International Journal of Geometric Methods in Modern Physics Vol. 13 (2016) 1640003 (11 pages) c  World Scientiﬁc Publishing Company DOI: 10.1142/S021988781640003X On heat equation in the framework of classic irreversible thermodynamics with internal variables Vincenzo Ciancio ∗ and Liliana Restuccia † Department of Mathematical and Computer Sciences Physical Sciences and Earth Sciences, University of Messina Viale F. Stagno d’Alcontres 31 Messina 98166, Italy ∗ ciancio@unime.it † lrestuccia@unime.it Received 30 December 2015 Accepted 11 April 2016 Published 9 June 2016 In this paper, we show that, using a procedure of classical irreversible thermodynamics (CIT) with internal variables, it is possible to describe the relaxation of thermal phe- nomena, obtaining some well known results of extended irreversible thermodynamics (EIT). In particular, we introduce as internal variables a vector and a second rank ten- sor, that inﬂuence the thermal transport phenomena, and we derive in the anisotropic and isotropic case, the phenomenological equations for these variables. In the case, in which the medium is isotropic, it is obtained that the total heat ﬂux can be split in two parts: a ﬁrst contribution J (0) , governed by Fourier law, and a second contribution J (1) , obeying Maxwell–Cattaneo–Vernotte (MCV) equation, in which a relaxation time is present. The obtained results may have applications in describing the thermal behav- ior in nanosystems (semiconductors, nanotubes,... ), where the phenomena are fast and there are high-frequency thermal waves. Keywords : Continuum thermodynamics; heat conduction; classical irreversible thermo- dynamics; entropy principle; extended irreversible thermodynamics. 1. Introduction. In 1931 Lars Onsager, with his pioneering works [1–4], established a very general approach to study diﬀerent interactions in irreversible thermodynamic processes, using classical exstensive thermodynamic variables. On the basis of these studies, Eckart [5–8], Prigogine [9, 10], De Groot [11], Mazur [12] and many others [13–18], developed the well known thermodynamic theory with internal variables, called classical irreversible thermodynamics (CIT). The ﬂexibility of the methodology, used in CIT, is due to the fact that “a priori ” the physical meaning of the internal variables is not speciﬁed, but only their inﬂuence on phenomena, occuring inside the material under consideration, is pointed out. 1640003-1 Int. J. Geom. Methods Mod. Phys. Downloaded from www.worldscientific.com by WSPC on 06/09/16. For personal use only.