388 BELL SYSTEM TECHNICAL JOURNAL 



In the quest for the "most probable distribution" this quantity is required to 

 vanish for variations which are controlled by a certain condition. 



On a previous page (368) where we were dealing with material atoms in 

 ordinary space, the sole condition was that the total number of atoms should 

 remain the same (and equal to N). This led to the uniform distribution, 

 N i — a\ here a stands for a constant, which turns out to be the product of N 

 by the ratio of the volume Fo of the region to the volume V of the box. 



On another previous page (369), where we were dealing with material 

 atoms in momentum space, the condition imposed was twofold: that the 

 number of atoms N and the total energy U of the atoms should remain the 

 same. This led to the distribution (20), which in the limit of extreme 

 rarefaction became the Maxwell-Boltzmann or canonical distribution Nj = 

 NA exp {—BEj); here Ej stands for the energy- value appropriate to the 

 jth region, and^l and ^ for two constants which were shown to be determined 

 by TV and U. 



In this case where we are dealing with photons in momentum space, the 

 condition which leads to the right result is simple but surprising. We must 

 admit only such variations as leave the total energy constant, but we must 

 not require that the total number of photons should likewise remain the same. 

 Applying this strange condition, we find it taking the form, 



In (iVy + C) - In Nj = BE,- (73) 



with only one constant, which is going to be controlled by the total energy U. 

 Rewriting this: 



N.- 1 



BE, 



C e"' - 1 



(74) 



One sees immediately that TV, which is the sum of all the quantities TV,-, is 

 no longer at liberty to take whatever value the experimenter pleases ! 



Hitherto I have assumed that all the regions are of equal volume, but I 

 can free myself from this assumption by pointing out that 'N jjC is the 

 average number of photons per cell in the portion of momentum-space 

 where E has the value £,. Now let us carve up the momentum-space into 

 regions separated by spherical shells all centred at the origin. The region 

 extending from the sphere of radius p to the sphere of radius p -\- dp will be 

 of volume 4Trp''dp, and will accordingly contain Awp'dp/qm cells, if by qm I 

 denote the volume of a cell. The appropriate value of E will be pc. The 

 number of photons in the region will accordingly be given thus: 



dN=^~^—^dp (75) 



5„ e^p^ - 1 ^ 



