378 BELL SYSTEM TECHNICAL JOURNAL 



like (44). This is one of the dominant reasons for preferring the new sta- 

 tistics to the old. 



Now I proceed to the theory of entropy and temperature derived from 

 the new statistics. 



New Statistical Theory of Entropy 



Entropy is identified with the quantity In W, multiplied by a constant k 

 which as yet is disposable: 



S = k\n W (47) 



It is now to be shown that for the picture of a gas which is a flock of mass- 

 points in the "most probable state" as defined by the new statistics, and in 

 the limit of extreme rarefaction, this expression becomes the same as (46), 

 with further consequences of much value. 



As in the previous article, I separate the entropy into Sc the "contribution 

 of volume to entropy" which springs from the sprinkling of the mass-points 

 in ordinary space, and Sm the "contribution of temperature to entropy" 

 which springs from the sprinkling of the mass-points in momentum-space. 

 This is an artificial separation and worse than artificial, for it leads to a 

 fault in a detail which is not trivial. Nevertheless I think that for ease of 

 exposition the procedure is justified, and the detail will be made correct at 

 the end of the argument. 



We must now take (16) down to the "limit of extreme rarefaction." 

 I repeat this equation: 



In Wj = {Nj + C) In (iVy + C) - CIn C - Nj In Nj (16) 



The journey toward the limit is menaced by some of the oddest pitfalls, 

 and must be travelled with care. I recall that by Taylor's expansion, In 

 {Nj + C) is equal in first approximation to {Nj/C + In C) when Nj is 

 small by comparison with C. Making this substitution into (16), one finds 

 that the right-hand member consists of six terms. The two largest of these, 

 C In C and —C In C, destroy one another. The smallest, N'/C, is to be 

 neglected (if we couldn't neglect it, the dependence of entropy upon N' 

 would be hopelessly misrepresented). All of the remaining three terms 

 must be kept, for even the smallest — which is Nj — will play a perceptible 

 part in the check of theory with experiment. We have: 



In Wj = Nj In C - Nj In Nj + Nj (48) 



The quantity In W is the summation of In W j over all the regions. 

 Notice that we are interpreting entropy in such a way, that the entropy of 

 the gas in the container is the sum of the entropies of the portions thereof 

 in the individual regions. This is why we are destined to come to a result 



