376 BELL SYSTEM TECHNICAL JOURNAL 



I propose to begin very soon on the proof that the new statistics, applied to 

 a flock of atoms which are merely mass-points, gives an excellent description 

 of just such a gas. However, there is a detail, or rather an element of the 

 structure, waiting to be inserted correctly — a trivial one in appearance, but 

 in all of thermodynamics and all of statistics there is nothing further re- 

 moved from the trivial. It is the dependence of the entropy on the quantity 

 of the substance, the dependence of S upon iV" the number of atoms. To 

 put a question seemingly so simple that it almost answers itself: given two 

 samples of the same kind of gas under identical conditions, one comprising 

 twice as many atoms as the other, what is the ratio of their entropies? 



This is a remarkable question, because it seems so absurdly simple and is 

 actually so very complex. 



Before the advent of statistical theory, anyone versed in thermodynamics 

 would probably have answered it by replying either that 6" is proportional 

 to N, or that the question has no meaning. The first reply is suggested by 

 the consequences of the fact that thermodynamics proposes no way of 

 measuring the entropy of a gas (or other substance), but only ways of 

 measuring the entropy-difference between two states. Let Vi, Ti and F2, 

 T2 stand for the values of V and T in two states of one gas. Equation (43) 

 informs us that the entropy-difference is L In {V2/V1) -\- Hv In (T2/T1). 

 The constant C has vanished; the remaining terms are proportional to N 

 because L and H^ are proportional to N. The entropy-difference is there- 

 fore proportional to TV". It seems reasonable to conclude that S is propor- 

 tional to N, but so long as there is no specific assertion about C the conclu- 

 sion is not binding; and the proper reply is actually, that the question has 

 no meaning. 



But the statistical theories do make assertions about C, and the question 

 is on the verge of acquiring a meaning; so it might be a good idea to ask in 

 advance what sort of answer we should like to have. It seems natural to 

 expect S to be proportional to N, so that the "double sample" shall have 

 twice the entropy of the "single sample" under identical conditions. But 

 what are "identical conditions?" Here is the catch. No more than two 

 of the three variables P, V, T can be made the same for both the samples. 

 I suppose that almost anyone would choose T for one of these two, so un- 

 plausible would it seem to expect the double sample to have twice the en- 

 tropy of the single sample if their temperatures differed. But after this is 

 decided, shall we make V the same for both, and accordingly give doubled 

 pressure to the double sample? or shall we make F the same for both, and 

 accordingly give doubled volume to the double sample? 



This is no mere quibble, for the choice will determine the dependence of 

 C onN. 



The first alternative requires that C be proportional to N. This is 



