Heat Production of Non-Interacting Fermions Subjected to Electric Fields

TL;DR

This paper mathematically defines and analyzes heat production in free fermions under electric fields, establishing a thermodynamic framework and verifying the first law for such quantum systems.

Contribution

It introduces a rigorous mathematical approach to quantify heat production in non-interacting fermions influenced by electric fields, using Araki's relative entropy.

Findings

Verified the first law of thermodynamics for the system.

Established a link between heat production and internal energy increase.

Analyzed the regime of small, slowly varying electric fields.

Abstract

Electric resistance in conducting media is related to heat (or entropy) production in presence of electric fields. In this paper, by using Araki's relative entropy for states, we mathematically define and analyze the heat production of free fermions within external potentials. More precisely, we investigate the heat production of the non-autonomous C*-dynamical system obtained from the fermionic second quantization of a discrete Schr\"odinger operator with bounded static potential in presence of an electric field that is time- and space-dependent. It is a first preliminary step towards a mathematical description of transport properties of fermions from thermal considerations. This program will be carried out in several papers. The regime of small and slowly varying in space electric fields is important in this context, and is studied the present paper. We use tree-decay bounds of the $n$-point, $n\in 2\mathbb{N}$, correlations of the many-fermion system to analyze this regime. We verify below the 1st law of thermodynamics for the system under consideration. The latter implies, for systems doing no work, that the heat produced by the electromagnetic field is exactly the increase of the internal energy resulting from the modification of the (infinite volume) state of the fermion system. The identification of heat production with an energy increment is, among other things, technically convenient. We initially focus our study on non-interacting (or free) fermions, but our approach will be later applied to weakly interacting fermions.