Relativistic heat conduction refers to the modelling of
heat conduction (and similar
diffusion processes) in a way compatible with
special relativity. In
special (and
general) relativity, the usual
heat equation for non-relativistic heat conduction must be modified, as it leads to faster-than-light signal propagation.[1][2] Relativistic heat conduction, therefore, encompasses a set of models for heat propagation in continuous media (solids, fluids, gases) that are consistent with relativistic
causality, namely the principle that an effect must be within the
light-cone associated to its cause. Any reasonable relativistic model for heat conduction must also be
stable, in the sense that differences in temperature propagate both slower than light and are damped over time (this stability property is intimately intertwined with relativistic causality[3]).
It can be shown that this definition of the heat flux vector also satisfies the second law of thermodynamics,[6]
where s is specific
entropy and σ is
entropy production. This mathematical model is inconsistent with special relativity: the
Green function associated to the heat equation (also known as
heat kernel) has support that extends outside the
light-cone, leading to faster-than-light propagation of information. For example, consider a pulse of heat at the origin; then according to Fourier equation, it is felt (i.e. temperature changes) at any distant point, instantaneously. The speed of propagation of heat is faster than the
speed of light in vacuum, which is inadmissible within the framework of relativity.
Hyperbolic model (relativistic)
The parabolic model for heat conduction discussed above shows that the Fourier equation (and the more general
Fick's law of diffusion) is incompatible with the theory of relativity[7] for at least one reason: it admits infinite speed of propagation of the continuum
field (in this case: heat, or temperature gradients). To overcome this contradiction, workers such as
Carlo Cattaneo,[2] Vernotte,[8] Chester,[9] and others[10] proposed that Fourier equation should be upgraded from the
parabolic to a
hyperbolic form, where the n, the temperature field is governed by:
For the HHC equation to remain compatible with the first law of thermodynamics, it is necessary to modify the definition of heat flux vector, q, to
where is a
relaxation time, such that This equation for the heat flux is often referred to as "Maxwell-Cattaneo equation". The most important implication of the hyperbolic equation is that by switching from a parabolic (
dissipative) to a hyperbolic (includes a
conservative term)
partial differential equation, there is the possibility of phenomena such as thermal
resonance[12][13][14] and thermal
shock waves.[15]
^
abCattaneo, C. R. (1958). "Sur une forme de l'équation de la chaleur éliminant le paradoxe d'une propagation instantanée". Comptes Rendus. 247 (4): 431.
^Barletta, A.; Zanchini, E. (1997). "Hyperbolic heat conduction and local equilibrium: a second law analysis". International Journal of Heat and Mass Transfer. 40 (5): 1007–1016.
doi:
10.1016/0017-9310(96)00211-6.
^Eckert, E. R. G.; Drake, R. M. (1972). Analysis of Heat and Mass Transfer. Tokyo: McGraw-Hill, Kogakusha.
^Vernotte, P. (1958). "Les paradoxes de la theorie continue de l'équation de la chaleur". Comptes Rendus. 246 (22): 3154.
^Mandrusiak, G. D. (1997). "Analysis of non-Fourier conduction waves from a reciprocating heat source". Journal of Thermophysics and Heat Transfer. 11 (1): 82–89.
doi:
10.2514/2.6204.
^Xu, M.; Wang, L. (2002). "Thermal oscillation and resonance in dual-phase-lagging heat conduction". International Journal of Heat and Mass Transfer. 45 (5): 1055–1061.
doi:
10.1016/S0017-9310(01)00199-5.
^Barletta, A.; Zanchini, E. (1996). "Hyperbolic heat conduction and thermal resonances in a cylindrical solid carrying a steady periodic electric field". International Journal of Heat and Mass Transfer. 39 (6): 1307–1315.
doi:
10.1016/0017-9310(95)00202-2.
^Tzou, D. Y. (1989). "Shock wave formation around a moving heat source in a solid with finite speed of heat propagation". International Journal of Heat and Mass Transfer. 32 (10): 1979–1987.
doi:
10.1016/0017-9310(89)90166-X.