The great majority of published work on the subject of quantum transport deals with conditions very near to thermal equilibrium, particularly with very small voltage drop across the transporting system. These conditions are known as the ``linear-response regime,'' because the currents induced are linear in the applied voltage. The reason that such circumstances have received so much attention is not due to the technological importance of the linear-response regime, but is rather due to the difficulty of theoretically describing significant departures from equilibrium. If these departures are negligible, then one may invoke the well-developed machinery of equilibrium statistical physics and simply treat the departure from equilibrium as a small perturbation on the equilibrium state.
One approach to linear response theory is represented by the Kubo formula for the conductivity [22,23]
where the brackets indicate an average over the equilibrium state. This is a form of the fluctuation-dissipation theorem, relating a transport coefficient, which necessarily characterizes a dissipative process, to the fluctuations about the equilibrium state. Another well-known form of the fluctuation-dissipation theorem is the Einstein relation connecting the mobility and the diffusivity in classical transport theory: . The Kubo formula expresses the conductivity in terms of the autocorrelation of the current density; if one can calculate this autocorrelation from the equations of motion, for example, one can evaluate the frequency-dependent conductivity.