Page:Elementary Principles in Statistical Mechanics (1902).djvu/186

Let us now suppose that we have a certain number of ensembles, $E_{0}$, $E_{1}$, $E_{2}$, etc., distributed canonically with the respective moduli $\Theta _{0}$, $\Theta _{1}$, $\Theta _{2}$, etc. By variation of the external coördinates of the ensemble $E_{0}$, let it be brought into connection with $E_{1}$, and then let the connection be broken. Let it then be brought into connection with $E_{2}$, and then let that connection be broken. Let this process be continued with respect to the remaining ensembles. We do not make the assumption, as in some cases before, that the work connected with the variation of the external coördinates is a negligible quantity. On the contrary, we wish especially to consider the case in which it is large. In the final state of the ensemble $E_{0}$, let us suppose that the external coördinates have been brought back to their original values, and that the average energy (${\overline {\epsilon }}_{0}$) is the same as at first.

In our usual notations, using one and two accents to distinguish original and final values, we get by repeated applications of the principle expressed in (463)

${\overline {\eta }}_{0}'+{\overline {\eta }}_{1}'+{\overline {\eta }}_{2}'+\mathrm {etc.} \geq {\overline {\eta }}_{0}''+{\overline {\eta }}_{1}''+{\overline {\eta }}_{2}''+\mathrm {etc.}$

(474)

But by Theorem III of Chapter XI,

${\overline {\eta }}_{0}''+{\frac {{\overline {\epsilon }}_{0}''}{\Theta _{0}}}\geq {\overline {\eta }}_{0}'+{\frac {{\overline {\epsilon }}_{0}'}{\Theta _{0}}},$

(475)

${\overline {\eta }}_{1}''+{\frac {{\overline {\epsilon }}_{1}''}{\Theta _{1}}}\geq {\overline {\eta }}_{1}'+{\frac {{\overline {\epsilon }}_{1}'}{\Theta _{1}}},$

(476)

${\overline {\eta }}_{2}''+{\frac {{\overline {\epsilon }}_{2}''}{\Theta _{2}}}\geq {\overline {\eta }}_{2}'+{\frac {{\overline {\epsilon }}_{2}'}{\Theta _{2}}},$

(477)

$\mathrm {etc.}$

Hence

${\frac {{\overline {\epsilon }}_{0}''}{\Theta _{0}}}+{\frac {{\overline {\epsilon }}_{1}''}{\Theta _{1}}}+{\frac {{\overline {\epsilon }}_{2}''}{\Theta _{2}}}+\mathrm {etc.} \geq {\frac {{\overline {\epsilon }}_{0}'}{\Theta _{0}}}+{\frac {{\overline {\epsilon }}_{1}'}{\Theta _{1}}}+{\frac {{\overline {\epsilon }}_{2}'}{\Theta _{2}}}+\mathrm {etc.}$

(478)

or, since

${\overline {\epsilon }}_{0}'={\overline {\epsilon }}_{0}'',$

$0\geq {\frac {{\overline {\epsilon }}_{1}'-{\overline {\epsilon }}_{1}''}{\Theta _{1}}}+{\frac {{\overline {\epsilon }}_{2}'-{\overline {\epsilon }}_{2}''}{\Theta _{2}}}+\mathrm {etc.}$

(479)

If we write ${\overline {W}}$ for the average work done on the bodies represented by the external coördinates, we have