44 CANONICAL DISTRIBUTION 



Now the average value in the ensemble of any quantity 

 (which we shall denote in general by a horizontal line above 

 the proper symbol) is determined by the equation 



r M C fc! 

 =J J u e & d Pl ... dq a . (108) 



phases 



Comparing this with the preceding equation, we have 



<ty = d - ~ d - A! da^ - 2 2 da 2 - etc. (109) 



(jj) (y 



Or, since fe J = ,, (110) 



and ^=^ 



d\f/ = yd AI da,i >Z 2 d2 etc. 

 Moreover, since (111) gives 



dty - c?e = cfy + ^, (113) 



we have also 



dk drj ^ ddi A 2 da 2 etc. (114) 



This equation, if we neglect the sign of averages, is identi- 

 cal in form with the thermodynamic equation 



de + A l da 1 + A z da z + etc. 

 drj= y -, (115) 



or 



de = Td-rj A! da L A z da 2 etc., (H6) 



which expresses the relation between the energy, .tempera- 

 ture, and entropy of a body in thermodynamic equilibrium, 

 and the forces which it exerts on external bodies, a relation 

 which is the mathematical expression of the second law of 

 thermodynamics for reversible changes. The modulus in the 

 statistical equation corresponds to temperature in the thermo- 

 dynamic equation, and the average index of probability with 

 its sign reversed corresponds to entropy. But in the thermo- 

 dynamic equation the entropy (77) is a quantity which is 



