The properties of canonically distributed ensembles of systems with respect to the equilibrium of the new ensembles which may be formed by combining each system of one ensemble with each system of another, are therefore not peculiar to them in the sense that analogous properties do not belong to some other distributions under special limitations in regard to the systems and forces considered. Yet the canonical distribution evidently constitutes the most simple case of the kind, and that for which the relations described hold with the least restrictions.

Returning to the case of the canonical distribution, we shall find other analogies with thermodynamic systems, if we suppose, as in the preceding chapters,^[1] that the potential energy (${\displaystyle \epsilon _{q}}$) depends not only upon the coördinates ${\displaystyle q_{1}\ldots q_{n}}$ which determine the configuration of the system, but also upon certain coördinates ${\displaystyle a_{1}}$, ${\displaystyle a_{2}}$, etc. of bodies which we call external meaning by this simply that they are not to be regarded as forming any part of the system, although their positions affect the forces which act on the system. The forces exerted by the system upon these external bodies will be represented by ${\displaystyle -d\epsilon _{q}/da_{1}}$, ${\displaystyle -d\epsilon _{q}/da_{2}}$, etc., while ${\displaystyle -d\epsilon _{q}/dq_{1},\ldots -d\epsilon _{q}/dq_{n}}$ represent all the forces acting upon the bodies of the system, including those which depend upon the position of the external bodies, as well as those which depend only upon the configuration of the system itself. It will be understood that ${\displaystyle e_{p}}$ depends only upon ${\displaystyle q_{1},\ldots q_{n}}$, ${\displaystyle p_{1},\ldots p_{n}}$, in other words, that the kinetic energy of the bodies which we call external forms no part of the kinetic energy of the system. It follows that we may write

${\displaystyle {\frac {d\epsilon }{da_{1}}}={\frac {d\epsilon _{q}}{da_{1}}}=-A_{1},}$

(104)

although a similar equation would not hold for differentiations relative to the internal coördinates.

^[2]

1. ↑ See especially Chapter I, p. 4.
2. ↑ the periods are the same they must be distributed canonically with same modulus in order that the compound ensemble with additional forces may be in statistical equilibrium.