Order to disorder transformation in an Ohmic resistor: Quantum theory of Joule heating

Abstract

The Joule–Lenz heating effect in a resistor is a transformation of ordered (electrochemical) into disordered (thermal) energy. The elementary quantitative account rests upon Ohm’s conduction law. The latter continues to be a theoretical challenge in the 21st century, just as the Joule heating effect at the microscopic level. This work first reviews thermodynamical prolegomena to near-equilibrium electrical conduction. The heating effect (under an applied force field) is argued to be a consequence of the thermalisation mechanism (acting under no force) underpinning the Zeroth Law of thermodynamics. The microscopic theory of thermalisation is worked out in a crystalline solid. Static disorder cannot account for thermalisation of the electron gas at the lattice temperature. The necessary dynamical disorder is handled perturbatively within a Wigner-function-like quantum-mechanical framework. Connection is made with the irreversible Boltzmann–Lorentz description of electron transport via a multiple-scale expansion ; Fermi’s golden rule is reappraised. Classical disorder fails to account for the thermalisation of electrons at the environmental temperature. Quantum disorder embodied in the quantization of lattice vibrations and the entanglement of electrons and phonons correctly account for the Zeroth Law. The mechanism of internal thermal equilibration of the environment is discussed.