364 BELL SYSTEM TECHNICAL JOURNAL 



greatest value of W. If now we baptize W with the name of "probability," 

 we may then say that the state which in Nature is the prevailing one is in 

 the Boltzmann statistics the most probable one. In the theory it is de- 

 scribed by the equation : 



Nj = N/M (2) 



Now think of "momentum-space" in which there is a dot for every atom, 

 and the Cartesian coordinates of the dot are the momentum-components 

 pT, py, pz of the atom. The coordinates of the dot determine the energy E 

 of the atom, by virtue (for material particles) of the relation: 



E = {l/2m)(pl ^ pli- pi) (3) 



This is a fact of the first importance, as will shortly appear. Let us divide 

 the momentum-space into regions of equal volume. Each of the regions 

 will correspond to a small range of energy-values, as a sample of which we 

 may take any particular one among them. Therefore when we distribute 

 the dots — or let me say simply, the atoms — in any manner among them, we 

 have perforce a certain value of the total energy U of the gas, which we may 

 consider as preassigned if we so wish. Now we are to compare this distribu- 

 tion only with such others as show the same value of U. Among these there 

 is one which is outstanding because it has the greatest value of W. This was 

 shown in the earlier article to be the canonical or Maxwell-Boltzmann 

 distribution, described by the formula: 



Nj = NA exp i-BEj) (4) 



in which Nj stands for the number of atoms in the region numbered j; E, 

 for the value of E appropriate to that region, i.e. obtained by substituting 

 into (3) the coordinates of some point in that cell; A and B for constants, 

 whereof A depends on B while B depends upon U/N the average energy of 

 the atoms of the gas. This distribution also is attested by experiment as 

 being truly that of a gas in its normal natural abiding state of equilibrium. 



Now I mention the concept of a six-dimensional space which comprehends 

 both the ordinary space and the momentum-space, and is divided into six- 

 dimensional regions of equal volume. By this device one is able to speak of 

 (2) and (4) as two aspects of a single distribution in the "M-space." This is 

 the distribution outstanding among those with which it may legitimately 

 be compared by reason of having the greatest Il^-value. It is the most 

 probable distribution, in the sense given in the Boltzmann statistics to the 

 word "probable." 



This is the first triumph of the Boltzmann statistics, attained by number- 

 ing the atoms. Its other triumphs, and its ultimate confusion, come when 



