﻿830 Messrs. C. Gr. Darwin and R. H. Fowler on 



Now it is established that actually the "thermodynamic 

 probability " does lead to the entropy, and so we must con^ 

 sider how it is to be interpreted in terms of true probability. 

 It is clear that the " thermodynamic probability " must be 

 divided by the total number of admissible complexions, and 

 that when we consider an assembly of given energy this 

 number is G. In so far as we have to calculate the most 

 probable state, only ratios are concerned and the de- 

 nominator is immaterial, for we only have to deal with 

 the equation * 



S'_S"=/clog(W7W"); . . . (4-4) 



there is no difference between using <( thermodynamic '.' 



and true probability. But when we attempt to determine 



the value of the entropy itself by (4*3), we shall in all 



cases find that when W is a true probability the maximum 



value gives always S = — a trivial determination of the 



arbitrary constant in the entropy. This result may be 



verified for any of the examples of the former paper. We 



have merely to substitute values for the a's and /As of the 



specifications, and make use of Stirling's theorem in 



the form 



log (x !) = sc log x — x (4*5) 



This is an approximation which is known by experience 

 to suffice in entropy calculations. The zero value of S, 

 roughly speaking, expresses the fact that a " normal "* 

 distribution is so enormously more probable than any other 

 that by comparison it is certain, and so for it W = l. 



It is thus clear that the straightforward process is useless, 

 and we must consider how it is to be modified so as to retain 

 the relation with true probability while giving the actual 

 value of the entropy — -in effect we must find a way of 

 justifiably omitting the denominator C. As long as we 

 consider the «hole assembly this is impossible, for C 

 depends on $ and cannot be regarded as an ignorable 

 constant when changes of temperature are contemplated. 

 But if instead we consider the entropy of a group of 

 systems immersed in a temperature bath, it becomes 

 simple. Take, for example, a group of M A's — systems 

 of the general quantized type described in § 2 — and suppose 

 them immersed in a bath of a very much larger number 

 of B's. We can now define the entropy of the A's when 

 their specification is a , a 1? . . . . as k times the logarithm 

 of the probability of that specification. In calculating 



* See Ehrenfest and Trkal, Proc. Acad. Amsterdam, vol. xxiii. p. 162 

 (1921). 



