The second order electrical effects in metals

The theory of transport phenomena in metals depends upon the solution of an integral equation for the velocity distribution function f of the conduction electrons. This integral equation is formed by equating the rate of change in f due to external fields and temperature gradients to the rate of change in f due to the mechanism which produces the resistance. If this latter rate of change is denoted by ∂f/∂tcoll it happens with some mechanisms that where f0 is the equilibrium distribution, and ℸ is the time of relaxation which does not depend on the external fields. When equation (1) is true, the problem is comparatively simple, but in general ∂/∂tcoll is an integral operator and it is not possible to define a time of relaxation and a free path. It is known that at high temperatures, such that (Θ/T)2 can be neglected, where Θ is the Debye temperature, a free path exists; but, in general, special methods have to be used to solve the integral equation.