THE NEW STATISTICAL MECHANICS 365 



it copes with the task of interpreting entropy. But if I continue longer 

 this review of the conclusions of the former article, the reader's tolerance 

 may be exhausted. Let us make haste to find out how the newer statistics 

 sets forth to find out the most probable distribution. 



One of the most appealing features of the new statistics is, that it does not 

 impose on atoms of a single kind that peculiar distinction which I described 

 above as "numbering" them. We therefore now remove the numbers from 

 the atoms, restoring thus to atoms of a single kind — it might for example 

 be helium — that quality of absolute indistinguishability which the classical 

 statistics took away from them in order to achieve its aims. 



Having de-numbered the atoms, we start anew to play that very game 

 with numbered balls and numbered baskets which we played in the classical 

 statistics with such remarkable but incomplete success. But now that the 

 atoms are de-numbered, they can no longer be the balls (nor, for that matter, 

 the baskets). Something drastically new must now be done, and is. 



In the new statistics, the balls stand for the compartments and the baskets 

 for the populations. 



I must define the word "population." It means the number of atoms in 

 a compartment, or as I will say from this point onward, in a "cell." The 

 balls which are tossed into the basket numbered stand for the cells con- 

 taining no atoms; the balls which go into the basket numbered 1 correspond 

 to the cells containing one atom apiece, and so forth indefinitely, d shall 

 be the symbol for the number of balls in the ith. basket, which is to say, the 

 number of cells containing i atoms apiece. C shall stand for the total 

 number of cells. 



Let the cells at first be compartments of equal volume in the ordinary 

 space, obtained by dividing up (mentally) the box containing the gas. For 

 the number of complexions or inventories corresponding to a given distribu- 

 tion, defined by given values of the quantities d, we have as before: 



W = C\/nCi\ (5) 



and taking the logarithm: 



\nW = C In C - ZCi In C^ (6) 



In using this expression I have again, as often in the previous article, as- 

 sumed the validity of what I there called "the super-Stirling approxi- 

 mation"; but notice that this no longer means that I assume each of the 

 cells to enclose an enormous number of atoms — it means instead that there 

 is an enormous number of cells having each particular population. 



1 Clearly this cannot be so for all populations no matter how great! This is a difficulty 

 which also pops up in the old statistics, though there it is not met until the ordinary space 

 is replaced by the momentum-space. 



