THE NEW STATISTICAL MECHANICS 387 



Theory of Radiation 



Black or total radiation, which is the electromagnetic radiation within a 

 cavity enclosed by walls at a uniform temperature, may be regarded as a 

 monatomic gas of which the atoms are called "photons." It has two 

 peculiarities. First, the relation between energy and momentum is not the 

 same for a photon as for a material atom. \ihy pi represent the magnitude 

 '^ypx -\- py -\- pz of the momentum, then the energy E is given no longer by 

 the familiar equation (3), but rather by this one: 



E= pc (70) 



c standing, of course, for the speed of light. This is no insignificant change, 

 but recedes into secondary importance when compared with the other 

 contrast. Not only the distribution-law for the photons, but the actual 

 total number of photons itself, is fixed by Nature when the temperature of 

 the walls of the cavity is fixed by the observer. To the quantity called N, 

 the number of atoms in a container of volume V, no specific value has ever 

 yet been assigned in these pages; for with a material gas it may be raised or 

 lowered at will, by pumping gas into or out of the box. In this section, 

 however, it will have to have a value, for Nature has given it one. 



Can the theory achieve what Nature demands of it? It can, and this is 

 the way. 



The momentum-space is divided as heretofore into regions of equal 

 volume, each containing C cells of volume qm- A distribution is described 

 by giving the number of photons in each region, Nj standing for the number 

 in the jth. region. The probability W of a distribution is given as always 

 by the formula (16) and this is it: 



In W = 2 In Wj = 2[{Nj + C) In (N,- -j- C) - Cln C - iVy In N,-] (71) 



We are not now proceeding to the limit of extreme rarefaction! Radiation 

 presents itself to us under conditions remote from this limit, and must be 

 treated without recourse to the approximation hitherto used in these pages. 

 When the quantities Nj are altered by the small amounts or "variations" 

 SNj, W undergoes the slight alteration or variation given thus to first 

 approximation: 



8W = S ('^j 8N; = - l!n (.V; + C) - In Nj] dN,- (72) 



of e^'^ (see the text preceding equation (35) of the prior article on page 134 of the January 

 issue of this Journal). Thus in both cases \vc arrive at e^'^ or e^'^, according as we pause 

 at a first approximation or go on to a second; but I discern no mathematical or physical 

 similarity whatever in the two situafions in which these approximations are made. 



