THE NEW STATISTICAL MECHANICS 363 



fallacious, and has to be altered in awkward and dubious ways to be made 

 completely successful. 



To consolidate these statements and introduce the new theory, I review 

 the Boltzmann statistics. 



The N atoms of a gas in a container are represented by N numbered balls, 

 identical in every way except the numbering. In the earlier article a game 

 was proposed in which a collection of numbered baskets was provided for 

 these balls, and the balls were tossed into the baskets in a predetermined 

 way: Ni of them into the first of the baskets, .V2 into the second, and so on 

 until each of the baskets contained its preassigned number of balls with N m 

 in the Mth or last basket. The set of numbers iVi, Ni, N3, • ■ • N^f was 

 called a "distribution," and the question was asked: in how many different 

 ways can this distribution be realized? It demands a previous answer to 

 another question: how can a given distribution be realized in more than 

 one way? It is the numbering of the balls which makes this possible. If 

 for instance we exchange two balls in different baskets, the distribution is 

 not changed, and yet there is a difference between the second situation and 

 the first, for an inventory of all the balls in all the baskets shows that in 

 those two particular baskets the assortment has been changed. We thus 

 have realized the same distribution in two different ways. If we had 

 exchanged two balls in one basket, this would not have been regarded as a 

 change; we should still be realizing the same distribution in the same way, 

 in the sense of the Boltzmann statistics. It was shown in the earlier article 

 that the number W of ways of realizing a distribution — or in more technical 

 language, the number of complexions in the distribution — is given by the 

 formula: 



W = Nl/UNjl (1) 



I have said that in the Boltzmann statistics, the balls stand for the atoms 

 of a gas. For what then shall the baskets stand? The baskets stand for 

 compartments in space; but "space" may have several different meanings. 



Giving "space" the ordinary meaning: imagine the gas contained in a 

 box, and the box divided mentally (not physically!) into M compartments 

 of equal volume. I called these by the name of "cells" in the previous 

 article, but now, for a reason which will shortly appear, I rebaptize them 

 "regions." These are the baskets. W has its smallest value, which is 

 unity, when all of the atoms are in the same region. It has its greatest value, 

 which is N\/({N/M)]]^, when in each of the regions there is the same 

 number N/M of atoms. But this corresponds, as nearly as the picture is 

 able to correspond, to the uniform spreading throughout the box which by 

 vast experience we recognize as the natural permanent state of the gas "in 

 equilibrium." The uniform distribution is outstanding because it has the 



