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Microscopic conductivity of lattice fermions at equilibrium. I: Non-interacting particles. (English) Zbl 1322.82015

The present paper belongs to a succession of works on Ohm’s and Joule’s laws starting with [J.-B. Bru et al., Commun. Pure Appl. Math. 68, No. 6, 964–1013 (2015; Zbl 1342.81120)], where heat production of free lattice fermions subjected to a static bounded potential and a time- and space-dependent electric field has been analysed in detail. Here also free lattice fermions subjected to a static bounded potential and a time- and space-dependent electric field are considered. When the electric field is switched on, it accelerates the charged particles and first induces diamagnetic currents. Thereat, a kind of “wave front” occurs, which destabilizes the whole system by changing its internal state. By the phenomenon of current viscosity, the presence of diamagnetic currents causes the appearance of paramagnetic currents, which are responsible for heat production and modify the electromagnetic potential energy of the charged particles.
Thus, in this work, in any bounded convex region \(R\subset \mathbb{R}^d\) (\(d\geq1\)) of space, electric fields \(\epsilon\) 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 \(\epsilon\) 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. Additionally it is shown 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 the 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 the parameters of the system and it is never the trivial measure \(0\;d\nu\). 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. (Text made up of the authors’ abstract and additional remarks by the reviewer)

MSC:

82C70 Transport processes in time-dependent statistical mechanics
82D25 Statistical mechanics of crystals
78A35 Motion of charged particles
78A25 Electromagnetic theory (general)
46F10 Operations with distributions and generalized functions
42A38 Fourier and Fourier-Stieltjes transforms and other transforms of Fourier type
44A15 Special integral transforms (Legendre, Hilbert, etc.)

Citations:

Zbl 1342.81120
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References:

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