Page:Elementary Principles in Statistical Mechanics (1902).djvu/182

by a sufficiently slow variation of the external coördinates, just as approximate thermodynamic equilibrium may usually be attained by sufficient slowness in the mechanical operations to which the body is subject.

We now pass to the consideration of the effect on an ensemble of systems which is produced by the action of other ensembles with which it is brought into dynamical connection. In a previous chapter^[1] we have imagined a dynamical connection arbitrarily created between the systems of two ensembles. We shall now regard the action between the systems of the two ensembles as a result of the variation of the external coördinates, which causes such variations of the internal coördinates as to bring the systems of the two ensembles within the range of each other's action.

Initially, we suppose that we have two separate ensembles of systems, $E_{1}$ and $E_{2}$. The numbers of degrees of freedom of the systems in the two ensembles will be denoted by $n_{1}$ and $n_{2}$ respectively, and the probability-coefficients by $e^{\eta _{1}}$ and $e^{\eta _{2}}$. Now we may regard any system of the first ensemble combined with any system of the second as forming a single system of $n_{1}+n_{2}$ degrees of freedom. Let us consider the ensemble ($E_{12}$) obtained by thus combining each system of the first ensemble with each of the second.

At the initial moment, which may be specified by a single accent, the probability-coefficient of any phase of the combined systems is evidently the product of the probability-coefficients of the phases of which it is made up. This may be expressed by the equation,

$e^{\eta _{12}'}=e^{\eta _{1}'}e^{\eta _{2}'},$

(455)

or

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

(456)

which gives

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

(457)

The forces tending to vary the internal coördinates of the combined systems, together with those exerted by either system upon the bodies represented by the coördinates called

1. See Chapter IV, page 37.