Page:Elementary Principles in Statistical Mechanics (1902).djvu/193

of the greater modulus to that of the less,^[1] or in case of equal moduli, that we have a condition of statistical equilibrium in regard to the distribution of energy,^[2]

Propositions have also been demonstrated analogous to those in thermodynamics relating to a Carnot's cycle,^[3] or to the tendency of entropy to increase,^[4] especially when bodies of different temperature are brought into contact.^[5]

We have thus precisely defined quantities, and rigorously demonstrated propositions, which hold for any number of degrees of freedom, and which, when the number of degrees of freedom ($n$) is enormously great, would appear to human faculties as the quantities and propositions of empirical thermodynamics.

It is evident, however, that there may be more than one quantity denned for finite values of $n$, which approach the same limit, when $n$ is increased indefinitely, and more than one proposition relating to finite values of $n$, which approach the same limiting form for $n=\infty$. There may be therefore, and there are, other quantities which may be thought to have some claim to be regarded as temperature and entropy with respect to systems of a finite number of degrees of freedom.

The definitions and propositions which we have been considering relate essentially to what we have called a canonical ensemble of systems. This may appear a less natural and simple conception than what we have called a microcanonical ensemble of systems, in which all have the same energy and which in many cases represents simply the time-ensemble, or ensemble of phases through which a single system passes in the course of time.

It may therefore seem desirable to find definitions and propositions relating to these microcanonical ensembles, which shall correspond to what in thermodynamics are based on experience. Now the differential equation

$d\epsilon =e^{-\phi }V\,d\log V-{\overline {A_{1}}}|_{\epsilon }\,da_{1}-{\overline {A_{2}}}|_{\epsilon }\,da_{2}-\mathrm {etc.} ,$

(485)

1. See Chapter XIII, page 160.
2. See Chapter IV, pages 35-37.
3. See Chapter XIII, pages 162, 163.
4. See Chapter XII, pages 143-151.
5. See Chapter XIII, page 159.