Heat Transport Equations with Phonons and Electrons
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Heat Transport Equations with Phonons and Electrons V.A. Cimmelli · D. Jou · A. Sellitto
Received: 14 November 2011 / Accepted: 7 February 2012 / Published online: 30 May 2012 © Springer Science+Business Media B.V. 2012
Abstract In the framework of Extended Irreversible Thermodynamics, a model of heat transport due to phonons and electrons is developed. Compatibility with Second Law of Thermodynamics is exploited through a generalized Coleman-Noll procedure. A system of nonlinear partial differential equations, ruling the evolution of the partial heat fluxes, is derived. Propagation of temperature waves through one-dimensional heat conductors is investigated as well. Keywords Extended Irreversible Thermodynamics · Nonlocal heat transport · Generalized Coleman-Noll procedure · Guyer-Krumhansl equation
1 Introduction The analysis of coupled processes is one of the outstanding features of nonequilibrium thermodynamics, and an active challenge in practical applications in materials sciences, highpower lasers, and optimization of energy generation [1, 2]. Much emphasis is put nowadays, for example, on the application of the coupling of heat and electricity. From the point of view of heat transport, the most important consequence of such a coupling is that the family of heat carriers is no longer constituted by the phonons only, as it happens in dielectric V.A. Cimmelli () · A. Sellitto Department of Mathematics and Computer Science, University of Basilicata, Campus Macchia Romana, 85100 Potenza, Italy e-mail: [email protected] A. Sellitto e-mail: [email protected] D. Jou Departament de Fisíca, Universitat Autònoma de Barcelona, 08193 Bellaterra, Catalonia, Spain e-mail: [email protected] D. Jou Institut d’Estudis Catalans, Carme 47, 08001 Barcelona, Catalonia, Spain
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crystals at low temperature [3, 4], since electrons and/or holes may contribute to the heat transport, too [1, 2]. In the present paper, we use the extended-thermodynamic approach to model heat transport due to phonons and electrons, under the hypothesis that the energy and entropy productions due to the presence of an electromagnetic field are negligible. As a consequence of this hypothesis, in building-up our model we may apply the classical local balances of energy and entropy for rigid heat conductors in Extended Irreversible Thermodynamics (EIT) [1, 2], together with a modified balance equation for the heat flux. The basic idea underlying EIT is to upgrade the physical fluxes of energy, matter and momentum to the status of independent variables at the same level as the classical variables like energy, mass, and momentum. The motivations for this selection is that these fluxes are typically nonequilibrium quantities vanishing at equilibrium. This choice finds also its roots in Grad’s thirteen-moments theory [5]. Particular emphasis will be given to nonlocal effects, which are evident in situations in which the mean-free path of the heat carriers is bigger than the characteristic length of the system. Since the pho
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