Page:Elementary Principles in Statistical Mechanics (1902).djvu/183

external, may be derived from a single force-function, which, taken negatively, we shall call the potential energy of the combined systems and denote by $\epsilon _{12}$. But we suppose that initially none of the systems of the two ensembles $E_{1}$ and $E_{2}$ come within range of each other's action, so that the potential energy of the combined system falls into two parts relating separately to the systems which are combined. The same is obviously true of the kinetic energy of the combined compound system, and therefore of its total energy. This may be expressed by the equation

$\epsilon _{12}'=\epsilon _{1}'+\epsilon _{2}',$

(458)

which gives

${\overline {\epsilon }}_{12}'={\overline {\epsilon }}_{1}'+{\overline {\epsilon }}_{2}'.$

(459)

Let us now suppose that in the course of time, owing to the motion of the bodies represented by the coördinates called external, the forces acting on the systems and consequently their positions are so altered, that the systems of the ensembles $E_{1}$ and $E_{2}$ are brought within range of each other's action, and after such mutual influence has lasted for a time, by a further change in the external coördinates, perhaps a return to their original values, the systems of the two original ensembles are brought again out of range of each other's action. Finally, then, at a time specified by double accents, we shall have as at first

${\overline {\epsilon }}_{12}''={\overline {\epsilon }}_{1}''+{\overline {\epsilon }}_{2}''.$

(460)

But for the indices of probability we must write^[1]

${\overline {\eta }}_{1}''+{\overline {\eta }}_{2}''\leqq {\overline {\eta }}_{12}''.$

(461)

The considerations adduced in the last chapter show that it is safe to write

${\overline {\eta }}_{12}''\leqq {\overline {\eta }}_{12}'.$

(462)

We have therefore

${\overline {\eta }}_{1}''+{\overline {\eta }}_{2}''\leqq {\overline {\eta }}_{1}'+{\overline {\eta }}_{2}',$

(463)

which may be compared with the thermodynamic theorem that

1. See Chapter XI, Theorem VII.