Page:Elementary Principles in Statistical Mechanics (1902).djvu/172

average value of the index of probability of phase, is consistent with an approach to a limiting condition in which that average value is less. We might perhaps fairly infer from such considerations as have been adduced that an approach to a limiting condition of statistical equilibrium is the general rule, when the initial condition is not of that character. But the subject is of such importance that it seems desirable to give it farther consideration.

Let us suppose that the total extension-in-phase for the kind of system considered to be divided into equal elements ($DV$) which are very small but not infinitely small. Let us imagine an ensemble of systems distributed in this extension in a manner represented by the index of probability $\eta$, which is an arbitrary function of the phase subject only to the restriction expressed by equation (46) of Chapter I. We shall suppose the elements $DV$ to be so small that $\eta$ may in general be regarded as sensibly constant within any one of them at the initial moment. Let the path of a system be defined as the series of phases through which it passes.

At the initial moment ($t'$) a certain system is in an element of extension $DV'$. Subsequently, at the time $t''$, the same system is in the element $DV''$. Other systems which were at first in $DV'$ will at the time $t''$ be in $DV''$, but not all, probably. The systems which were at first in $DV'$ will at the time $t''$ occupy an extension-in-phase exactly as large as at first. But it will probably be distributed among a very great number of the elements ($DV$) into which we have divided the total extension-in-phase. If it is not so, we can generally take a later time at which it will be so. There will be exceptions to this for particular laws of motion, but we will confine ourselves to what may fairly be called the general case. Only a very small part of the systems initially in $DV'$ will be found in $DV''$ at the time $t''$, and those which are found in $DV''$ at that time were at the initial moment distributed among a very large number of elements $DV$.

What is important for our purpose is the value of $\eta$, the index of probability of phase in the element $DV''$ at the time