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We conclude our analysis of the linear response of charge transport in lattice systems of free fermions subjected to a random potential by deriving general mathematical properties of its conductivity at the macroscopic scale. The present paper belongs to a succession of studies on Ohm and Joules laws from a thermodynamic viewpoint. We show, in particular, the existence and finiteness of the conductivity measure $mu _{mathbf{Sigma }}$ for macroscopic scales. Then we prove that, similar to the conductivity measure associated to Drudes model, $mu _{mathbf{Sigma }}$ converges in the weak$^{ast } $-topology to the trivial measure in the case of perfect insulators (strong disorder, complete localization), whereas in the limit of perfect conductors (absence of disorder) it converges to an atomic measure concentrated at frequency $ u =0$. However, the AC--conductivity $mu _{mathbf{Sigma }}|_{mathbb{R}backslash {0}}$ does not vanish in general: We show that $mu _{mathbf{Sigma }}(mathbb{R}backslash {0})>0$, at least for large temperatures and a certain regime of small disorder.
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 th
Free fermions on Johnson graphs $J(n,k)$ are considered and the entanglement entropy of sets of neighborhoods is computed. For a subsystem composed of a single neighborhood, an analytical expression is provided by the decomposition in irreducible sub
Electromagnetic Casimir stresses are of relevance to many technologies based on mesoscopic devices such as MEMS embedded in dielectric media, Casimir induced friction in nano-machinery, micro-fluidics and molecular electronics. Computation of such st
Electric resistance in conducting media is related to heat (or entropy) production in presence of electric fields. In this paper, by using Arakis relative entropy for states, we mathematically define and analyze the heat production of free fermions w
We investigate the level density for several ensembles of positive random matrices of a Wishart--like structure, $W=XX^{dagger}$, where $X$ stands for a nonhermitian random matrix. In particular, making use of the Cauchy transform, we study free mult