Proceedings of the Estonian Academy of Sciences, 2008, 57, 3, 118–126 doi: 10.3176/proc.2008.3.01 Available online at www.eap.ee/proceedings Extended irreversible thermodynamics of heat transport A brief introduction David Jou a* , José Casas-Vázquez a , and Georgy Lebon b a Universitat Autònoma de Barcelona, 08193 Bellaterra, Catalonia, Spain b Université de Liège, Sart Tilman, B-4000 Liège, Belgique Received 5 February 2008, in revised form 3 April 2008 Abstract. Current frontiers of technology require generalized transport equations incorporating memory, non-local effects, and non-linear effects. Extended Irreversible Thermodynamics provides such transport equations in a form compatible with the second law of thermodynamics, and that, for low frequency and short mean-free paths, reduce to the classical transport equations. Here we present the basic concepts of extended irreversible thermodynamics, namely, the fluxes as independent variables, and their evolution equations as generalized transport equations obeying the second law of thermodynamics. We show that these equations cover a rich phenomenology in heat transport, including thermal waves, phonon hydrodynamics, ballistic transport, and saturation in the fluxes for high values of the thermodynamic forces. Key words: non-equilibrium thermodynamics, ballistic heat transport, extended irreversible thermodynamics, thermal waves. 1. INTRODUCTION. PHYSICAL MOTIVATIONS * Current frontiers in technology, e.g. in materials sciences and nanotechnology, require generalized trans- port equations beyond the classical theory. For instance, transport equations for miniaturized systems whose size is comparable to the mean free path have become an important topic because of the surge of nanotechnology. Analogously, the behaviour of systems submitted to high-frequency perturbations, comparable to the reciprocal of internal relaxation times, is necessary to optimize the operation of high-frequency devices. Equa- tions for heat, mass, charge, and momentum transport have been actively explored in several situations: in miniaturized electronic devices, in nanotubes and nanowires, in theoretical models of energy transport in one-dimensional chains, in rarefied gases, etc. How far thermodynamic formalisms are helpful or necessary in this endeavour is an open question, because the mentioned situations clearly exceed the limits of validity of the classical local-equilibrium thermo- * Corresponding author, David.Jou@uab.cat dynamics. Therefore, their study is a challenge for non- equilibrium thermodynamics to better understand its basic concepts, its limits of application, and its frontiers. Among the several thermodynamic theories going beyond the local-equilibrium approach, Extended Irre- versible Thermodynamics (EIT) provides generalized transport equations that incorporate memory and non- local effects, that reduce to the classical transport equations for low frequency and short mean-free paths, and that are able to incorporate the high-frequency behaviour of dissipative signals as, for instance, a finite speed of propagation of heat pulses, and general features of ballistic behaviour for long mean-free paths. Thus, they provide not only small corrections to the usual transport equations, but they turn out to be useful even under extreme non-equilibrium situations where the classical equations completely fail. In this paper, we introduce the basic concepts of EIT, namely, the idea of the fluxes as independent variables of the entropy and entropy flux, and the procedure to obtain their evolution equations, which provide generalized transport equations obeying the second law of thermodynamics. The reader interested in this topic and in its relation with other current frontiers