682 BELL SYSTEM TECHNICAL JOURNAL 



and from equations (4) we deduce, for particles of light: 



E = pc. (6) 



The difference between these two relations is responsible for some of 

 the contrasts between radiation-gas on the one hand, electron-gas 

 and material gases on the other; but by no means for the major part. 

 The major difference lies in the statistical theory, as we shall now 

 find out. 



The Classical Statistics 



We are going to represent three kinds of objects — ordinary or 

 material gases, radiation in enclosures, negative electricity in metals — 

 as assemblages of particles possessing location and momentum. We 

 may visualize such an assemblage first as a swarm of points in ordinary 

 space, with a coordinate-frame along the axes of which the coordinates 

 X, y, z of the particles are measured; then as a swarm of points in 

 momentum-space with a frame along the axes of which the momenta 

 px, py, pz are measured. 



I will first illustrate the method of classical statistics by using it to 

 ascertain the most likely distribution of particles in ordinary space, 

 a case where seemingly the result may be foreseen. For it seems a 

 truth of intuition that inside a box of ordinary space, with nothing 

 {e.g. no variations of potential) to distinguish one region from another, 

 the particles must tend to distribute themselves uniformly. This is 

 a conclusion to which the statistical method must lead. The uniform 

 distribution must be the most probable. How then should we define 

 the "probability" of a distribution so that it shall be greatest for 

 the uniform one? 



But in the first place, what is a uniform distribution? We must 

 divide the space — mentally, of course — into conpartments of equal 

 volume. The distribution will then be called uniform, if the numbers 

 of particles in the various compartments are about the same. But 

 this clearly requires that these subdivisions be of a certain size. 

 Their linear dimensions cannot for example be smaller than the 

 average distance between particles, as then a "uniform distribution" 

 would be impossible. To partition the space too finely would be 

 like studying a painting with a microscope. The quality which we 

 wish to define evades too sharp a scrutiny. The compartments 

 should contain large numbers of particles, both for the stated reason 

 and for the convenience of a certain mathematical approximation 

 which is made. 



Denote then by N the total number of particles, by m the number 



