THERMODYNAMIC ANALOGIES. 169 



of the greater modulus to that of the less, * or in case of equal 

 moduli, that we have a condition of statistical equilibrium in 

 regard to the distribution of energy, f 



Propositions have also been demonstrated analogous to 

 those in thermodynamics relating to a Carnot's cycle,:]: or to 

 the tendency of entropy to increase, especially when bodies 

 of different temperature are brought into contact. || 



We have thus precisely defined quantities, and rigorously 

 demonstrated propositions, which hold for any number of 

 degrees of freedom, and which, when the number of degrees 

 of freedom (n) is enormously great, would appear to human 

 faculties as the quantities and propositions of empirical ther- 

 modynamics. 



It is evident, however, that there may be more than one 

 quantity denned for finite values of n, which approach the 

 same limit, when n is increased indefinitely, and more than one 

 proposition relating to finite values of n, which approach the 

 same limiting form for n = oo. There may be therefore, 

 and there are, other quantities which may be thought to have 

 some claim to be regarded as temperature and entropy with 

 respect to systems of a finite number of degrees of freedom. 



The definitions and propositions which we have been con- 

 sidering relate essentially to what we have called a canonical 

 ensemble of systems. This may appear a less natural and 

 simple conception than what we have called a microcanonical 

 ensemble of systems, in which all have thex-sa"me^energy v and 

 which in many cases represents simply tltte time-ensemble, or 

 ensemble of phases through which a single system passes in 

 the course of time. 



It may therefore seem desirable to find definitions and 

 propositions relating to these microcanonical ensembles, which 

 shall correspond to what in thermodynamics are based on 

 experience. Now the differential equation 



de = e~* Vd log F- ZTle fai - 3^] 6 da z - etc., (485) 



* See Chapter XIII, page 160. t See Chapter IV, pages 35-37. 



J See Chapter XIII, pages 162, 163. See Chapter XII, pages 143-151. 

 || See Chapter XIII, page 159. 



