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

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.^[1]

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

$d\epsilon ={\bigg (}{\frac {d\epsilon }{d\phi }}{\bigg )}_{a}\,d\phi +{\bigg (}{\frac {d\epsilon }{da_{1}}}{\bigg )}_{\phi ,a}\,da_{1}+{\bigg (}{\frac {d\epsilon }{da_{2}}}{\bigg )}_{\phi ,a}\,da_{2}+\mathrm {etc.} ,$

(489)

or

$d\phi ={\frac {d\phi }{d\epsilon }}\,d\epsilon +{\frac {d\phi }{da_{1}}}\,da_{1}+{\frac {d\phi }{da_{2}}}\,da_{2}+\mathrm {etc.}$

(490)

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.^[2] 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.

1. See Chapter X, pages 120, 121.
2. See Chapter IX, equations (321), (327).