Page:Elementary Principles in Statistical Mechanics (1902).djvu/41

The condition of statistical equilibrium may be expressed by equating to zero the second member of either of these equations.

The same substitutions in (22) give

${\bigg (}{\frac {dP}{dt}}{\bigg )}_{a,\ldots h}=0,$

(43)

and

${\bigg (}{\frac {d\eta }{dt}}{\bigg )}_{a,\ldots h}=0.$

(44)

That is, the values of $P$ and $\eta$, like those of $D$, are constant in time for moving systems of the ensemble. From this point of view, the principle which otherwise regarded has been called the principle of conservation of density-in-phase or conservation of extension-in-phase, may be called the principle of conservation of the coefficient (or index) of probability of a phase varying according to dynamical laws, or more briefly, the principle of conservation of probability of phase. It is subject to the limitation that the forces must be functions of the coördinates of the system either alone or with the time.

The application of this principle is not limited to cases in which there is a formal and explicit reference to an ensemble of systems. Yet the conception of such an ensemble may serve to give precision to notions of probability. It is in fact customary in the discussion of probabilities to describe anything which is imperfectly known as something taken at random from a great number of things which are completely described. But if we prefer to avoid any reference to an ensemble of systems, we may observe that the probability that the phase of a system falls within certain limits at a certain time, is equal to the probability that at some other time the phase will fall within the limits formed by phases corresponding to the first. For either occurrence necessitates the other. That is, if we write $P'$ for the coefficient of probability of the phase $p_{1}',\ldots q_{n}'$ at the time $t'$, and $P''$ for that of the phase $p_{1}'',\ldots q_{n}''$ at the time $t''$,