1663:
959:
646:
379:
46:
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
771:, 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. The speed of propagation of heat is faster than the
741:
1101:
826:
532:
240:
523:
141:
653:
1007:
436:
59:, 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).
1201:
1203:
This equation for the heat flux is often referred to as "Maxwell-Cattaneo equation". The most important implication of the hyperbolic equation is that by switching from a parabolic (
444:
1743:
78:
1134:
231:
821:
954:{\displaystyle {\frac {1}{C^{2}}}~{\frac {\partial ^{2}\theta }{\partial t^{2}}}~+~{\frac {1}{\alpha }}~{\frac {\partial \theta }{\partial t}}~=~\nabla ^{2}\theta .}
641:{\displaystyle \nabla ~=~\mathbf {i} ~{\frac {\partial }{\partial x}}~+~\mathbf {j} ~{\frac {\partial }{\partial y}}~+~\mathbf {k} ~{\frac {\partial }{\partial z}}.}
1704:
374:{\displaystyle \nabla ^{2}~=~{\frac {\partial ^{2}}{\partial x^{2}}}~+~{\frac {\partial ^{2}}{\partial y^{2}}}~+~{\frac {\partial ^{2}}{\partial z^{2}}}.}
1607:
Barletta, A.; Zanchini, E. (1996). "Hyperbolic heat conduction and thermal resonances in a cylindrical solid carrying a steady periodic electric field".
984:
800:
796:
72:
998:
For the HHC equation to remain compatible with the first law of thermodynamics, it is necessary to modify the definition of heat flux vector,
987:
heat conduction" (HHC) equation. Mathematically, the above equation is called "telegraph equation", as it is formally equivalent to the
392:
1697:
787:) is incompatible with the theory of relativity for at least one reason: it admits infinite speed of propagation of the continuum
792:
736:{\displaystyle \nabla \cdot \left({\frac {\mathbf {q} }{\theta }}\right)~+~\rho ~{\frac {\partial s}{\partial t}}~=~\sigma ,}
1758:
1212:
1634:
Tzou, D. Y. (1989). "Shock wave formation around a moving heat source in a solid with finite speed of heat propagation".
1289:
Cattaneo, C. R. (1958). "Sur une forme de l'équation de la chaleur éliminant le paradoxe d'une propagation instantanée".
1753:
1738:
1690:
1733:
439:
234:
784:
988:
1723:
1143:
1405:
Barletta, A.; Zanchini, E. (1997). "Hyperbolic heat conduction and local equilibrium: a second law analysis".
992:
783:
The parabolic model for heat conduction discussed above shows that the
Fourier equation (and the more general
650:
It can be shown that this definition of the heat flux vector also satisfies the second law of thermodynamics,
383:
This
Fourier equation can be derived by substituting Fourier’s linear approximation of the heat flux vector,
1728:
1096:{\displaystyle \tau _{_{0}}~{\frac {\partial \mathbf {q} }{\partial t}}~+~\mathbf {q} ~=~-k~\nabla \theta ,}
199:
1553:
Mandrusiak, G. D. (1997). "Analysis of non-Fourier conduction waves from a reciprocating heat source".
1527:
1477:
1331:
1252:
183:
1580:
Xu, M.; Wang, L. (2002). "Thermal oscillation and resonance in dual-phase-lagging heat conduction".
1662:
760:
175:
48:
1748:
1355:
1321:
756:
56:
39:
35:
31:
791:(in this case: heat, or temperature gradients). To overcome this contradiction, workers such as
518:{\displaystyle \rho ~c~{\frac {\partial \theta }{\partial t}}~+~\nabla \cdot \mathbf {q} ~=~0,}
1347:
1268:
1106:
1674:
806:
209:
55:
associated to its cause. Any reasonable relativistic model for heat conduction must also be
1643:
1616:
1589:
1562:
1535:
1485:
1414:
1339:
1260:
1208:
203:
795:, Vernotte, Chester, and others proposed that Fourier equation should be upgraded from the
1468:
1309:
1137:
788:
156:
23:
1447:
Vernotte, P. (1958). "Les paradoxes de la theorie continue de l'Ă©quation de la chaleur".
1531:
1481:
1335:
1256:
1670:
772:
1593:
1240:
1717:
1647:
1620:
1418:
1359:
1264:
976:
972:
68:
43:
136:{\displaystyle {\frac {\partial \theta }{\partial t}}~=~\alpha ~\nabla ^{2}\theta ,}
1343:
968:
1220:
1204:
764:
148:
768:
52:
1539:
1489:
1272:
1216:
27:
1351:
759:. This mathematical model is inconsistent with special relativity: the
748:
191:
775:
in vacuum, which is inadmissible within the framework of relativity.
980:
1566:
1326:
526:
67:
Heat conduction in a
Newtonian context is modelled by the
51:, namely the principle that an effect must be within the
1308:
Gavassino, L.; Antonelli, M.; Haskell, B. (2022-01-06).
1215:, there is the possibility of phenomena such as thermal
1678:
1146:
1109:
212:
1010:
829:
809:
656:
535:
447:
395:
243:
81:
1518:Joseph, D. D.; Preziosi, L. (1989). "Heat waves".
1195:
1128:
1095:
953:
815:
735:
640:
517:
430:
373:
225:
135:
431:{\displaystyle \mathbf {q} ~=~-k~\nabla \theta ,}
1636:International Journal of Heat and Mass Transfer
1609:International Journal of Heat and Mass Transfer
1582:International Journal of Heat and Mass Transfer
1407:International Journal of Heat and Mass Transfer
763:associated to the heat equation (also known as
1466:Chester, M. (1963). "Second sound in solids".
1698:
8:
1377:(Second ed.). Oxford: University Press.
1310:"Thermodynamic Stability Implies Causality"
1705:
1691:
1555:Journal of Thermophysics and Heat Transfer
1196:{\textstyle C^{2}~=~\alpha /\tau _{_{0}}.}
1744:Hyperbolic partial differential equations
1325:
1241:"A model for relativistic heat transport"
1182:
1178:
1169:
1151:
1145:
1118:
1114:
1108:
1061:
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1030:
1019:
1015:
1009:
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906:
893:
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850:
839:
830:
828:
808:
803:form, where the n, the temperature field
698:
669:
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655:
620:
612:
588:
580:
556:
548:
534:
495:
460:
446:
396:
394:
389:, as a function of temperature gradient,
359:
345:
339:
321:
307:
301:
283:
269:
263:
248:
242:
217:
211:
121:
82:
80:
16:Model compatible with special relativity
1432:Eckert, E. R. G.; Drake, R. M. (1972).
1231:
767:) has support that extends outside the
529:operator, ∇, is defined in 3D as
73:parabolic partial differential equation
1373:Carslaw, H. S.; Jaeger, J. C. (1959).
7:
1659:
1657:
1284:
1282:
30:processes) in a way compatible with
1503:Morse, P. M.; Feshbach, H. (1953).
1677:. You can help Knowledge (XXG) by
1434:Analysis of Heat and Mass Transfer
1084:
1043:
1033:
936:
917:
909:
868:
854:
709:
701:
657:
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471:
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419:
352:
342:
314:
304:
276:
266:
245:
214:
118:
93:
85:
63:Parabolic model (non-relativistic)
14:
983:). The equation is known as the "
1661:
1436:. Tokyo: McGraw-Hill, Kogakusha.
1239:van Kampen, N. G. (1970-03-02).
1062:
1037:
670:
613:
581:
549:
496:
397:
779:Hyperbolic model (relativistic)
1505:Methods of Theoretical Physics
1344:10.1103/PhysRevLett.128.010606
1207:) to a hyperbolic (includes a
1:
1594:10.1016/S0017-9310(01)00199-5
1213:partial differential equation
1648:10.1016/0017-9310(89)90166-X
1621:10.1016/0017-9310(95)00202-2
1419:10.1016/0017-9310(96)00211-6
1375:Conduction of Heat in Solids
1265:10.1016/0031-8914(70)90231-4
991:, which can be derived from
20:Relativistic heat conduction
440:first law of thermodynamics
22:refers to the modelling of
1775:
1656:
1520:Reviews of Modern Physics
1129:{\textstyle \tau _{_{0}}}
1540:10.1103/RevModPhys.61.41
1507:. New York: McGraw-Hill.
1490:10.1103/PhysRev.131.2013
226:{\textstyle \nabla ^{2}}
42:) relativity, the usual
1314:Physical Review Letters
989:telegrapher's equations
967:is called the speed of
816:{\displaystyle \theta }
785:Fick's law of diffusion
1673:-related article is a
1387:Some authors also use
1197:
1130:
1097:
995:of electrodynamics.
955:
817:
737:
642:
519:
432:
375:
227:
200:specific heat capacity
137:
1198:
1131:
1098:
956:
818:
738:
643:
520:
433:
376:
235:Cartesian coordinates
228:
138:
1759:Thermodynamics stubs
1144:
1107:
1008:
971:(that is related to
827:
807:
654:
533:
445:
393:
241:
210:
184:thermal conductivity
79:
1754:Transport phenomena
1739:Concepts in physics
1532:1989RvMP...61...41J
1482:1963PhRv..131.2013C
1336:2022PhRvL.128a0606G
1257:1970Phy....46..315V
993:Maxwell’s equations
176:thermal diffusivity
1734:Special relativity
1193:
1126:
1093:
963:In this equation,
951:
813:
757:entropy production
733:
638:
515:
428:
371:
223:
133:
32:special relativity
1686:
1685:
1642:(10): 1979–1987.
1476:(15): 2013–2015.
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849:
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726:
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697:
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633:
619:
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601:
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579:
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328:
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116:
110:
104:
100:
1766:
1707:
1700:
1693:
1665:
1658:
1652:
1651:
1631:
1625:
1624:
1615:(6): 1307–1315.
1604:
1598:
1597:
1588:(5): 1055–1061.
1577:
1571:
1570:
1550:
1544:
1543:
1515:
1509:
1508:
1500:
1494:
1493:
1463:
1457:
1456:
1444:
1438:
1437:
1429:
1423:
1422:
1413:(5): 1007–1016.
1402:
1396:
1385:
1379:
1378:
1370:
1364:
1363:
1329:
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843:
831:
823:is governed by:
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233:, is defined in
232:
230:
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204:Laplace operator
142:
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114:
108:
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99:
91:
83:
69:Fourier equation
1774:
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1724:Heat conduction
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1469:Physical Review
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1141:
1138:relaxation time
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1110:
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1032:
1016:
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1005:
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117:
92:
84:
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65:
24:heat conduction
17:
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1729:Thermodynamics
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1695:
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1671:thermodynamics
1666:
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1626:
1599:
1572:
1567:10.2514/2.6204
1545:
1510:
1495:
1458:
1449:Comptes Rendus
1439:
1424:
1397:
1380:
1365:
1300:
1291:Comptes Rendus
1278:
1251:(2): 315–332.
1230:
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977:quasiparticles
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834:
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793:Carlo Cattaneo
780:
777:
773:speed of light
761:Green function
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1323:
1320:(1): 010606.
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75:of the kind:
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50:
45:
44:heat equation
41:
37:
33:
29:
26:(and similar
25:
21:
1679:expanding it
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1639:
1635:
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1612:
1608:
1602:
1585:
1581:
1575:
1561:(1): 82–89.
1558:
1554:
1548:
1526:(1): 47–71.
1523:
1519:
1513:
1504:
1498:
1473:
1467:
1461:
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1219:and thermal
1209:conservative
1140:, such that
1000:
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747:is specific
744:
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1205:dissipative
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985:hyperbolic
801:hyperbolic
769:light-cone
525:where the
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