Nonlocal and nonlinear transport theories at nanoscale: applications to wave propagation

Abstract

Many books and/or papers have been published on linear generalizations of Fourier's equation in order to introduce relaxational and nonlocal e ects for the heat ux [1{9]. Describing a heat-pulse propagation with a nite speed [10{15], in agreement with experimental observations, such works are of much conceptual interest both because they may be applied to small systems (the characteristic size of which is comparable to the mean-free path of the heat carriers) [16{23] or to fast processes (as for instance response to short laser pulses) [24{28], and because they have stimulated generalized formulations of non-equilibrium thermodynamics, with generalized expressions of the entropy and of the entropy ux incorporating heat- ux contributions [1{4, 7, 22, 29, 30]. The linear generalizations of Fourier's equation should be only employed to analyze the propagation of small-amplitude waves. When the amplitude of temperature waves (or of heat- ux waves) is not negligible, in fact, nonlinear e ects cannot be neglected: this is the case, for example, when short and intense laser pulses are applied to heat a given material. Therefore, there is much interest in generalizing the linear theory of heat waves which has been, up to now, a fruitful stimulus to generalizations of non-equilibrium thermodynamics [1{4, 7, 9, 11, 12, 14, 29, 31{34] to nonlinear situations, namely, for waves with su ciently high amplitudes [18, 20, 35{39]; indeed, there are many possible nonlinear generalizations and, from a thermodynamic point of view, it is of special interest selecting the forms which t in a most direct way with the requirements of the second law of thermodynamics. The present thesis aims at being a contribution to the study of heat waves when nonlinear and/or nonlocal generalizations of the Maxwell-Cattaneo equation in the context of extended thermodynamics [1, 4, 7, 35, 40] are introduced. Whereas nonlocal e ects in heat transport have led to fruitful analogies with hydrodynamics, especially in the so-called phonon hydrodynamics, in the present thesis we also show how some particular nonlinear e ects lead to fruitful analogies with nonlinear optics. We think that these analogies of heat transport with hydrodynamics and with optics are a nice illustration of the deep unity of physics, where results in some eld may also be helpful to other elds, provided that the connection between both elds is found. The present thesis is a contribution in that direction, and the results contained in it may be of interest to current researches aiming to nd new ways of control and applications of the heat ux, which is the main goal of the socalled phononics [25, 27]. In particular, the interaction of intense laser pulses with heat-conducting solids has motivated nonlinear phononics [24, 26, 28], requiring a combination of nonlinear optics and nonlinear heat transport. The plan of this thesis is the following. In Chapter 1 we recall the basic mathematical de nitions and concepts which will be employed in this thesis, and brie y summarize the theoretical thermodynamic background. ... [edited by Author]