THE NEW STATISTICAL MECHANICS 391 



ways can C numbered balls be distributed among two baskets, these to 

 contain {C — N j) and N j of the balls respectively, two ways being con- 

 sidered as different unless the inventory of each basket is just the same for 

 both ways? But this is the problem set up and solved in the earlier article, 

 though there the balls stood for atoms and the baskets for regions. The 

 answer is: 



Wui = C\/{C- Ni)\Ni\ (85) 



instead of equation (22). Using the first or second order Stirling approxima- 

 tion — it doesn't matter which — one comes to the analogue of (23), which is: 



In IFtot = Cln C - (C - Ni) In (C - Nj) - N, In N, (86) 



Different as this looks from (23), the two become alike in the limit of extreme 

 rarefaction, and in this limit equation (48) expresses the result both of the 

 Bose-Einstein and of the Fermi-Dirac statistics. Since equation (48) is the 

 parent of the Maxwell-Boltzmann distribution-law and of the expression 

 (66) for the entropy of a monatomic gas, both of these flow from either type 

 of statistics, and experiment does not decide for either over the other. 



When we avoid the limit of extreme rarefaction, the two forms of statistics 

 do depart from one another. If photons obeyed the Fermi-Dirac statistics, 

 the distribution-law for black radiation would not be (75). We should be 

 obliged, in the denominator on the right-hand side of that equation, to re- 

 place the negative sign of the second term by the positive sign. In so doing 

 we should contradict the data of experiment in an unmistakable way; and 

 for photons accordingly, the Fermi-Dirac statistics is to be rejected. 



This form of the new statistics being no better than the other for material 

 gases, and definitely wrong for radiation, where is it to be preferred and why? 



To answer the first question, I point to the "electron-gas" which pervades 

 the metals and is accountable for their quality of being excellent conductors. 

 Experiment (as I recounted in these pages fourteen years ago^") confirms that 

 these intra-metallic electrons form a gas which obeys the Fermi-Dirac 

 statistics. It is not, however, the limit of extreme rarefaction which here 

 we meet but the opposite one, the limit of extreme condensation. These 

 electrons are as densely concentrated as the atoms of the solid itself, a degree 

 of condensation never even approached by any ordinary gases. In this 

 limit the distribution-law attains a form entirely different from both the 

 Maxwell-Boltzmann law and the black-radiation law, and very remarkable. 

 I dare not, however, expose this article to the risk of a doubling in length, 

 which a treatment of this topic would probably entail; and I can avoid it 

 with a fairly clear conscience, for the experimental evidence that electrons 



'"This Journal, -?, 672 (1929); also Phxsical Revicu.- Supplement (Reviews of Modern 

 Physics) 7,90(1929) . 



