Microscopic conductivity of lattice fermions at equilibrium. I. Non-interacting particles

Autor(es)

Bru, J. -B. ^{[1, 2, 3]} ; de Siqueira Pedra, W. ^[4] ; Hertling, C. ^[5]

Afiliação do(s) autor(es)

^[1] Univ Basque Country, Dept Matemat, Fac Ciencia & Tecnol, E-48080 Bilbao - Spain

^[2] BCAM, Bilbao 48009 - Spain

^[3] Basque Fdn Sci, Ikerbasque, Bilbao 48011 - Spain

^[4] Univ Sao Paulo, Inst Fis, Dept Fis Matemat, BR-05314970 Sao Paulo, SP - Brazil

^[5] Johannes Gutenberg Univ Mainz, D-55099 Mainz - Germany

• Tipo de documento: Artigo Científico
• Fonte: Journal of Mathematical Physics; v. 56, n. 5 MAY 2015.
• Citações Web of Science: 5

Resumo

We consider free lattice fermions subjected to a static bounded potential and a time-and space-dependent electric field. For any bounded convex region R subset of R-d (d >= 1) of space, electric fields epsilon within R drive currents. At leading order, uniformly with respect to the volume vertical bar R vertical bar of R and the particular choice of the static potential, the dependency on epsilon of the current is linear and described by a conductivity (tempered, operator-valued) 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 time-correlation 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 so-called 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 in presence of currents. The conductivity measure is uniformly bounded with respect to parameters of the system and it is never the trivial measure 0 d.. Therefore, electric fields generally produce heat in such systems. In fact, the conductivity measure defines a quadratic form in the space of Schwartz functions, the Legendre-Fenchel transform of which describes the resistivity of the system. This leads to Joule's law, i.e., the heat produced by currents is proportional to the resistivity and the square of currents. (C) 2015 AIP Publishing LLC. (AU)

• Processo FAPESP: 13/13215-5 - Produção de calor em sistemas fermiônicos infinitos sujeitos a campos elétricos
• Beneficiário: Walter Alberto de Siqueira Pedra
• Modalidade de apoio: Auxílio à Pesquisa - Regular