The Physical Interpretation of Mean Free Path and the Integral Method

In previous papers, general expressions for the linear electronic transport constants of solids were obtained in terms of a conjugate function ψ†related, by a linear inhomogeneous integral equation, to the function (of electron state) ψ measured by the “flux.” It is now shown that τψ†=∫0∞ψ|t) dt, where the integrand is the expectation of ψ for an electron which at time t earlier was in the specified state (of which ψ†is a function) and 1/τ is the collision frequency. In particular, the vector mean free path τv†is: “the limit, after a virtually infinite time, of the mean displacement, in Brownian motion, of the position of an electron initially in the specified state.” If there is a force (e.g. that due to a magnetic field) accelerating the electrons between collisions, then a linear transport constant is the same functional of an “extended conjugate” ψ†as it is of ψ†in the absence of the force. It is shown that τψ†eis obtained (instead of τψ†) when the integrand in the integral above is replaced by the “expectation after time t ” as modified by the accelerations between collisions. The relation of the present formalism to the Shockley-Chambers theory is discussed.