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.

Ali and Zhang claim their model of relativistic heat conduction is the only one compatible with the theory of special relativity, the second law of thermodynamics, electrodynamics, and quantum mechanics, simultaneously. The main features of their model are:

1. It admits a finite speed of heat propagation, and allows for relativistic effects when heat flux transients approach that speed.
2. It removes the possibility of paradoxical situations that may violate the second law of thermodynamics.
3. It, implicitly, admits the wave–particle duality of the heat-carrying “phonon”.

These outcomes are achieved by (1) upgrading the Fourier equation of heat conduction to the form of a Telegraph equation of electrodynamics, and (2) introducing a new definition of the heat flux vector. Consequently, their model gives rise to a number of phenomena, such as thermal resonance and thermal shock waves, which are possible during high-frequency pulsed laser heating of thermal insulators.

Background

Classical model

For most of the last two centuries, heat conduction has been modelled by the well-known Fourier equation:

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

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,$\scriptstyle \nabla ^{2}$, is defined in Cartesian coordinates as

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

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

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

into the first law of thermodynamics

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

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

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

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

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

where s is specific entropy and σ is entropy production. Alternatively, the second law can be written as

$\sigma ~=~{\frac {-1}{\theta ^{2}}}~\mathbf {q} \cdot \nabla \theta ,$

which leads to the condition

$\sigma ~=~{\frac {k}{\theta ^{2}}}~\left[{\left({\frac {\partial \theta }{\partial x}}\right)}^{2}~+~{\left({\frac {\partial \theta }{\partial y}}\right)}^{2}~+~{\left({\frac {\partial \theta }{\partial z}}\right)}^{2}\right],$

which is always true, because k is a non-negative material property.

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 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, Vernotte, Chester, and others proposed that Fourier equation should be upgraded from the parabolic to a hyperbolic form,

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

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 (HHC) 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

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

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

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 and thermal shock waves.

Criticism to the HHC model

• The relaxation time, $\scriptstyle \tau _{_{0}}$, is justified based on microscopic aspects of lattice vibration and electron transport; is an extension of kinetic theory calculations and Boltzmann equation for rarefied gases to the case of solids; and is calculated from statistical Newtonian mechanics. Further, the speed C is only a collection of terms, α and $\scriptstyle \tau _{_{0}}$, and has no physical reality or significance similar to that associated with the speed of light. Hence, the hyperbolic equation is compatible with relativity artificially (in form only), but is still fundamentally classical Newtonian.
• The new definition of heat flux vector is an ad hoc mathematical approximation of a far more complicated expression; this raises some doubts about the whole approach.
• The most serious criticism is that the hyperbolic equation can violate the second law of thermodynamics. For example, consider an infinitely long wire conductor, with a heat source at the origin, and measure temperature at distances significantly remote from origin. If the heat source at origin varies with a frequency much higher than the relaxation time (i.e. faster than the speed of second sound) then the hyperbolic equation admits a temperature field in which heat would appear to be moving from cold to hot, in violation of the second law. This contradiction was demonstrated in more mathematically rigorous form.

Ali and Zhang's theory of RHC attempts to resolve the controversies surrounding the hyperbolic equation, while maintaining the form of that equation. This is achieved by:

• Deriving the hyperbolic equation starting from space–time duality of a Minkowski space, and simple Lorentz transformations, that are basic to the theory of special relativity. This is done without any reference to microstructure or statistical mechanics.
• Treating the speed of second sound, C, as a fundamental property of the temperature field, although still fundamentally inferior to the speed of light.
• Modifying the definition of the heat flux vector so that it is simpler, more elegant, and bring it in compliance with the second law of thermodynamics.

Derivation of the RHC equation

Transformations

In a Euclidean space, distance between any two points, ds, is measured by

$ds^{2}~=~dx^{2}~+~dy^{2}~+~dz^{2},$

where dx, dy, and dz are displacements along three orthogonal axes.

In a Minkowski space, distance between two events, ds, is measured by

$ds^{2}~=~d\tau ^{2}~+~dx^{2}~+~dy^{2}~+~dz^{2},$

where, τ, is space-like-time and is related to real time, t, by

$\tau ~=~i~C~t,$

where C is speed of light in vacuum and $\scriptstyle i~=~{\sqrt {-1}}$. Hence,

$ds^{2}~=~dx^{2}~+~dy^{2}~+~dz^{2}~-~C^{2}~dt^{2}.$

Consequently, the 3D del, ∇,operator is upgraded to the 4D quad, $\scriptstyle \square$, operator (also known as the Four-gradient)

$\square ~=~{\frac {\partial }{\partial \tau }}~\mathbf {o} ~+~{\frac {\partial }{\partial x}}~\mathbf {i} ~+~{\frac {\partial }{\partial y}}~\mathbf {j} ~+~{\frac {\partial }{\partial z}}~\mathbf {k} ~=~{\frac {\partial }{\partial \tau }}~\mathbf {o} ~+~\nabla ~=~{\frac {-i}{C}}~{\frac {\partial }{\partial t}}~\mathbf {o} ~+~\nabla .$

Likewise, the 3D Laplacian, $\scriptstyle \nabla ^{2}$, operator is upgraded to the 4D d'Alembert operator, $\scriptstyle \square ^{2},$

$\square ^{2}~=~{\frac {\partial ^{2}}{\partial \tau ^{2}}}~+{\frac {\partial ^{2}}{\partial x^{2}}}~+~{\frac {\partial ^{2}}{\partial y^{2}}}~+~{\frac {\partial ^{2}}{\partial z^{2}}}~=~{\frac {\partial ^{2}}{\partial \tau ^{2}}}~+~\nabla ^{2}~=~{\frac {-1}{C^{2}}}~{\frac {\partial ^{2}}{\partial t^{2}}}~+~\nabla ^{2}.$

Any physical quantity that is Galilean invariant in Euclidean space can be made Lorentz invariant in a Minkowski space, by upgrading from 3D to 4D operators. Consequently, Fourier’s equation can be upgraded to 4D as

${\frac {\partial \theta }{\partial t}}~=~\alpha ~\square ^{2}\theta ~=~{\frac {-\alpha }{C^{2}}}~{\frac {\partial ^{2}\theta }{\partial t^{2}}}~+~\alpha ~\nabla ^{2}\theta ,$

which is called the relativistic heat conduction equation. Likewise, the definition of the heat-flux vector, q, is upgraded to the 4D form as

$\mathbf {q} =-k\,\square \,\theta ~=-k\,\nabla \theta ~+~{\frac {ik}{C}}~{\frac {\partial \theta }{\partial t}}~\mathbf {o} .$

Implications

It can be shown that this definition of q is compatible with the first law of thermodynamics,

$\rho ~c~{\frac {\partial \theta }{\partial t}}~+~\square \cdot \mathbf {q} ~=~0~=~\rho ~c~{\frac {\partial \theta }{\partial t}}~+~\nabla \cdot \mathbf {q} ~+~{\frac {-i}{C}}~{\frac {\partial \mathbf {q} }{\partial t}}\cdot \mathbf {o} ,$

as well as the second law of thermodynamics,

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

in their 4D upgraded form. The imaginary terms in these equations are direct manifestation of the wave nature of heat, and are essential for the heat equation to become compatible with all laws of physics. The real terms in these equations are identical to those in the classical heat model.

It has been observed that RHC reduces the second law of thermodynamics to a statement of the form

${\left({\frac {dt}{dx}}\right)}^{2}~+~{\left({\frac {dt}{dy}}\right)}^{2}~+~{\left({\frac {dt}{dz}}\right)}^{2}~\geqslant ~{\frac {1}{C^{2}}},$

which is the “no action at a distance” principle of special relativity. Essentially, the RHC asserts that relativity and the second law of thermodynamics are two alternative, but equal statements about the nature of time. Both physical principles are mutually derivable from each other and are complementary.

Criticism to RHC

As far as heat conduction is concerned, the RHC equation is identical in form to the hyperbolic equation, and all analytical and experimental results that are relevant to one are equally applicable to the other. The definition of heat flux vector, however, is different; but the RHC definition is merely a 4D upgrade of the original linear Fourier approximation. The mathematics of RHC is much simpler and more elegant. However, RHC raises some significant conceptual challenges:

1. This weak interpretation of relativity, in which the speed of second sound plays a role similar to that of the speed of light, can be viewed as downgrading or degrading to the universality of the theory of relativity. Notice how the symbol c in standard relativity theory is replaced with C without much interpretation.
2. The implied wave nature of heat is controversial. Some workers reject the wave nature of heat on dogmatic grounds. Moreover, RHC implies that a phonon is a full-fledged objective quantum particle whose physical reality is no lesser than that of a photon. Existing experimental evidences are not enough to support for or against such views.
3. Heat quantities become complex numbers, with values including "imaginary temperature", which are cannot be interpreted experimentally.
4. The equivalence of relativity and the second law is shocking, because it implies that one of them can be a derivative of the other.

In summary, while the RHC is mathematically simple and elegant, and experimentally practical and relevant, it raises a number of conceptual issues that are highly controversial.

Applications

The RHC theory is applicable for any physical problem in which the hyperbolic equation is relevant: when speed of heat propagation is small, e.g. thermal insulators, or when speed of heat-flux variation is very large, e.g. pulsed-laser heating. Applications for those types of problems are abundant, and there is plenty of published work (see Notes, below). Most of these results remain relevant to RHC, but because the definition of heat flux vector is different, final closed-form solutions may not be the same. In many cases, RHC provides closed-form solutions that are not possible using the HHC model. A number of useful fundamental solutions for 1D and 2D relativistic moving heat sources are available in closed-form.

References

• Y.M. Ali, L.C. Zhang, Relativistic heat conduction, Int. J. Heat Mass Trans. 48 (2005) 2397.
• Y.M. Ali, L.C. Zhang, Relativistic moving heat source, Int. J. Heat Mass Trans. 48 (2005) 2741.