We give a brief historical account on microscopic explanations of electricalconduction. One aim of this short review is to show that Thermodynamics isfundamental to the theoretical understanding of the phenomenon. We discuss howthe 2nd law, implemented in the scope of Quantum Statistical Mechanics, can benaturally used to give mathematical sense to conductivity of very generalquantum many-body models. This is reminiscent of original ideas of J.P. Joule.We start with Ohm and Joule's discoveries and proceed by describing the Drudemodel of conductivity. The impact of Quantum Mechanics and the Anderson modelare also discussed. The exposition is closed with the presentation of ourapproach to electrical conductivity based on the 2nd law of Thermodynamics aspassivity of systems at thermal equilibrium. It led to new rigorous results onlinear conductivity of interacting fermions. One example is the existence ofso-called AC-conductivity measures for such a physical system. These measuresare, moreover, Fourier transforms of time correlations of current fluctuationsin the system. I.e., the conductivity satisfies, for a large class of quantummechanical microscopic models, Green-Kubo relations.
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