On the nature of the relaxation time, the Maxwell–Cattaneo and Fourier law in the thermodynamics of a continuous medium, and the scale effects in thermal conductivity

Heat Transport on Ultrashort Time and Space Scales in Nanosized Systems: Diffusive or Wave-like?

“…where τ is the relaxation time to the local equilibrium. Equation (1) has been obtained from both macroscopic and microscopic approaches (see [12,[16][17][18]22,25,26,[29][30][31] and references therein). The values of the relaxation time τ range from tens of seconds in systems with heterogeneous inner structure and biosystems [49][50][51][52][53][54][55][56][57][58] to picoseconds in metals and dielectric solids [4][5][6]12,[32][33][34]38,39,46,47].…”

“…The interest is largely motivated by technological needs such as thermal management in microelectronics and the ultra-fast laser processing of advanced metamaterials [1,4], i.e., artificial materials and media of designed properties, such as layered correlated materials [5,6]. Moreover, the heat transport at ultrashort space and time scales leads to unusual non-Fourier phenomena such as wavelike temperature propagation [4][5][6]12,13], size [14,15] and distance [15] dependent thermal conductivity, and boundary temperature jumps [1,14,15], which have raised an extensive body of literature concerning different conceptual questions of these phenomena [15][16][17][18][19][20][21][22][23][24][25][26][27][28][29][30]. The problem is that when the characteristic length of the process is of the order of the mean free path (MFP) of energy carriers and/or the characteristic time scale of the process is of the order of the mean free time (MFT) of energy carriers, the thermal dynamics occur under far from local equilibrium conductions and cannot be described by classical Fourier law based on the local equilibrium assumption [31].…”

“…In this case, one needs to use an approach which is not based on the local equilibrium assumption. The most simple and well-known generalization of the classical Fourier heat conduction equation of the parabolic type is the hyperbolic heat conduction equation (HHCE), which combines diffusive (dissipative) and wave modes of heat transport and transmits temperature discontinuity with a constant velocity without smoothing [12,25,26,[29][30][31]. To our knowledge, the HHCE was first obtained under an assumption that the energy carriers move with a well-defined finite velocity (see discussion in [29]).…”

“…To our knowledge, the HHCE was first obtained under an assumption that the energy carriers move with a well-defined finite velocity (see discussion in [29]). While the classical heat conduction equation of the parabolic type results in the infinite propagation velocity of thermal disturbances, the HHCE removes this paradox leading to a finite value of propagation [12,22,25,31]. This property of the HHCE has attracted a wide range of interest mainly in the context of heat waves in liquid helium II (the so-called "second sound") [12].…”

Analytical Solution of Stationary Coupled Thermoelasticity Problem for Inhomogeneous Structures

“…Subsequently, these contradictions were removed as a result of the development of a generalization of the theory of heat conduction [2][3][4]. The idea of studying connected processes of thermodynamics of deformations and thermal conductivity problems led to the development of a generalized model of hyperbolic thermal conductivity in [5][6][7][8].…”