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.