of the quantities in the most simple analytical form. Yet so far as the notions are concerned, it is perhaps more natural to regard $$\phi$$ as derived from $$e^\phi$$ by integration. At all events, $$e^\phi$$ may be defined independently of $$V$$, and its definition may be regarded as more simple as not requiring the determination of the zero from which $$V$$ is measured, which sometimes involves questions of a delicate nature. In fact, the quantity $$e^\phi$$ may exist, when the definition of $$V$$ becomes illusory for practical purposes, as the integral by which it is determined becomes infinite.

The case is entirely different, when we regard the