42 CANONICAL DISTRIBUTION 



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 en- 

 semble 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,* that the potential 

 energy (e q ) depends not only upon the coordinates q l . . . q n 

 which determine the configuration of the system, but also 

 upon certain coordinates i, 2 , etc. of bodies which we call 

 external? meaning by this simply that they are not to be re- 

 garded 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 de q jda v de q fda 2 , etc., while de q jdq v 

 ... de 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 under- 

 stood that p depends only upon qi , . . . q n , p\ , . . . 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 



although a similar equation would not hold for differentiations 

 relative to the internal coordinates. 



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. 

 * See especially Chapter I, p. 4. 



