Relativistic heat conduction

Relativistic heat conduction refers to the modelling of heat conduction (and similar diffusion processes) in a way compatible with special relativity. This article discusses models using a wave equation with a dissipative term.

Heat conduction in a Newtonian context is modelled by the Fourier equation:[1]

${\displaystyle {\frac {\partial \theta }{\partial t}}~=~\alpha ~\nabla ^{2}\theta ,}$

where θ is temperature,[2] t is time, α = k/(ρ c) is thermal diffusivity, k is thermal conductivity, ρ is density, and c is specific heat capacity. The Laplace operator,${\displaystyle \scriptstyle \nabla ^{2}}$, is defined in Cartesian coordinates as

${\displaystyle \nabla ^{2}~=~{\frac {\partial ^{2}}{\partial x^{2}}}~+~{\frac {\partial ^{2}}{\partial y^{2}}}~+~{\frac {\partial ^{2}}{\partial z^{2}}}.}$

This Fourier equation can be derived by substituting Fourier’s linear approximation of the heat flux vector, q, as a function of temperature gradient,

${\displaystyle \mathbf {q} ~=~-k~\nabla \theta ,}$

into the first law of thermodynamics

${\displaystyle \rho ~c~{\frac {\partial \theta }{\partial t}}~+~\nabla \cdot \mathbf {q} ~=~0,}$

where the del operator, ∇, is defined in 3D as

${\displaystyle \nabla ~=~{\frac {\partial }{\partial x}}~\mathbf {i} ~+~{\frac {\partial }{\partial y}}~\mathbf {j} ~+~{\frac {\partial }{\partial z}}~\mathbf {k} .}$

It can be shown that this definition of the heat flux vector also satisfies the second law of thermodynamics,[3]

${\displaystyle \nabla \cdot \left({\frac {\mathbf {q} }{\theta }}\right)~+~\rho ~{\frac {\partial s}{\partial t}}~=~\sigma ,}$

where s is specific entropy and σ is entropy production.

Hyperbolic model

It is well known that the Fourier equation (and the more general Fick's law of diffusion) is incompatible with the theory of relativity[4] for at least one reason: it admits infinite speed of propagation of heat signals within the continuum field. 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 information propagation is faster than the speed of light in vacuum, which is inadmissible within the framework of relativity.

To overcome this contradiction, workers such as Cattaneo,[5] Vernotte,[6] Chester,[7] and others[8] proposed that Fourier equation should be upgraded from the parabolic to a hyperbolic form,

${\displaystyle {\frac {1}{C^{2}}}~{\frac {\partial ^{2}\theta }{\partial t^{2}}}~+~{\frac {1}{\alpha }}~{\frac {\partial \theta }{\partial t}}~=~\nabla ^{2}\theta }$.

In this equation, C is called the speed of second sound (i.e. the fictitious quantum particles, phonons). The equation is known as the hyperbolic heat conduction (HCC) equation. Mathematically, it is the same as the telegrapher's equation, which is derived from Maxwell’s equations of electrodynamics.

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

${\displaystyle \tau _{_{0}}~{\frac {\partial \mathbf {q} }{\partial t}}~+~\mathbf {q} ~=~-k~\nabla \theta ,}$

where ${\displaystyle \scriptstyle \tau _{_{0}}}$ is a relaxation time, such that ${\displaystyle \scriptstyle C^{2}~=~\alpha /\tau _{_{0}}.}$

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[9][10][11] and thermal shock waves.[12]

Notes

1. Carslaw, H. S.; Jaeger, J. C. (1959). Conduction of Heat in Solids (Second ed.). Oxford: University Press.
2. Some authors also use T, φ,...
3. 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.
4. Eckert, E. R. G.; Drake, R. M. (1972). Analysis of Heat and Mass Transfer. Tokyo: McGraw-Hill, Kogakusha.
5. Cattaneo, C. R. (1958). "Sur une forme de l'équation de la chaleur éliminant le paradoxe d'une propagation instantanée". Comptes Rendus. 247 (4): 431.
6. Vernotte, P. (1958). "Les paradoxes de la theorie continue de l'équation de la chaleur". Comptes Rendus. 246 (22): 3154.
7. Chester, M. (1963). "Second sound in solids". Physical Review. 131 (15): 2013–2015. Bibcode:1963PhRv..131.2013C. doi:10.1103/PhysRev.131.2013.
8. Morse, P. M.; Feshbach, H. (1953). Methods of Theoretical Physics. New York: McGraw-Hill.
9. 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.
10. 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.
11. 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.
12. 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.

References