Page:Elementary Principles in Statistical Mechanics (1902).djvu/207

As a general theorem, the conclusion may be expressed in the words:—If a system of a great number of degrees of freedom is microcanonically distributed in phase, any very small part of it may be regarded as canonically distributed.^[1]

It would seem, therefore, that a canonical ensemble of phases is what best represents, with the precision necessary for exact mathematical reasoning, the notion of a body with a given temperature, if we conceive of the temperature as the state produced by such processes as we actually use in physics to produce a given temperature. Since the anomalies of the body increase with the quantity of the bath, we can only get rid of all that is arbitrary in the ensemble of phases which is to represent the notion of a body of a given temperature by making the bath infinite, which brings us to the canonical distribution.

A comparison of temperature and entropy with their analogues in statistical mechanics would be incomplete without a consideration of their differences with respect to units and zeros, and the numbers used for their numerical specification. If we apply the notions of statistical mechanics to such bodies as we usually consider in thermodynamics, for which the kinetic energy is of the same order of magnitude as the unit of energy, but the number of degrees of freedom is enormous, the values of $\Theta$, $d\epsilon /d\log V$, and $d\epsilon /d\phi$ will be of the same order of magnitude as $1/n$, and the variable part of ${\overline {\eta }}$, $\log V$, and $\phi$ will be of the same order of magnitude as $n$.^[2] If these quantities, therefore, represent in any sense the notions of temperature and entropy, they will nevertheless not be measured in units of the usual order of magnitude,—a fact which must be borne in mind in determining what magnitudes may be regarded as insensible to human observation.

Now nothing prevents our supposing energy and time in our statistical formulae to be measured in such units as may

1. It is assumed—and without this assumption the theorem would have no distinct meaning—that the part of the ensemble considered may be regarded as having separate energy.
2. See equations (124), (288), (289), and (314); also page 106.