676 BELL SYSTEM TECHNICAL JOURNAL 



theory of matter and radiation, I may in effect be insisting on the 

 contrast between a faulty way of visuaHzing some phenomena, and a 

 correct way of visualizing all. 



A function of the sort which I just mentioned, a so-called distribution- 

 function, is the goal of every statistical theory. I have said that it 

 states how many among the multitude of particles we assume to be 

 located in each small element of space and to have momentum com- 

 prised in each small range of values of momentum. So it does; but 

 the purpose of a statistical theory is, to derive it from assumptions 

 still more fundamental, in preference to assuming it outright. Of 

 course one might say instead, that the reason for deriving a distribu- 

 tion-function is to put the fundamental assumptions to their test. 

 Whichever viewpoint one prefers, it is the distribution-function which 

 is tested by experiment: indirectly, in that it supplies numerical 

 values for such things as conductivity, viscosity, specific heat; and 

 directly, for there are now immediate ways of observing it in certain 

 cases. 



A distribution-function commonly appears in an equation of this 

 form: 



dN = f(x, y, z, px, py, pz)-dxdydzdpj.dpydpz. (1) 



Such an equation will as a rule refer to some particular assemblage 

 of particles, say N altogether, occupying some definite region of space: 

 a gas in a tube, radiation in a cavity, electrons in a wire. It is to be 

 read as follows: ''dN, equal to f-dxdydzdp^dpydpz, stands for the 

 number of particles having coordinates in dx at x, in dy at y, in dz 

 at z, and components of momentum in dp^ at px, in dpy at py, in dp^ 

 at pz." The phrasing "in dx at x" is a succinct alternative for 

 "between x and x + dx." 



The function / is the distribution-function in the variables in 

 question— here the coordinates of the particles referred to some 

 Cartesian frame in the ordinary or "coordinate" space, and the 

 components of momentum resolved along the axes of that frame, the 

 momenta. Heretofore it has been customary to use the components 

 of velocity rather than the momenta, but these are much to be pre- 

 ferred: partly because it is they which figure in the canonical equations, 

 but chiefly because we shall find when we pass over to the study of 

 assemblages of photons that the momenta play the same role in these 

 as they do in assemblages of atoms, while the speeds of all photons 

 are the same. There is a well-known formula for translating a 

 distribution-function from one set of variables to another set dependent 



