Page:Elementary Principles in Statistical Mechanics (1902).djvu/200

176 closely together that the most probable division may fairly represent the whole. This is in general the case, to a very close approximation, when $$n$$ is enormously great; it entirely fails when $$n$$ is small.

If we regard $$d\phi/d\epsilon$$ as corresponding to the reciprocal of temperature, or, in other words, $$d\epsilon/d\phi$$ as corresponding to temperature, $$\phi$$ will correspond to entropy. It has been defined as $$\log(dV/d\epsilon)$$. In the considerations on which its definition is founded, it is therefore very similar to $$\log V$$. We have seen that $$d\phi/d\log V$$ approaches the value unity when $$n$$ is very great.

To form a differential equation on the model of the thermodynamic equation (482), in which $$d\epsilon/d\phi$$ shall take the place of temperature, and $$\phi$$ of entropy, we may write or  With respect to the differential coefficients in the last equation, which corresponds exactly to (482) solved with respect to $$d\eta$$, we have seen that their average values in a canonical ensemble are equal to $$1/\Theta$$, and the averages of $$A_1/\Theta$$, $$A_2/\Theta$$, etc. We have also seen that $$d\epsilon/d\phi$$ (or $$d\phi/d\epsilon$$) has relations to the most probable values of energy in parts of a microcanonical ensemble. That $$(d\epsilon/da_1)_{\phi,a}$$, etc., have properties somewhat analogous, may be shown as follows.

In a physical experiment, we measure a force by balancing it against another. If we should ask what force applied to increase or diminish $$a_1$$ would balance the action of the systems, it would be one which varies with the different systems. But we may ask what single force will make a given value of $$a_1$$ the most probable, and we shall find that under certain conditions $$(d\epsilon/da_1)_{\phi,a}$$, a represents that force.