678 BELL SYSTEM TECHNICAL JOURNAL 



nine-tenths of the elements the number of participles is zero while / 

 is greater than zero, and in the others the number is generally much 

 greater than / times the size of the element. To subdivide the space 

 so finely would be to make the atomic picture too definite, and ruin 

 it for the purposes for which we now require it. 



It is not too early in this paper for me to say emphatically that the 

 differential elements which figure in equations such as (1) must not 

 he identified with the elementary compartments of the phase-space, 

 which we shall presently encounter, and which are so important in 

 the new statistics and in the old alike. It takes a great many of these 

 latter to make up an element large enough to be employed in an 

 equation like (1). Otherwise expressed: the subdivision of the phase- 

 space into the elementary cells or compartments of the forthcoming 

 theory is much too fine to be used in connection with the distribution- 

 function. Much confusion may arise from failing to realize this.^ 



In speaking of the distribution-function, I have been tacitly assuming 

 that there is such a thing as a stable, self-sustaining, changeless 

 distribution of the atoms of a gas, the photons in a cavity, the electrons 

 in a wire. This assumption must now be examined. It is scarcely 

 self-evident; one might guess at first that the more numerous the 

 particles, the more abruptly would the distribution vary from one 

 moment to the next, and that an assemblage of 10^" particles would 

 be in such unceasing turmoil that it would be senseless to imagine 

 one single distribution for it. 



Experience however shows the reverse. The gas in a tube remains 

 uniformly dense and stationary, it does not surge forever to and fro 

 nor huddle in a corner nor become spontaneously hot at one end and 

 cold at the other. In the radiation in a cavity with heated walls the 

 intensity comprised within any portion of the spectral range remains 

 unchanged so long as the temperature of the walls is constant. The 

 distribution-in-velocity of the electrons streaming from a heated 

 filament does not appear to change. Moreover, when by artifice the 

 gas in a tube is forced to assume uneven density, non-uniform temper- 

 ature, or any sort of flow or turbulence, it settles down very quickly 

 into a stagnant uniformity as soon as it is left to itself. 



Now we know that while a gas is passing from an unstable state — 

 a state of non-uniform temperature, for instance — to its stable and 

 permanent condition, a property which we call its entropy and denote 



3 1 am thinking particularly of tiie fact that in most expositions of the classical 

 statistics one is adjured that there must be many particles in each compartment, 

 and then in taking up the Fermi statistics one is told that there must be not more 

 than one in each compartment; yet the two lead to formulae which in the limiting 

 case are the same. 



