﻿the Partition of Energy. 839 



that they may possess. The abandonment of: absolute entropy 

 involves of course the acceptance of the paradox that the 

 entropy of 2 grammes of gas may not be twice that of 

 1 gramme ; but this paradox causes no real difficulty *. 



§ 9. The " canonical ensemble " of Gibbs, and its relation 

 to a temperature bath. 



It is important to consider also the general question of 

 the truth of the Second Law as deduced statistically for 

 assemblies which are of some more complex type for which 

 the energy cannot be separated up in the way that has 

 hitherto been possible : any general proof must cover such 

 cases. Now Gibbs 1 work deals with these generalized 

 assemblies, and he establishes with great simplicity the 

 necessary theorems, provided that he may start by postulating 

 the conditions of the '* canonical ensemble." But the idea 

 of " canonical " is not very easily defined, and it leaves 

 a slight feeling that there might be somewhere in it a 

 jpetitio principii. He later turns over to the " micro- 

 canonical " conditions, but the calculus becomes rather 

 heavy. With our present method we can very quickly 

 show that Gibbs' "canonical ensemble of phases" is, lor 

 the purpose of averaging, equivalent to having our assembly 

 of systems in a temperature bath. 



Consider an assembly composed of a mechanical system 

 of n degrees of freedom, with coordinates Qi...Q n and 

 momenta P^.-P^, together with a very large number M 

 of systems of any of the types we have treated. The 

 mechanical system exchanges energy with the others, but 

 for the greater part of its motion we may, as usual, think of 

 it undisturbed and in possession of a definite energy of its 

 own. For simplicity we may suppose the temperature to be 

 the only variable in the partition function f of the systems 

 of the bath, though this is quite immaterial. Let the various 

 weight factors be j^,^, ... and energies e , e 1? ... so that the 

 partition function is f(S)=2 lr p r S er . For the mechanical 

 system we mi 

 as having wei< 

 our former paper. 



Now consider arrangements in which the mechanical 

 system is in dil, while for the bath there are a , a u a 2 , 

 systems respectively in states 0, 1, 2, ... . By the methods 



lust take any element of phase d£l( = dQ l .. .dP n ) 

 sight dCl/h n , by the principles described in § 2 of 



* Ehreufest and Trkal, loc. cit. 



