Microscopic conductivity of lattice fermions at equilibrium. I. Noninteracting particles
We consider free lattice fermions subjected to a static bounded potential and a time and spacedependent electric field. For any bounded convex region ℛ ⊂ ℝ{sup d} (d ≥ 1) of space, electric fields E within R drive currents. At leading order, uniformly with respect to the volume R of R and the particular choice of the static potential, the dependency on E of the current is linear and described by a conductivity (tempered, operatorvalued) distribution. Because of the positivity of the heat production, the real part of its Fourier transform is a positive measure, named here (microscopic) conductivity measure of R, in accordance with Ohm’s law in Fourier space. This finite measure is the Fourier transform of a timecorrelation function of current fluctuations, i.e., the conductivity distribution satisfies Green–Kubo relations. We additionally show that this measure can also be seen as the boundary value of the Laplace–Fourier transform of a socalled quantum current viscosity. The real and imaginary parts of conductivity distributions are related to each other via the Hilbert transform, i.e., they satisfy Kramers–Kronig relations. At leading order, uniformly with respect to parameters, the heat production is the classical work performed by electric fields on the system inmore »
 Authors:

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 Departamento de Matemáticas, Facultad de Ciencia y Tecnología, Universidad del País Vasco, Apartado 644, 48080 Bilbao, Spain, and BCAM–Basque Center for Applied Mathematics, Mazarredo, 14, 48009 Bilbao, Spain, and Ikerbasque, Basque Foundation for Science, 48011 Bilbao (Spain)
 Departamento de Física Matemåtica, Instituto de Física, Universidade de São Paulo, Caixa Postal 66318, São Paulo, SP 05314970 (Brazil)
 Johannes Gutenberg University Mainz, D 55099 Mainz (Germany)
 Publication Date:
 OSTI Identifier:
 22403138
 Resource Type:
 Journal Article
 Resource Relation:
 Journal Name: Journal of Mathematical Physics; Journal Volume: 56; Journal Issue: 5; Other Information: (c) 2015 AIP Publishing LLC; Country of input: International Atomic Energy Agency (IAEA)
 Country of Publication:
 United States
 Language:
 English
 Subject:
 71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSICS; CORRELATION FUNCTIONS; ELECTRIC FIELDS; FERMIONS; FOURIER TRANSFORMATION; HEAT PRODUCTION; SPACE DEPENDENCE