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

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