Page:Elementary Principles in Statistical Mechanics (1902).djvu/194

170 which has been demonstrated in Chapter X, and which relates to a microcanonical ensemble, $$\overline{A_1}|_{\epsilon}$$ denoting the average value of $$A_1$$ in such an ensemble, corresponds precisely to the thermodynamic equation, except for the sign of average applied to the external forces. But as these forces are not entirely determined by the energy with the external coördinates, the use of average values is entirely germane to the subject, and affords the readiest means of getting perfectly determined quantities. These averages, which are taken for a microcanonical ensemble, may seem from some points of view a more simple and natural conception than those which relate to a canonical ensemble. Moreover, the energy, and the quantity corresponding to entropy, are free from the sign of average in this equation.

The quantity in the equation which corresponds to entropy is $$\log V$$, the quantity $$V$$ being defined as the extension-in-phase within which the energy is less than a certain limiting value ($$\epsilon$$). This is certainly a more simple conception than the average value in a canonical ensemble of the index of probability of phase. $$\operatorname{Log} V$$ has the property that when it is constant which closely corresponds to the thermodynamic property of entropy, that when it is constant  The quantity in the equation which corresponds to temperature is $$e^{-\phi}V$$, or $$d\epsilon/d\log V$$. In a canonical ensemble, the average value of this quantity is equal to the modulus, as has been shown by different methods in Chapters IX and X.

In Chapter X it has also been shown that if the systems of a microcanonical ensemble consist of parts with separate energies, the average value of $$e^{-\phi}V$$ or any part is equal to its average value for any other part, and to the uniform value of the same expression for the whole ensemble. This corresponds to the theorem in the theory of heat that in case of thermal equilibrium the temperatures of the parts of a body are equal to one another and to that of the whole body.