Microscopic Foundations of Ohm and Joule's Laws - The Relevance of Thermodynamics

TL;DR

This paper emphasizes the fundamental role of thermodynamics, especially the second law, in understanding electrical conduction at the microscopic level, connecting classical laws with quantum statistical mechanics and presenting new rigorous results.

Contribution

It introduces a thermodynamics-based approach to electrical conductivity, providing rigorous results and new measures for quantum many-body models, linking classical and quantum theories.

Findings

Existence of AC-conductivity measures for quantum systems

Green-Kubo relations hold for a broad class of models

Thermodynamics underpins microscopic explanations of conduction

Abstract

We give a brief historical account on microscopic explanations of electrical conduction. One aim of this short review is to show that Thermodynamics is fundamental to the theoretical understanding of the phenomenon. We discuss how the 2nd law, implemented in the scope of Quantum Statistical Mechanics, can be naturally used to give mathematical sense to conductivity of very general quantum 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 Drude model of conductivity. The impact of Quantum Mechanics and the Anderson model are also discussed. The exposition is closed with the presentation of our approach to electrical conductivity based on the 2nd law of Thermodynamics as passivity of systems at thermal equilibrium. It led to new rigorous results on linear conductivity of interacting fermions. One example is the existence of so-called AC-conductivity measures for such a physical system. These measures are, moreover, Fourier transforms of time correlations of current fluctuations in the system. I.e., the conductivity satisfies, for a large class of quantum mechanical microscopic models, Green-Kubo relations.