132 BELL SYSTEM TECHNICAL JOURNAL 



"entropy of mixing" though as we have seen it is really the "entropy of 

 expansion". It is the alteration in the second term of the righthand member 

 of (31). 



But now suppose both of the boxes hold helium. One may indeed con- 

 tinue to suppose that when the partition is opened each one of the two 

 samples of helium undergoes an expansion, doubling its volume. The 

 entropy would then go up by 2 Nk In 2. However this looks so silly a thing 

 to say that no one, I feel almost secure in affirming, has ever said it. The 

 natural thing to say is, that the 2N atoms of helium distributed through 

 the two boxes at uniform temperature and uniform pressure have just the 

 same entropy -value whether or not the partition is broken. 



What does the classical statistics say about this situation? Its answer can 

 be foretold. Since the two samples of helium are different by virtue of the 

 different numberings of the two sets of atoms, the classical statistics insists 

 that the entropy increase by 2Nk In 2 when the partition is broken, even 

 though the gases are the same. This is indeed, if I may pervert the poem, 

 "the little rift within the lute, which makes the classical statistics mute." 

 The achievement of predicting the uniform distribution in ordinary space, 

 the achievement of predicting the Maxwell-Boltzmann distribution-law in 

 momentum-space, the achievement of providing the proper relation between 

 temperature and mean kinetic energy — all of these are unsettled by this 

 calamity. 



Were I writing a strictly logical article I should quit at this point. No- 

 thing further can apparently be done, except to tamper with the classical 

 statistics in an effort to remove the unwanted result which has sprung forth 

 to plague us. To violate the logic of the classical statistics in order to 

 banish the undesired while keeping the desired results is a very questionable 

 act. In theoretical physics, it is not admissible that the end justifies any 

 and all means. Nevertheless so successful a feat of tampering has been 

 done, that I cannot refrain from mentioning it as I close. 



Let me first express in a slightly different way the nature of the "rift". 

 Compare two samples of the same gas at the same temperature, one con- 

 sisting of N atoms in a volume V, the other consisting of xN atoms in a 

 volume xV. That which is called entropy in thermodynamics — and there- 

 fore that which is entropy, since it is the privilege of thermodynamics to 

 give the definition of entropy — is x times as great for the latter as for the 

 former. But that which the classical statistics calls entropy — or, as we must 

 admit, miscalls entropy — is not x times as great for the latter as for the 

 former. It would be, if there were x times as many atoms but just the same 

 number of cells. However, there are x times as many atoms but also x 

 times as many cells into which to put them. The number of complexions is 

 approximately W^ m the former case and {xMy^ in the latter, M standing 



