From the 2nd Law of Thermodynamics to AC-Conductivity Measures of Interacting Fermions in Disordered Media

Abstract

We study the dynamics of interacting lattice fermions with random hopping amplitudes and random static potentials, in presence of time-dependent electromagnetic fields. The interparticle interaction is short-range and translation invariant. Electromagnetic fields are compactly supported in time and space. In the limit of infinite space supports (macroscopic limit) of electromagnetic fields, we derive Ohm and Joule's laws in the AC-regime. An important outcome is the extension to interacting fermions of the notion of macroscopic AC-conductivity measures, known so far only for free fermions with disorder. Such excitation measures result from the 2nd law of thermodynamics and turn out to be L\'{e}vy measures. As compared to the Drude (Lorentz--Sommerfeld) model, widely used in Physics, the quantum many-body problem studied here predicts a much smaller AC-conductivity at large frequencies. This indicates (in accordance with experimental results) that the relaxation time of the Drude model, seen as an effective parameter for the conductivity, should be highly frequency-dependent. We conclude by proposing an alternative effective description - using L\'{e}vy Processes in Fourier space - of the phenomenon of electrical conductivity.