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Concept
Relativistic heat conduction
Summary
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. 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).
Heat conduction in a Newtonian context is modelled by the Fourier equation, namely a parabolic partial differential equation of the kind:
where θ is temperature, 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,, is defined in Cartesian coordinates as
This Fourier equation can be derived by substituting Fourier’s linear approximation of the heat flux vector, q, as a function of temperature gradient,
into the first law of thermodynamics
where the del operator, ∇, is defined in 3D as
It can be shown that this definition of the heat flux vector also satisfies the second law of thermodynamics,
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.
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Related concepts (6)
Heat equation
In mathematics and physics, the heat equation is a certain partial differential equation. Solutions of the heat equation are sometimes known as caloric functions. The theory of the heat equation was first developed by Joseph Fourier in 1822 for the purpose of modeling how a quantity such as heat diffuses through a given region. As the prototypical parabolic partial differential equation, the heat equation is among the most widely studied topics in pure mathematics, and its analysis is regarded as fundamental to the broader field of partial differential equations.
Thermal conduction
Conduction is the process by which heat is transferred from the hotter end to the colder end of an object. The ability of the object to conduct heat is known as its thermal conductivity, and is denoted k. Heat spontaneously flows along a temperature gradient (i.e. from a hotter body to a colder body). For example, heat is conducted from the hotplate of an electric stove to the bottom of a saucepan in contact with it.
Diffusion
Diffusion is the net movement of anything (for example, atoms, ions, molecules, energy) generally from a region of higher concentration to a region of lower concentration. Diffusion is driven by a gradient in Gibbs free energy or chemical potential. It is possible to diffuse "uphill" from a region of lower concentration to a region of higher concentration, like in spinodal decomposition. Diffusion is a stochastic process due to the inherent randomness of the diffusing entity and can be used to model many real-life stochastic scenarios.