MEMORIAL TO CLASSICAL STATISTICS 131 



different ways into a set of N' atoms bearing one set of numbers and a set 

 of N" atoms bearing another set of numbers. Each set would obviously 

 have to have the same distribution, with the same A and the same B, as any 

 other set or as the totality of all the atoms. This conclusion, which is self- 

 evident in the case in which all the atoms are called "mercury", remains true 

 when some of the atoms are called "mercury" and others are called "helium". 

 We have done nothing but change the names of some of the atoms; we have 

 not imported into our theory anything which differentiates one kind of 

 atom from another kind. No wonder we have arrived at the conclusion that 

 all kinds have the same distribution-in-energy, the same A, the same B and 

 the same temperature! The rabbit was indeed in the hat, but it does not 

 look like so much of a rabbit. 



The classical statistics therefore doesn't recognize any of the real dif- 

 ferences between atoms of different kinds, except for alterations in the last 

 term of (31); but it does make an artificial difference which creates the 

 astonishing result, that any two samples of the same gas are different gases! 

 At this point we may begin to wonder whether this peculiarity, which has 

 led to so apparently brilliant a result in respect of the equality of tempera- 

 tures in thermal equilibrium, might elsewhere lead us astray. It does; and 

 here appears the rift in the lute of classical statistics. 



The Rift in the Lute 



Let us imagine two boxes of equal size separated by a common partition, 

 each containing a gas consisting of N atoms, both gases at the same tempera- 

 ture. We will baptize one gas "mercury" and the other gas "helium". 

 Let an opening be made through the partition. It is known that in such a 

 situation in Nature, the two gases diffuse into one another, the final and 

 permanent condition being that in which the mercury and the helium are 

 equally distributed between the two boxes. The process of diffusion is an 

 example of what in thermodynamics is called an "irreversible" process. The 

 state of uniform mixing ought to correspond to the most probable state in 

 the statistical picture. But what does the statistical theory say? 



The statistical theory says nothing about diffusion and nothing about 

 mixing. The statistical theory takes account of nothing but the facts that 

 the mercury had at its disposal the volume V before and the volume 2V 

 after the breaking of the partition, and ditto for the helium. The value V 

 contains M cells {M = V/Vo) and the volume 2V contains 2M cells. The 

 (approximate) probabilities of the uniform distribution are M^ before and 

 (2M)^ after. The latter is greater than the former; the entropy goes up by 

 Nk In 2 for each gas, by 2 Nk In 2 for the two of them, when the private pre- 

 serve of each is thrown open to the other. This gain is what is called the 



