AC-Conductivity Measure from Heat Production of Free Fermions in Disordered Media

TL;DR

This paper develops a macroscopic AC-conductivity measure for free fermions in disordered media, linking heat production, thermodynamics, and Green-Kubo relations, extending previous mathematical physics results.

Contribution

It introduces a new AC-conductivity measure derived from thermodynamic principles and Green-Kubo relations for disordered free fermion systems.

Findings

Positivity of heat production for cyclic processes

Continuity and boundedness of the conductivity function

Green-Kubo relations involving current fluctuation correlations

Abstract

We extend [Bru-de Siqueira Pedra-Hertling, J. Math. Phys. 56 (2015) 051901] in order to study the linear response of free fermions on the lattice within a (independently and identically distributed) random potential to a macroscopic electric field that is time- and space-dependent. We obtain the notion of a macroscopic AC-conductivity measure which only results from the second principle of thermodynamics. The latter corresponds here to the positivity of the heat production for cyclic processes on equilibrium states. Its Fourier transform is a continuous bounded function which is naturally called (macroscopic) conductivity. We additionally derive Green-Kubo relations involving time-correlations of bosonic fields coming from current fluctuations in the system. This is reminiscent of non-commutative central limit theorems.