Page:Elementary Principles in Statistical Mechanics (1902).djvu/56

CHAPTER IV.

ON THE DISTRIBUTION IN PHASE CALLED CANONICAL, IN WHICH THE INDEX OF PROBABILITY IS A LINEAR FUNCTION OF THE ENERGY.

Let us now give our attention to the statistical equilibrium of ensembles of conservation systems, especially to those cases and properties which promise to throw light on the phenomena of thermodynamics.

The condition of statistical equilibrium may be expressed in the form^[1]

$\Sigma {\bigg (}{\frac {dP}{dp_{1}}}{\dot {p}}_{1}+{\frac {dP}{dq_{1}}}{\dot {q}}_{1}{\bigg )}=0,$

(88)

where $P$ is the coefficient of probability, or the quotient of the density-in-phase by the whole number of systems. To satisfy this condition, it is necessary and sufficient that $P$ should be a function of the $p$'s and $q$'s (the momenta and coördinates) which does not vary with the time in a moving system. In all cases which we are now considering, the energy, or any function of the energy, is such a function.

$P=\operatorname {func.} (\epsilon )$

will therefore satisfy the equation, as indeed appears identically if we write it in the form

$\Sigma {\Bigg (}{\frac {dP}{dp_{1}}}{\frac {d\epsilon }{dp_{1}}}-{\frac {dP}{dq_{1}}}{\frac {d\epsilon }{dq_{1}}}{\Bigg )}=0.$

There are, however, other conditions to which $P$ is subject, which are not so much conditions of statistical equilibrium, as conditions implicitly involved in the definition of the coeffi--

1. See equations (20), (41), (42), also the paragraph following equation (20). The positions of any external bodies which can affect the systems are here supposed uniform for all the systems and constant in time.