Page:Elementary Principles in Statistical Mechanics (1902).djvu/30

or more briefly by

$D\,dp_{1}\ldots dp_{n}\,dq_{1}\ldots dq_{n},$

(11)

where $D$ is a function of the $p$'s and $q$'s and in general of $t$ also, for as time goes on, and the individual systems change their phases, the distribution of the ensemble in phase will in general vary. In special cases, the distribution in phase will remain unchanged. These are cases of statistical equilibrium.

If we regard all possible phases as forming a sort of extension of $2n$ dimensions, we may regard the product of differentials in (11) as expressing an element of this extension, and $D$ as expressing the density of the systems in that element. We shall call the product

$dp_{1}\ldots dp_{n}\,dq_{1}\ldots dq_{n}$

(12)

an element of extension-in-phase, and $D$ the density-in-phase of the systems.

It is evident that the changes which take place in the density of the systems in any given element of extension-in-phase will depend on the dynamical nature of the systems and their distribution in phase at the time considered.

In the case of conservative systems, with which we shall be principally concerned, their dynamical nature is completely determined by the function which expresses the energy ($\epsilon$) in terms of the $p$'s, $q$'s, and $a$'s (a function supposed identical for all the systems); in the more general case which we are considering, the dynamical nature of the systems is determined by the functions which express the kinetic energy ($\epsilon _{p}$) in terms of the $p$'s and $q$'s, and the forces in terms of the $q$'s and $a$'s. The distribution in phase is expressed for the time considered by $D$ as function of the $p$'s and $q$'s. To find the value of $dD/dt$ for the specified element of extension-in-phase, we observe that the number of systems within the limits can only be varied by systems passing the limits, which may take place in $4n$ different ways, viz., by the $p_{1}$ of a system passing the limit $p_{1}'$, or the limit $p_{1}''$, or by the $q_{1}$ of a system passing the limit $q_{1}'$ or the limit $q_{1}''$, etc. Let us consider these cases separately.