On the theory of electric dc-conductivity : linear and non-linear microscopic evolution and macroscopic behaviour

We consider the Schrodinger time evolution of charged particles subject to a static substrate potential and to a homogeneous, macroscopic electric field (a magnetic field may also be present). We investigate the microscopic velocities and the resulting macroscopic current. We show that the microscopic velocities are in general non-linear with respect to the electric field. One kind of non-linearity arises from the highly non-linear adiabatic evolution and (or) from an admixture of parts of it in so-called intermediate states, and the other kind from non-quadratic transition rates between adiabatic states. The resulting macroscopic dc-current may or may not be linear in the field. Three cases can be distinguished : (a) The microscopic non-linearities can be neglected. This is assumed to be the case in linear response theory (Kubo formalism, ...). We give arguments which make it plausible that often such an assumption is indeed justified, in particular for the current parallel to the field. (b) The microscopic non-linearitites lead to macroscopic non-linearities. An example is the onset of dissipation by increasing the electric field in the breakdown of the quantum Hall effect. (c) The macroscopic current is linear although the microscopic non-linearities constitute an essential part of it and cannot be neglected. We show that the Hall current of a quantized Hall plateau belongs to this case. This illustrates that macroscopic linearity does not necessarily result from microscopic linearity. In the second and third cases linear response theory is inadequate. We elucidate also some other problems related to linear response theory.