740 BELL SYSTEM TECHNICAL JOURNAL 



Heretofore we have tacitly assumed that n remains the same while 

 T and therefore v vary along the axis of x, so that : 



d(nv)/dx = ndv/dx = n{dv/dT){dT/dx), (128) 



and the reader can verify the expressions for a and PF given in (119) 

 by starting from this point. But now we will assume that T and v 

 remain the same while n varies along the x-direction with a gradient 

 dn/dx. Then : 



d{nv)/dx = V- dn/dx. (129) 



Putting this into (127), and recalling that in the Maxwell distribution 

 the mean values of v and v~'^ are thus related, 



V = {2kT/m)tF'\ (130) 



one perceives that v disappears by division from the two sides of the 

 equation, leaving this: 



an = {eElm)n = {kTlm)dnjdx, (131) 



the desired equation for the necessary electric field. Integrating it, 

 we obtain another of very familiar aspect: 



n = no exp {eEjkT){x — Xo) = «o exp (— [F — Fo]/^r). (132) 



This is the celebrated equation of Boltzmann embodying the 

 statement that if in an assemblage of particles at uniform temperature 

 there are variations in the number-per-unit-volume from place to 

 place, then there must also be a field of force against which work 

 must be done to move a particle from place to place — and vice versa. 

 Specifically: if at any two points P and the number-per-unit-volume 

 of the particles has values n and Wo, there must be a field of force 

 such that when a particle is moved from to P its potential energy 

 is increased by — kT -login /no). If the particles are electrons and 

 the field of force is electric and derived from a potential having values 

 F at P and Vo at 0, then of course this change in potential energy is 

 expressed by e(F — Fo). 



Boltzmann's equation is so deeply rooted in modern physics, that 

 it seems strange and suspicious that the new statistics should substitute 

 another but it does. The reason for the innovation stands out very 

 clearly in (127) when the absolute-zero extreme of the Fermi distri- 

 bution is applied. Owing to the dependence of v on n, owing to the 

 interrelation between average speed and number per -unit-volume which 



