170 THERMODYNAMIC ANALOGIES. 



which has been demonstrated in Chapter X, and which relates to 

 a microcanonical ensemble, A^ denoting the average value of 

 A 1 in such an ensemble, corresponds precisely to the thermody- 

 namic equation, except for the sign of average applied to the 

 external forces. But as these forces are not entirely deter- 

 mined by the energy with the external coordinates, the use of 

 average values is entirely germane to the subject, and affords 

 the readiest means of getting perfectly determined quantities. 

 These averages, which are taken for a microcanonical ensemble, 

 may seem from some points of view a more simple and natural 

 conception than those which relate to a canonical ensemble. 

 Moreover, the energy, and the quantity corresponding to en- 

 tropy, are free from the sign of average in this equation. 



The quantity in the equation which corresponds to entropy 

 is log FJ the quantity V being defined as the extension-in- 

 phase within which the energy is less than a certain limiting 

 value (e). This is certainly a more simple conception than the 

 average value in a canonical ensemble of the index of probabil- 

 ity of phase. Log V has the property that when it is constant 



de = - 21]. dat - A^\ f da z + etc., (486) 



which closely corresponds to the thermodynamic property of 

 entropy, that when it is constant 



de = Aj_ da^ A 2 da z + etc. (487) 



The quantity in the equation which corresponds to tem- 

 perature is e~* F", or dejd log V. In a canonical ensemble, the 

 average value of this quantity is equal to the modulus, as has 

 been shown by different methods in Chapters IX and X. 



In Chapter X it has also been shown that if the systems 

 of a microcanonical ensemble consist of parts with separate 

 energies, the average value of e~* Vi or any part is equal to its 

 average value for any other part, and to the uniform value 

 of the same expression for the whole ensemble. This corre- 

 sponds to the theorem in the theory of heat that in case of 

 thermal equilibrium the temperatures of the parts of a body 

 are equal to one another and to that of the whole body. 



