THE NEW STATISTICAL MECHANICS 373 



Yet any policy which leaves this basic law unsaid, or even fails to emphasize 

 it, is (I think) a bad one for all but the very few to whom it is already obvious. 



Another question: may the old statistics be regarded as the limiting form 

 of the new statistics, in the limiting case of "extreme rarefaction" where 

 in every region the number of atoms is very much smaller than the number 

 of cells? 



It may seem that this question has already been answered with a yes, 

 in view of the fact that in the limiting case the new statistics gives the same 

 distribution-law — in momentum-space as in ordinary space — as the old one 

 does. Nevertheless the answer is no. Mathematically this appears in the 

 following way. In the old statistics the Maxwell-Boltzmann law springs 

 from the denominator IT N j\ in the right-hand member of equation (1), 

 which turns up as (— SiVy In Nj) in the expression for InW. Now we look 

 at the equation (23) and see the term —Njln Nj appearing with several 

 other terms in the right-hand member. In the limit of extreme rarefaction 

 it outweighs all the others and survives by itself. We summed it over all 

 the regions and so arrived again at {—'LNj In Nj), from which again the 

 Maxwell-Boltzmann law emerged. But in this method of the new statistics 

 each term of the summation comes by itself from the corresponding region, 

 whereas in the method of the old statistics the whole summation arrived 

 upon the scene en bloc or all in a single piece. The former method does not 

 pass into the latter method in the limiting case. The conclusions agree 

 in the limit, but the methods do not. 



I have mentioned this because not infrequently one finds in print the 

 careless statement that the old statistics is the limiting case of the new 

 statistics, or words to that effect. Actually one can find more potent ways of 

 contradicting that statement, as for example by emphasizing that the old 

 statistics numbers the atoms and the new one leaves them un-numbered, 

 and in no way can the one policy be regarded as a limiting case of the other. 

 More convincing yet would it be to show that the new statistics and the old 

 lead to results which definitely differ even in the limiting case of extreme 

 rarefaction. This is what I next undertake to show as an incident of the 

 explanation of entropy which the new statistics affords. 



Theory or Entropy 



For a substance of a single kind in a single phase, the basis of thermo- 

 dynamics is the single equation, 



dU = TdS - PdV (31) 



in which there are five variables: pressure P, volume V, absolute tempera- 

 ture T, energy U and entropy S. Two may be varied independently, and 



