Page:Elementary Principles in Statistical Mechanics (1902).djvu/174

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MOTION OF SYSTEMS AND ENSEMBLES

systems in $DV''$ , for example, by the total number of systems, and also by the extension-in-phase of the element, and take the logarithm of the quotient, we would get a number which would be less than the average value of $\eta$ for the systems within $DV''$ based on the distribution in phase at the time $t'$ .^[1] Hence the average value of $\eta$ for the whole ensemble of systems based on the distribution at $t''$ will be less than the average value based on the distribution at $t'$ .

We must not forget that there are exceptions to this general rule. These exceptions are in cases in which the laws of motion are such that systems having small differences of phase will continue always to have small differences of phase.

It is to be observed that if the average index of probability in an ensemble may be said in some sense to have a less value at one time than at another, it is not necessarily priority in time which determines the greater average index. If a distribution, which is not one of statistical equilibrium, should be given for a time $t'$ , and the distribution at an earlier time $t''$ should be defined as that given by the corresponding phases, if we increase the interval leaving $t'$ fixed and taking $t''$ at an earlier and earlier date, the distribution at $t''$ will in general approach a limiting distribution which is in statistical equilibrium. The determining difference in such cases is that between a definite distribution at a definite time and the limit of a varying distribution when the moment considered is carried either forward or backward indefinitely.^[2]

But while the distinction of prior and subsequent events may be immaterial with respect to mathematical fictions, it is quite otherwise with respect to the events of the real world. It should not be forgotten, when our ensembles are chosen to illustrate the probabilities of events in the real world, that

↑ See Chapter XI, Theorem IX.
↑ One may compare the kinematical truism that when two points are moving with uniform velocities, (with the single exception of the case where the relative motion is zero,) their mutual distance at any definite time is less than for $t=\infty$ , or $t=-\infty$ .

[1] See Chapter XI, Theorem IX.

[2] One may compare the kinematical truism that when two points are moving with uniform velocities, (with the single exception of the case where the relative motion is zero,) their mutual distance at any definite time is less than for $t=\infty$ , or $t=-\infty$ .

[1]

[2]