Quantum Transport Theory

Abstract

Boltzmann transport theory (BTT) is an ideal theory. It has the twin virtues of conceptual and mathematical simplicity. It also works far better than one could reasonably expect from its origin as a graft from the classical theory of dilute gases. Quantum transport theory (QTT) (Kohn and Luttinger, 1957, 1958; Kubo, 1957; Dresden, 1961; Chester, 1963; Kubo, 1966; Luttinger, 1968) enjoys no such status: it is neither conceptually nor mathematically simple; it is very hard to make it work; and it often reduces, after considerable labour, to the Boltzmann picture (Peierls, 1974; Cohen and Thirrig, 1973). But even if there were no manifestly quantum transport phenomena (and we might single out transport in quantizing magnetic fields, but equally: hopping conduction, impurity conduction, polaron transport, high-frequency transport, quasi-1 D transport, the Kondo effect, size limited transport, and of course superconductivity) (Barker, 1978, 1979), we would still require QTT as an ab initio theory to explain how the phenomenological BTT picture and its related concepts actually arise from the underlying framework of reversible quantum statistical mechanics. QTT is thus concerned with: (1) an explanatory and supportive theory for the Boltzmann picture, where that exists; (2) setting confidence limits for the application of BTT; (3) developing the necessary novel concepts and transport kinetics for genuine quantum transport phenomena (the latter may be loosely defined as those effects which depend explicitly on the quantum mechanical nature of the electron, and/or those processes for which the simple relaxive local Boltzmann description fails.