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| The theory of '''relativistic heat conduction (RHC)'''<ref name="Ali Zhang 2005a">{{Cite journal |first=Y. M. |last=Ali |first2=L. C. |last2=Zhang |title=Relativistic heat conduction |journal=Int. J. Heat Mass Trans. |volume=48 |year=2005 |issue=12 |pages=2397 |doi=10.1016/j.ijheatmasstransfer.2005.02.003 }}</ref> claims to be the only model for [[heat conduction]] (and similar [[diffusion]] processes) that is compatible with the theory of [[special relativity]], the [[second law of thermodynamics]], [[electrodynamics]], and [[quantum mechanics]], simultaneously. The main features of RHC are:
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| # It admits a [[Wikt:finite|finite]] [[speed]] of [[heat]] [[wave propagation|propagation]], and allows for relativistic effects when [[heat flux]] [[Transient state|transients]] approach that speed.
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| # It removes the possibility of [[paradoxical]] situations that may violate the second law of thermodynamics.
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| # It, implicitly, admits the [[wave–particle duality]] of the heat-carrying “[[phonon]]”.
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| These outcomes are achieved by (1) upgrading the [[heat conduction|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, RHC gives rise to a number of interesting phenomena, such as thermal [[resonance]] and thermal [[shock waves]], which are possible during high-[[frequency]] pulsed [[laser]] heating of thermal [[Thermal insulation|insulators]]. The main appealing feature of the theory is its mathematical elegance and simplicity.
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| ==Background==
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| ===Classical model===
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| For most of the last two centuries, heat conduction has been modelled by the well-known [[heat equation|Fourier equation]]:<ref>{{Cite book |first=H. S. |last=Carslaw |first2=J. C. |last2=Jaeger |title=Conduction of Heat in Solids |edition=Second |publisher=University Press |location=Oxford |year=1959 |isbn= }}</ref>
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| :<math>\frac{\partial\theta}{\partial t}~=~\alpha~\nabla^2\theta ,</math>
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| where ''θ'' is [[temperature]],<ref>Some authors also use ''T'', φ,...</ref> ''t'' is [[time in physics|time]], ''α'' = ''k''/(''ρ'' ''c'') is [[thermal diffusivity]], ''k'' is [[thermal conductivity]], ''ρ'' is [[density]], and ''c'' is [[specific heat capacity]]. The [[Laplace operator]],<math>\scriptstyle\nabla^2</math>, is defined in [[Cartesian coordinate system|Cartesian coordinates]] as
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| :<math>\nabla^2~=~\frac{\partial^2}{\partial x^2}~+~\frac{\partial^2}{\partial y^2}~+~\frac{\partial^2}{\partial z^2} .</math>
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| This Fourier equation can be derived by substituting Fourier’s linear approximation of the heat flux vector, '''''q''''', as a function of temperature gradient,
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| :<math>\mathbf{q}~=~-k~\nabla\theta ,</math>
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| into the [[first law of thermodynamics]]
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| :<math>\rho~c~\frac{\partial \theta}{\partial t}~+ ~\nabla \cdot \mathbf{q}~=~ 0 ,</math>
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| where the [[del]] operator, ∇, is defined in 3D as
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| :<math>\nabla ~=~\frac{\partial}{\partial x}~\mathbf{i}~+~\frac{\partial}{\partial y}~\mathbf{j}~+~\frac{\partial}{\partial z}~\mathbf{k} .</math>
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| It can be shown that this definition of the heat flux vector also satisfies the second law of thermodynamics,<ref>{{Cite journal |first=A. |last=Barletta |first2=E. |last2=Zanchini |title=Hyperbolic heat conduction and local equilibrium: a second law analysis |journal=Int. J. Heat Mass Trans. |volume=40 |issue=5 |year=1997 |pages=1007–1016 |doi=10.1016/0017-9310(96)00211-6 }}</ref>
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| :<math>\nabla\cdot\left(\frac{\mathbf{q}}{\theta}\right)~+~\rho~\frac{\partial s}{\partial t}~=~\sigma,</math>
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| where ''s'' is specific [[entropy]] and ''σ'' is entropy production. Alternatively, the second law can be written as
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| :<math>\sigma~=~\frac{-1}{\theta^2}~\mathbf{q}\cdot\nabla\theta ,</math>
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| which leads to the condition<ref name="Ali Zhang 2005a" />
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| :<math>\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] ,</math>
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| which is always true, because ''k'' is a non-negative material property.
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| ===Hyperbolic model===
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| For most of the last century, it was recognized that Fourier equation (and its more general [[Fick's law of diffusion]]) is in contradiction with the theory of relativity,<ref>{{Cite book |first=E. R. G. |last=Eckert |first2=R. M. |last2=Drake |title=Analysis of Heat and Mass Transfer |publisher=McGraw-Hill, Kogakusha |location=Tokyo |year=1972 |isbn= }}</ref> for ''at least'' one reason: it admits infinite speed of propagation of heat signals within the continuum [[field (physics)|field]]. For example, consider a pulse of heat at the origin; then according to Fourier equation, it is felt (i.e. temperature changes) at [[infinity]], instantaneously. The speed of information propagation is faster than the [[speed of light]] in vacuum, which is physically inadmissible within the framework of relativity.
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| To overcome this contradiction, workers such as Cattaneo,<ref>{{Cite journal |first=C. R. |last=Cattaneo |title={{lang|fr|Sur une forme de l'équation de la chaleur éliminant le paradoxe d'une propagation instantanée}} |journal=Comptes Rendus |volume=247 |issue=4 |year=1958 |pages=431 |doi= }}</ref> Vernotte,<ref>{{Cite journal |first=P. |last=Vernotte |title=Les paradoxes de la theorie continue de l'équation de la chaleur |journal=Comptes Rendus |volume=246 |issue=22 |year=1958 |pages=3154 |doi= }}</ref> Chester,<ref>{{Cite journal |first=M. |last=Chester |title=Second sound in solids |journal=[[Physical Review|Phys. Rev.]] |volume=131 |issue=15 |year=1963 |pages=2013–2015 |doi=10.1103/PhysRev.131.2013 |bibcode = 1963PhRv..131.2013C }}</ref> and others<ref>{{Cite book |first=P. M. |last=Morse |first2=H. |last2=Feshbach |title=Methods of Theoretical Physics |publisher=McGraw-Hill |location=New York |year=1953 }}</ref> proposed that Fourier equation should be upgraded from the [[parabolic partial differential equation|parabolic]] to a [[hyperbolic partial differential equation|hyperbolic]] form,
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| :<math>\frac{1}{C^2}~\frac{\partial^2\theta}{\partial t^2}~+~\frac{1}{\alpha}~\frac{\partial\theta}{\partial t}~=~\nabla^2\theta ,</math>
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| also known as the [[Telegrapher's equation]]. Interestingly, the form of this equation traces its origins to [[Maxwell’s equations]] of electrodynamics; hence, the wave nature of heat is implied. In this equation, ''C'' is called the speed of [[second sound]] (i.e. the fictitious quantum particles, phonons) and this equation is known as the [[Hyperbolic Heat Conduction]] (HHC) equation.
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| 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
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| :<math>\tau_{_0}~\frac{\partial\mathbf{q}}{\partial t}~+~\mathbf{q}~=~-k~\nabla\theta,</math>
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| where <math>\scriptstyle\tau_{_0}</math> is a [[relaxation time]], such that <math>\scriptstyle C^2~=~ \alpha/ \tau_{_0} .</math>
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| The most important implication of the hyperbolic equation is that by switching from a parabolic ([[dissipation|dissipative]]) to a hyperbolic (includes a [[conservation law|conservative]] term) [[partial differential equation]], there is the possibility of phenomena such as thermal [[resonance]]<ref>{{Cite journal |first=G. D. |last=Mandrusiak |title=Analysis of non-Fourier conduction waves from a reciprocating heat source |journal=J. Thermophys. Heat Trans. |volume=11 |issue=1 |year=1997 |pages=82 |doi= }}</ref><ref>{{Cite journal |first=M. |last=Xu |first2=L. |last2=Wang |title=Thermal oscillation and resonance in dual-phase-lagging heat conduction |journal=Int. J. Heat Mass Trans. |volume=45 |issue=5 |year=2002 |pages=1055 |doi=10.1016/S0017-9310(01)00199-5 }}</ref><ref>{{Cite journal |first=A. |last=Barletta |first2=E. |last2=Zanchini |title=Hyperbolic heat conduction and thermal resonances in a cylindrical solid carrying a steady periodic electric field |journal=Int. J. Heat Mass Trans. |volume=39 |issue=6 |year=1996 |pages=1307 |doi=10.1016/0017-9310(95)00202-2 }}</ref> and thermal [[shock waves]].<ref>{{Cite journal |first=D. Y. |last=Tzou |title=Shock wave formation around a moving heat source in a solid with finite speed of heat propagation |journal=Int. J. Heat Mass Trans. |volume=32 |issue=10 |year=1989 |pages=1979 |doi=10.1016/0017-9310(89)90166-X }}</ref>
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| ===Criticism to the HHC model===
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| * The relaxation time, <math>\scriptstyle\tau_{_0}</math>, is justified based on [[microscopic]] aspects of [[phonon|lattice vibration]] and electron [[transport phenomena|transport]]; is an extension of [[kinetic theory]] calculations and [[Boltzmann equation]] for rarefied gases to the case of solids;<ref>{{Cite journal |first=H. |last=Grad |title=On the kinetic theory of rarefied gases |journal=[[Communications on Pure and Applied Mathematics|Commun. Pure Appl. Math.]] |volume=2 |issue=4 |year=1949 |pages=331–407 |doi=10.1002/cpa.3160020403 }}</ref> and is calculated from [[statistical mechanics|statistical]] [[classical mechanics|Newtonian]] mechanics.<ref>{{Cite journal |first=A. H. |last=Ali |title=Statistical mechanical derivation of Cattaneo's heat flux law |journal=J. Thermophys. Heat Trans. |volume=13 |issue=4 |year=1999 |pages=544–546 |doi= }}</ref> Further, the speed ''C'' is only a collection of terms, α and <math>\scriptstyle\tau_{_0}</math>, 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.
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| * 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.
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| * 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.<ref name="Ali Zhang 2005a" /><ref>{{Cite journal |first=C. |last=Koerner |first2=H. W. |last2=Bergmann |title=Physical defects of the hyperbolic heat conduction equation |journal=Appl. Phys. A: Mater. Sci. Process. |volume=67 |issue=4 |year=1998 |pages=397–401 |doi=10.1007/s003390050792 |bibcode = 1998ApPhA..67..397K }}</ref><ref>{{Cite journal |first=C. |last=Bai |first2=A. S. |last2=Lavine |title=On hyperbolic heat conduction and the second law of thermodynamics |journal=J. Heat Trans., Trans. ASME |volume=117 |issue=2 |year=1995 |pages=256 |doi= }}</ref><ref>{{Cite journal |first=M. B. |last=Rubin |title=Hyperbolic heat conduction and the second law |journal=Int. J. Eng. Sci. |volume=30 |issue=11 |year=1992 |pages=1665–1676 |doi=10.1016/0020-7225(92)90134-3 }}</ref>
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| The theory of RHC attempts to resolve the controversies surrounding the hyperbolic equation, while maintaining the form of that equation. This is achieved by:
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| * Deriving the hyperbolic equation starting from [[space–time]] duality of a [[Minkowski space]], and simple [[Lorentz transformation]]s, that are basic to the theory of special relativity. This is done without any reference to [[microstructure]] or [[statistical mechanics]].
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| * Treating the speed of second sound, ''C'', as a fundamental property of the temperature field, although still fundamentally inferior to the speed of light.
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| * 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.
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| ==Derivation of the RHC equation==
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| ===Transformations===
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| In a [[Euclidean space]], distance between any two points, ''ds'', is measured by
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| :<math>ds^2~=~dx^2~+~dy^2~+~dz^2 ,</math>
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| where ''dx'', ''dy'', and ''dz'' are displacements along three [[orthogonal]] axes.
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| In a [[Minkowski space]], distance between two events, ''ds'', is measured by
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| :<math> ds^2~=~d\tau^2~+~dx^2~+~dy^2~+~dz^2 ,</math>
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| where, ''τ'', is space-like-time and is related to real time, ''t'', by
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| :<math>\tau~=~i~C~t ,</math>
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| where ''C'' is speed of light in vacuum and <math>\scriptstyle i~=~ \sqrt{-1}</math>. Hence,
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| :<math> ds^2~=~dx^2~+~dy^2~+~dz^2~-~C^2~dt^2 .</math>
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| Consequently, the 3D del, ∇,operator is upgraded to the 4D quad, <math>\scriptstyle\square</math>, operator (also known as the [[Four-gradient]])
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| :<math>\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 .</math>
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| Likewise, the 3D Laplacian, <math>\scriptstyle\nabla^2</math>, operator is upgraded to the 4D [[d'Alembert operator]],<ref>Some authors use <math>\scriptstyle\square</math> for the d'Alembert operator, but to maintain the analogy with the Laplacian operator, using <math>\scriptstyle\square^2</math> is more consistent.</ref> <math>\scriptstyle\square^2 ,</math>
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| :<math>\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 .</math>
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| 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
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| :<math>\frac{\partial\theta}{\partial t}~=~\alpha~\square^2\theta~=~\frac{-\alpha}{C^2}~\frac{\partial^2\theta}{\partial t^2}~+~\alpha~\nabla^2\theta ,</math>
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| which is called the relativistic heat conduction equation. Likewise, the definition of the heat-flux vector, '''''q''''', is upgraded to the 4D form as
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| :<math>\mathbf{q}~=~-k~\square\theta~=~-k~\nabla\theta~+~\frac{i~k}{C}~\frac{\partial\theta}{\partial t}~\mathbf{o} .</math>
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| ===Implications===
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| It can be shown that this definition of '''''q''''' is compatible with the first law of thermodynamics,
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| :<math>\rho~c~\frac{\partial \theta}{\partial t}~+ ~\square \cdot \mathbf{q}~=~0~=~\rho~c~\frac{\partial \theta}{\partial t}~+~
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| \nabla \cdot \mathbf{q}~+~\frac{-i}{C}~\frac{\partial \mathbf{q}}{\partial t}\cdot\mathbf{o} ,</math>
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| as well as the second law of thermodynamics,
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| :<math>\square\cdot\left(\frac{\mathbf{q}}{\theta}\right)~+~\rho~\frac{\partial s}{\partial t}~=~\sigma ,</math>
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| in their 4D upgraded form.<ref name="Ali Zhang 2005a" /> 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.
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| The most interesting observation about RHC is that it reduces the second law of thermodynamics to a statement of the form
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| :<math>{\left(\frac{dt}{dx}\right)}^2~+~{\left(\frac{dt}{dy}\right)}^2~+~{\left(\frac{dt}{dz}\right)}^2~\geqslant~\frac{1}{C^2} ,</math>
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| 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.<ref name="Ali Zhang 2005a" />
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| ==Criticism to RHC==
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| 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:
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| # 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.
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| # 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.
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| # Heat quantities become complex numbers, with values including "imaginary temperature", which are hard to interpret experimentally.
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| # The equivalence of relativity and the second law is shocking, because it implies that one of them can be a derivative of the other.
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| 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.
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| ==Applications==
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| 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.<ref>{{Cite journal |first=Y. M. |last=Ali |first2=L. C. |last2=Zhang |title=Relativistic moving heat source |journal=Int. J. Heat Mass Trans. |volume=48 |issue= |year=2005 |pages=2741 |doi= }}</ref>
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| ==Notes==
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| {{Reflist|colwidth=30em}}
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| ==References==
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| * Y.M. Ali, L.C. Zhang, Relativistic heat conduction, Int. J. Heat Mass Trans. 48 (2005) 2397.
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| * Y.M. Ali, L.C. Zhang, Relativistic moving heat source, Int. J. Heat Mass Trans. 48 (2005) 2741.
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| ==External links==
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| * [http://www.sciencedirect.com/science?_ob=ArticleURL&_udi=B6V3H-4FRHDBN-3&_user=10&_rdoc=1&_fmt=&_orig=search&_sort=d&view=c&_acct=C000050221&_version=1&_urlVersion=0&_userid=10&md5=2e0b4d59934ae15c868a330faaa21b16 Original theory paper]
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| * [http://www.sciencedirect.com/science?_ob=ArticleURL&_udi=B6V3H-4FS23H2-4&_user=10&_rdoc=1&_fmt=&_orig=search&_sort=d&view=c&_acct=C000050221&_version=1&_urlVersion=0&_userid=10&md5=6b00cb524e176d7c1bc11e35b212fabb Applications paper]
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| [[Category:Heat conduction]]
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| [[Category:Thermodynamics]]
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| [[Category:Special relativity]]
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| [[Category:Concepts in physics]]
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| [[Category:Hyperbolic partial differential equations]]
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| [[Category:Diffusion]]
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| [[Category:Transport phenomena]]
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