RECENT STATISTICAL THEORIES 735 



As Sommerfeld has shown, all of the reasoning by which (108) was 

 reached remains intact even when it is supposed that Z is a function 

 of v; in that case, I remains under the integral sign in (111), and the 

 integral itself is equal to w/47r times the mean value of v~H{lv~)ldv. 

 This generalization may be of some value. 



Uniform Metal with a Temperature-Gradient; Thermal Conduction 

 We have now to find a function g such that the distribution (/o + ^g) 

 is stable in a metal in which there is a constant gradient of temperature 

 along the axis of x. When we find it, we shall be able to evaluate 

 the integral 



W = hm j i j d^d7jd^-g{vye, (115) 



which is like the integral in (219) except for the differently-chosen 

 form of g and the substitution of ^mv^ for e, and therefore represents 

 the net rate of flow of kinetic energy borne by electrons through unit 

 area perpendicular to the gradient — the contribution of the electrons 

 to the flow of heat, under the circumstances stated. 



The standard distribution-function /o, involving as it does the 

 temperature T, is in this case itself a function of x. One might expect 

 that this dependence of fo on x would be sufficient to fulfil the require- 

 ments. An isotropic distribution which varies from point to point 

 cannot however be stable; the particles conforming to such a one at 

 any given moment would proceed to drift off down the gradient. 

 A stable distribution cannot be isotropic. We must repeat the 

 process of balancing the rates at which particles enter and leave each 

 compartment of the phase-space through collisions and through drift. 

 I say the phase-space now, instead of the velocity-space; for this 

 case is made more complicated than the previous one by reason of 

 the fact, that we now must make the balance separately for the 

 electrons contained in each of the six-dimensional cells d^drjd^dxdydz, 

 whereas previously we could make it en bloc for all of the particles in 

 the entire metal comprised within any velocity-cell d^drjd^. 



Consider then the six-dimensional cell d^d-qd'^dxdydz = ds, and the 

 /(^, V, f , X, y, z)ds electrons in it. The first three factors in ds denote 

 the range of velocity, the last three the range in position, within 

 which an electron must lie if it is to belong to ds. Electrons in the 

 proper range of position are continually entering or leaving the proper 

 range of velocity, because of impacts. The net rate at which ds 

 gains electrons in this way is given, as before, by (106). Electrons 

 in the proper range of velocity are continually drifting into the proper 



