The European Physical Journal H

, Volume 40, Issue 2, pp 205–240 | Cite as

Early history of extended irreversible thermodynamics (1953–1983): An exploration beyond local equilibrium and classical transport theory

  • G. Lebon
  • D. Jou


This paper gives a historical account of the early years (1953–1983) of extended irreversible thermodynamics (EIT). The salient features of this formalism are to upgrade the thermodynamic fluxes of mass, momentum, energy, and others, to the status of independent variables, and to explore the consistency between generalized transport equations and a generalized version of the second law of thermodynamics. This requires going beyond classical irreversible thermodynamics by redefining entropy and entropy flux. EIT provides deeper foundations, closer relations with microscopic formalisms, a wider spectrum of applications, and a more exciting conceptual appeal to non-equilibrium thermodynamics. We first recall the historical contributions by Maxwell, Cattaneo, and Grad on generalized transport equations. A thermodynamic theory wide enough to cope with such transport equations was independently proposed between 1953 and 1983 by several authors, each emphasizing different kinds of problems. In 1983, the first international meeting on this theory took place in Bellaterra (Barcelona). It provided the opportunity for the various authors to meet together for the first time and to discuss the common points and the specific differences of their previous formulations. From then on, a large amount of applications and theoretical confirmations have emerged. From the historical point of view, the emergence of EIT has been an opportunity to revisit the foundations and to open new avenues in thermodynamics, one of the most classical and well consolidated physical theories.


Entropy Kinetic Theory Entropy Production Irreversible Thermodynamic Nonequilibrium Thermodynamic 
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  1. Alts, T. and I. Müller. 1972. Relativistic thermodynamics of simple heat conducting fluids. Arch. Rat. Mech. Anal. 48: 245–273.zbMATHGoogle Scholar
  2. Alvarez, F.X., A.V. Cimmelli, D. Jou and A. Sellitto. 2012. A mesoscopic description of boundary effects in nanoscale heat transport. Nanoscale Syst. MMTA 1: 112–142.zbMATHGoogle Scholar
  3. Anile, A.M. and S. Pluchino. 1984. Linear waves modes for dissipative fluids with rate type constitutive equations. Meccanica 19: 204–110.Google Scholar
  4. Arnold, V. 1966. Sur la géometrie différentielle des groupes de Lie de dimension infinie et ses applications à l’hydrodynamique des fluides parfaits. Ann. Inst. Fourier 16: 319–361.Google Scholar
  5. Bampi, F. and A. Morro. 1984. Non equilibrium thermodynamics: a hidden variable theory. In: Lecture Notes in Physics, Vol. 199. Springer, Berlin.Google Scholar
  6. Barbera, E., I. Müller, D. Reitebuch and N.R. Zhao. 2004. Determination of boundary conditions in extended thermodynamics via fluctuation theory. Continuum Mech. Thermodyn. 16: 411–425.zbMATHADSGoogle Scholar
  7. Belinskii, V.A., S. Nikomarov and I.S Khalatnikov. 1979. Investigation of the cosmological evolution of a viscoelastic matter with causal thermodynamics. Sov. Phys. JETP 50: 213–221.ADSGoogle Scholar
  8. Belousov B.P. 1958. A periodic reaction and its mechanism. Shornik Referatov po Radiacionnoii Medicine za. (Collection of Abstracts on Radiation Medicine), pp. 145–147. Moscow.Google Scholar
  9. Bénard, H. 1900. Les tourbillons cellulaires dans une nappe liquide transportant la chaleur par conduction en régime permanent. Rev. Gén. Sci. Pures et Appliquées 11: 1261–1271.Google Scholar
  10. Beris, A.N. and B.J. Edwards. 1994. Thermodynamics of Flowing Systems with Internal Microstructures. Oxford Sci. Pub., Oxford.Google Scholar
  11. Bird, R.B., C.F. Curtiss, R.C. Armstrong and D. Hassager. 1987. Dynamics of Polymer Liquids, 2nd edition, Vol. 2. Wiley, New York.Google Scholar
  12. Boltzmann, L. 1872. Weiteren Studien über das Wärmegleichgewicht zwischen Gasmolekülen. Sitzungsberichte der Akad. der Wissensch. Wien Abt. II: 275–370.Google Scholar
  13. Boltzmann, L. 1895, 1898. Vorlesungen über Gastheorie I und II. Verlag Metzger und Wittig, Leipzig.Google Scholar
  14. Bubnov, V.A. 1976. Wave concepts in the theory of heat. Int. J. Heat Mass Transfer 19: 175–184.zbMATHADSGoogle Scholar
  15. Carnot, S. 1824. Réflexions sur la puissance motrice du feu et sur des machines propres à développer cette puissance. Librairie Bachelier, Paris.Google Scholar
  16. Casas-Vazquez, J., D. Jou and G. Lebon. (eds.). 1984. Recent Developments in Non- Equilibrium Thermodynamics. In: Lecture Notes in Physics, Vol. 199. Springer, Berlin.Google Scholar
  17. Casas-Vazquez, J. and D. Jou. 2003. Temperature in nonequilibrium states: a review of open problems and current proposals. Rep. Prog. Phys. 66: 1937–2023.ADSGoogle Scholar
  18. Casimir, H.B.G. 1945. On Onsager’s principle of microscopic irreversibility. Rev. Mod. Phys. 17: 343–350.ADSGoogle Scholar
  19. Cattaneo, C. 1948. Sulla conduzione del calore. Atti Seminario Mat. Fis. Univ. Modena 3: 83–101.MathSciNetGoogle Scholar
  20. Chapman, S. and T.G. Cowling. 1970. The Mathematical Theory of Non-uniform Gases. Cambridge Univ. Press, Cambridge.Google Scholar
  21. Chester, M. 1963. Second sound in solids. Phys. Rev. 131: 2013–2015.ADSGoogle Scholar
  22. Chester, M. 1966. High-frequency thermometry. Phys. Rev. 145: 76–80.MathSciNetADSGoogle Scholar
  23. Cimmelli, A., D. Jou, T. Ruggeri and P. Van. 2014. Entropy principle and recent results in non-equilibrium theories. Entropy 16: 1756–1807.MathSciNetADSGoogle Scholar
  24. Clausius, R. 1854. Über eine veränderte Form des zweiten Hauptsatzes der mechanischen Wärmetheorie. Poggendorff’s Annalen der Physik 93: 481–506.ADSGoogle Scholar
  25. Clausius, R. 1865. Über verschiedene für die Anwendung bequeme Formen der Hauptgleichungen der mechanischen Wärmetheorie. Poggendorf’s Annalen der Physik 125: 353–400.ADSGoogle Scholar
  26. Clebsch, A. 1859. Über die Integration der hydrodynamischen Gleichungen. J. Reine Angew. Math. 56: 1–10.zbMATHMathSciNetGoogle Scholar
  27. Coleman, B.D. and C. Truesdell. 1960. On the reciprocal relations of Onsager. J. Chem. Phys. 33: 28–31.MathSciNetADSGoogle Scholar
  28. Coleman, B.D. and W. Noll. 1963. The thermodynamics of elastic materials with heat conduction and viscosity. Arch. Rat. Mech. Anal. 13: 167–178.zbMATHMathSciNetGoogle Scholar
  29. Coleman, B.D. 1964. Thermodynamics of materials with memory. Arch. Rat. Mech. Anal. 17: 1–46.Google Scholar
  30. Coleman, B.D. and M.E. Gurtin. 1967. Thermodynamics with internal state variables. J. Chem. Phys. 47: 597–613.ADSGoogle Scholar
  31. Courant, R. and D. Hilbert. 1962. Methods of Mathematical Physics. J. Wiley, New York.Google Scholar
  32. de Groot, S.R. 1951. Thermodynamics of Irreversible Processes. North-Holland, Amsterdam.Google Scholar
  33. de Groot, S.R. and P. Mazur. 1962. Non-equilibrium Thermodynamics. North-Holland, Amsterdam.Google Scholar
  34. Denbigh, R.G. 1950. The Thermodynamics of the Steady state. Wiley, New York.Google Scholar
  35. Dufour, L. 1873. Über die Diffusion der Gase durch poröse Wände und die sie begleitenden Termperaturveränderungen. Ann. der Phys. 148: 490–492 (translated from the original article published in French in Arch. Sci. Phys. Nat. Genève 45: 9–11).ADSGoogle Scholar
  36. Eckart, C. 1940. The thermodynamics of irreversible processes. Phys. Rev. 58: 267–269 and 58: 269–275.ADSGoogle Scholar
  37. Eu, B.C. 1980. A modified moment method and irreversible thermodynamics. J. Chem.Phys. 73: 2958–2969.MathSciNetADSGoogle Scholar
  38. Eu, B.C. 1992. Kinetic Theory and Irreversible Thermodynamics. Wiley, New York.Google Scholar
  39. Fick, A. 1855. Über Diffusion. Ann. Phys. 94: 59–86.Google Scholar
  40. Finlayson, B. 1972. The Method of Weighted Residuals and Variational Principles. Acad. Press, New York.Google Scholar
  41. Fourier, J.B. 1822. Théorie Analytique de la Chaleur. F. Didot, Paris.Google Scholar
  42. Garcia-Colin, L.S. and M. Lopez de Haro. 1982. The Burnett equations in extended irreversible thermodynamics. J. Non-Equilib. Thermodyn. 7: 95–104.zbMATHADSGoogle Scholar
  43. Garcia-Colin, L.S., R.F. Rodriguez, M. Lopez de Haro, D. Jou and J. Casas-Vazquez. 1984. On the foundations of extended irreversible thermodynamics. J. Stat. Phys. 17: 465–484.MathSciNetADSGoogle Scholar
  44. Garcia-Colin, L.S. and R.F. Rodriguez. 1988a. On the relationship between extended thermodynamics and the wave approach in thermodynamics. J. Non-Equilib. Thermodyn. 13: 81–94.zbMATHADSGoogle Scholar
  45. Garcia-Colin, L.S. 1988b. Extended non-equilibrium thermodynamics, scope and limitations. Rev. Mexicana Fisica 34: 344–366.Google Scholar
  46. Garcia-Colin, L.S. 1995. Extended irreversible thermodynamics: an unfinished task. Mol. Phys. 86: 697–706.ADSGoogle Scholar
  47. Gaspard, P. 1998. Chaos, Scattering and Statistical Mechanics. Cambridge Univ. Press, Cambridge.Google Scholar
  48. Gibbs, W. 1875, 1878. On the equilibrium of heterogeneous substances. Transactions of the Connecticut Academy, pp. 108–248 and pp. 342–524.Google Scholar
  49. Glansdorff, P. and I. Prigogine. 1964. On a general evolution criterion in macroscopic physics. Physica. 30: 351–374.MathSciNetADSGoogle Scholar
  50. Glansdorff, P. and I. Prigogine. 1971. Thermodynamics of Structures, Stability and Fluctuations. Wiley, New York.Google Scholar
  51. Grad, H. 1949. On the kinetic theory of rarefied gases. Commun. Pure Appl. Math. 2: 331–407.zbMATHMathSciNetGoogle Scholar
  52. Grad, H. 1958. Principles of the kinetic theory of gases. In: Flugge S. (ed.) Hd. der Physik, Vol. XII. Springer, Berlin.Google Scholar
  53. Griffin, J.J. and K.K. Kan. 1976. Colliding heavy ions: nuclei as dynamical fluids. Rev. Mod. Phys. 48: 467–477.ADSGoogle Scholar
  54. Grmela, M. and C. Lye. 1987. Shear flow induced structural changes in polymeric liquid crystals. Phys. Lett. A 120: 282–285.ADSGoogle Scholar
  55. Grmela, M. 1984. Particle and bracket formulation of kinetic equations. Contemp. Math. AMS 28: 125–132.zbMATHMathSciNetGoogle Scholar
  56. Grmela, M. and G. Lebon 1990. Hamiltonian extended thermodynamics. J. Phys. A23: 3341–3351.MathSciNetADSGoogle Scholar
  57. Grmela, M. 2010. Multiscale equilibrium and nonequilibrium thermodynamics in chemical engineering. Adv. Chem. Eng. 36: 75–128.Google Scholar
  58. Grmela, M. 2014. Contact geometry of mesoscopic thermodynamics and dynamics. Entropy 16: 1652–1686.MathSciNetADSGoogle Scholar
  59. Gurtin, M.E. and A.C Pipkin. 1968. A general theory of heat conduction with finite wave speed. Arch. Rational Mech. Anal. 31: 116–126.MathSciNetADSGoogle Scholar
  60. Guyer, R.A. and J.A. Krumhansl. 1966. Solution of the linearized Boltzmann phonon equation. Phys. Rev. 148: 766–778 and 148: 778–788.ADSGoogle Scholar
  61. Gyarmati, I. 1970. Non-equilibrium Thermodynamics. Springer, Berlin.Google Scholar
  62. Gyarmati, I. 1977. On the wave approach of thermodynamics and some problems of non- linear theories. J. Non-Equilib. Thermodyn. 2: 233–260.zbMATHADSGoogle Scholar
  63. Haase, R. 1969. Thermodynamics of Irreversible Processes. Addison-Wesley, Reading MA.Google Scholar
  64. Hand, G.L. 1962. A theory of anisotropic fluids. J. Fluid. Mech. 13: 33–46.zbMATHMathSciNetADSGoogle Scholar
  65. Hess, S. 1977. On nonlocal constitutive relations, continued fraction expansion for the wave vector dependent diffusion coefficient. Z. Naturforsch. 32a: 678–684.MathSciNetADSGoogle Scholar
  66. Hiscock, W.A. and L. Lindblom. 1985. Generic instabilities in first-order dissipative relativistic fluids theories. Phys. Rev. D 31: 725–733.MathSciNetADSGoogle Scholar
  67. Hutter, K. 1977. The foundations of thermodynamics, its basic postulate and implications. A review of modern thermodynamics. Acta Mechanica 27: 1–54.MathSciNetADSGoogle Scholar
  68. Hutter, K. and I. Müller. 1975. On mixture of relativistic fluids. Helvetica Physica Acta 48: 1–24.Google Scholar
  69. Israel, W. 1976. Non stationary irreversible thermodynamics: a causal relativistic theory. Ann. Phys. (New York) 100: 310–331.MathSciNetADSGoogle Scholar
  70. Israel, W. and J.M. Stewart. 1979. Transient relativistic thermodynamics and kinetic theory. Ann. Phys. (NY) 118: 341–372.MathSciNetADSGoogle Scholar
  71. Jaynes, E.T. 1963. Information theory and statistical mechanics. In: Statistical Physics (Ford, W.K. ed.). Benjamin, New York.Google Scholar
  72. Jou, D., J. Casas-Vazquez and G. Lebon. 1979. A dynamical interpretation of second-order constitutive equations of hydrodynamics. J. Non-Equilib. Thermodyn. 4: 349–362.ADSGoogle Scholar
  73. Jou, D., J.E. Llebot and J. Casas-Vazquez. 1982. Thermodynamic aspects of non- equilibrium fluctuations. Phys. Rev. A 25: 3277–3281.ADSGoogle Scholar
  74. Jou, D. 1983. Equacions de Gibbs generalitzades i extensió de la termodinamica de processos irreversibles. Institut d’Estudis Catalans, Barcelona (in Catalan language).Google Scholar
  75. Jou, D., J. Casas-Vazquez and G. Lebon. 1993. Extended Irreversible Thermodynamics. First edition. Second edition 1996. Third edition 2001. Fourth edition 2010. Springer, Berlin.Google Scholar
  76. Jou, D., J. Casas-Vazquez and M. Criado-Sancho. 2000. Thermodynamics of Fluids under Flow. Second edition 2011. Springer, Berlin.Google Scholar
  77. Jou, D., G. Lebon and M. Criado-Sancho. 2010. Variational principles in thermal transport in nanosystems with heat slip flow. Phys. Rev. E 82: 031128.ADSGoogle Scholar
  78. Jou, D., J. Casas-Vazquez, G. Lebon and M. Grmela. 2005. A phenomenological scaling approach for heat transport in nanosystems. Appl. Math. Lett. 18: 963–967.zbMATHGoogle Scholar
  79. Joule, J.P. 1841. On the heat evolved by metallic conductors of electricity, and in the cells of a battery during electrolysis. Phil. Mag. 19: 275.Google Scholar
  80. Kestin, J. and J. Bataille. 1980. Thermodynamics of solids. In: Continuum Models of Discrete Systems. University of Waterloo Press, Waterloo.Google Scholar
  81. Kirkwood, J.C. 1967. Selected Topics in Statistical Mechanics. Gordon and Breach, New York.Google Scholar
  82. Koide, T., G.S. Denicol, P. Mota and T. Kodama. 2007. Relatisvistic dissipative hydrodynamics: a minimal causal theory. Phys. Rev. C 75: 034909.ADSGoogle Scholar
  83. Kranys, M. 1972. Kinetic derivation of non-stationary general relativistic thermodynamics. Nuovo Cimento B 8: 417–441.ADSGoogle Scholar
  84. Kranys, M. 1977. Hyperbolic elasticity of dissipative media and its wave propagation modes. J. Phys. A: Math.Gen. 10: 689–709.zbMATHMathSciNetADSGoogle Scholar
  85. Kranys, M. 1989. Casual theories of evolution and wave propagation in mathematical physics. Appl. Mech. Rev. 42: 305–322.MathSciNetADSGoogle Scholar
  86. Lambermont, J. and G. Lebon. 1973. On a generalization of the Gibbs equation for heat conduction. Phys. Lett. A 42: 499–500.ADSGoogle Scholar
  87. Landau, L.D. and E.M. Lifshitz. 1958. Mechanics of Fluids. Addison Wesley, Reading, Mass.Google Scholar
  88. Landau, L.D. and E.M. Lifshitz. 1980. Statistical Physics, 3rd edition, Pergamon, Oxford.Google Scholar
  89. Lavenda, B. 1979. Thermodynamics of Irreversible Processes. McMillan, London.Google Scholar
  90. Lebon, G. and J. Lambermont. 1973. Generalization of Hamilton’s principle to continuous dissipative systems. J. Chem. Phys. 5: 2929–2936.MathSciNetADSGoogle Scholar
  91. Lebon, G. 1978. Derivation of generalized Fourier and Stokes-Newton equations based on the thermodynamics of irreversible processes. Bull. Acad. Roy. Belgique 64: 456–472.ADSGoogle Scholar
  92. Lebon, G., D. Jou and J. Casas-Vazqez. 1980a. An extension of the local equilibrium hypothesis. J. Phys. A: Math. Gen. 13: 275–290.ADSGoogle Scholar
  93. Lebon, G. 1980b. Variational Principles in Thermomechanics. In: Recent Developments in Thermomechanics of Solids (Lebon, G. and Perzyna, P. eds.). CISM Courses and Lectures, Vol. 282, pp. 221–415. Springer, Wien, New York.Google Scholar
  94. Lebon, G. and A. Cloot. 1989. Propagation of ultrasonic sound waves in dissipative dilute gases and extended irreversible thermodynamics. Wave Motion 11: 23–32.zbMATHGoogle Scholar
  95. Lebon, G and P.C. Dauby. 1990. Heat transport in dielectric crystals at low temperature; A variational formulation based on extended irreversible thermodynamics. Phys. Rev. A 42: 4710–4715.ADSGoogle Scholar
  96. Lebon, G., M. Ruggieri and A. Valenti. 2008a. Extended thermodynamics revisited: renormalized flux variables and second sound in rigid solids. J. Phys. C 20: 025223.Google Scholar
  97. Lebon, G., D. Jou and J. Casas-Vazquez. 2008b. Understanding Non-equilibrium Thermodynamics, Foundations, Applications, Frontiers. Springer, Berlin, Heidelberg.Google Scholar
  98. Lebon, G. 2014. Heat conduction at micro and nanoscales: a review through the prism of extended irreversible thermodynamics. J. Non-Equilib. Thermodyn. 39: 35–59.Google Scholar
  99. Lhuillier, D. 1979. Stress tensor of dilute polymer solutions. Phys. Fluids 22: 2033–2035.ADSGoogle Scholar
  100. Liu, I.S. 1972. Method of Lagrange multipliers for exploitation of the entropy principle. Arch. Rat. Mech. Anal. 46: 131–148.zbMATHGoogle Scholar
  101. Liu, I.S. and I. Müller. 1983. Extended thermodynamics of classical and degenerate gases. Arch. Rat. Mech. Anal. 83: 285–332.zbMATHGoogle Scholar
  102. Liu, I.S., I. Müller and T. Ruggeri. 1986. Relativistic thermodynamics of gases. Ann. Phys. (New York) 169: 191–219.ADSGoogle Scholar
  103. Luikov, A.V. 1969. Analytical Heat Diffusion Theory. Acad. Press, New York.Google Scholar
  104. Luzzi, R., A.R. Vasconcellos, J. Casas-Vázquez, and D. Jou. 1997. Characterization and measurement of a nonequilibrium temperature-like variable in irreversible thermodynamcs. Physica A 234: 699–714.ADSGoogle Scholar
  105. Luzzi R., A.R. Vasconcellos and J.S. Ramos. 2001. On the statistical foundations of irreversible thermodynamics. Teubner Verlag, Berlin.Google Scholar
  106. Luzzi R., A.R. Vasconcellos and J.S. Ramos. 2002. Predictive statistical mechanics: a non-equilibrium ensemble formalism. Kluwer, Dordrecht.Google Scholar
  107. Machlup, S. and L. Onsager. 1953. Fluctuations and irreversible process. II Systems with kinetic energy. Phys. Rev. 91: 1512–1515.zbMATHMathSciNetADSGoogle Scholar
  108. Mandel, J. 1978. Propriétés Mécaniques des Matériaux. Eyrolles, Paris.Google Scholar
  109. Maugin, G. and W. Muschik. 1994. Thermodynamics with internal variables. J. Non- Equilib. Thermodyn. 19: 217–249 and 19: 250–289.zbMATHADSGoogle Scholar
  110. Maugin, G. 1999. The Thermodynamics of Nonlinear Irreversible Behaviours. World Scientific, Singapore.Google Scholar
  111. Maxwell, J.C. 1867. On the dynamical theory of gases. Philos. Trans. Roy. Soc. London, 157: 49–88.Google Scholar
  112. Meixner, J. and H.G. Reik. 1959. Thermodynamik der Irreversible Prozesse. In: Handbuch der Physik, Bd 3/ II. Springer, Berlin.Google Scholar
  113. Meixner, J. 1968. TIP has many faces. In: Proceed. IUTAM Symposium Vienna, 1966. Springer, Berlin.Google Scholar
  114. Meixner, J. 1974. Coldness and temperature, Arch. Rat. Mech. Anal. 3: 281–290.Google Scholar
  115. Mongiovi, M.S. 1991. Superfluidity and the entropy conservation in extended thermodynamics. J. Non- Equilib. Thermodyn. 16: 225–239.zbMATHADSGoogle Scholar
  116. Mongiovi, M.S. 1992. Thermomechanical phenomena in extended thermodynamics of an ideal monatomic superfluid. J. Non-Equilib. Thermodyn. 17: 183–186.ADSGoogle Scholar
  117. Müller, I. 1966. Zur Ausbreitungsgeschwindigkeit von Störungen in kontinuierlichen Medien. Ph.D. Thesis, Technical University, Aachen.Google Scholar
  118. Müller, I. 1967. Zum Paradox der Wärmetheorie. Z. Phys. 198: 329–344.zbMATHADSGoogle Scholar
  119. Müller, I. 1969. Toward relativistic thermodynamics. Arch. Rat. Mech. Anal. 34: 259–282.zbMATHGoogle Scholar
  120. Müller, I. 1970. Die Kälte funktion, eine univeselle Funktion in der Thermodynamik in der viskoser warmeleitender Flüssigkeiten. Arch. Rat. Mech. Anal. 40: 1–36.Google Scholar
  121. Müller, I. 1971. The coldness, a universal function in thermoelastic bodies. Arch. Rat. Mech. Anal. 41: 319–332.zbMATHGoogle Scholar
  122. Müller, I. and T. Ruggeri. 1993. Extended Thermodynamics. Springer, New York.Google Scholar
  123. Müller, I. and Ruggeri, T. 1998. Rational Extended thermodynamics. Springer, New York.Google Scholar
  124. Müller, I., D. Reitebuch and W. Weiss. 2003. Extended thermodynamics consistent in order of magnitude. Continuum Mech. Thermodyn. 15: 113–146.zbMATHADSGoogle Scholar
  125. Müller, I. and W. Weiss. 2012. Thermodynamics of irreversible processes-past and present. Eur. Phys. J. H 37: 139–236.Google Scholar
  126. Muschik, W. 1977. Empirical foundation and axiomatic treatment of non-equilibrium temperature. Arch. Rat. Mech. Anal. 66: 379–400.MathSciNetGoogle Scholar
  127. Muschik, W. 2007. Why so many “Schools” of thermodynamics? Forsch. Ingenieurwes. 71: 149–161.Google Scholar
  128. Nettleton, R.E. 1959. Thermodynamics of viscoelasticity in liquids. Phys. Fluids 2: 256–263.zbMATHMathSciNetADSGoogle Scholar
  129. Nettleton, R.E. 1960. Relaxation theory of thermal conduction in liquids. Phys. Fluids 3: 216–225.MathSciNetADSGoogle Scholar
  130. Nettleton, R.E. 1984. Early applications of extended irreversible thermodynamics. In: Lecture Notes in Physics, Vol. 199, pp. 1–31. Springer, Berlin.Google Scholar
  131. Nettleton, R.E. and S.I Sobolev. 1995. Applications of extended thermodynamics to chemical, rheological, and transport process; a special survey Parts I and II. J. Non-Equilib. Thermodyn. 20: 205–229 and 20: 297–331.Google Scholar
  132. Newton, I. 1701. Scala graduum caloris. Calorum descriptions &signa. Phil. Trans. Royal Society London 22: 824–829. English translation in: Newton, I. 1809. Phil. Trans. Royal Society London 4: 572–575.Google Scholar
  133. Newton, I. 1726. Philosophiae Naturalis Principia Mathematica, 3rd edition. Londini Juffia Societatis Regis ac Typis.Google Scholar
  134. Nicolis, G. and I. Prigogine. 1977. Self-organization in Nonequilibrium Systems. Wiley, New York.Google Scholar
  135. Nicolis, G. and I. Prigogine. 1989. Exploring Complexity. Freeman, New York.Google Scholar
  136. Noll, W. and C. Truesdell. 1965. The non-linear field theories of mechanics. Springer, New York.Google Scholar
  137. Ohm, G.S. 1827. Die galvanische Kette, mathematisch bearbeitet. T.H. Riemann, Berlin.Google Scholar
  138. Onsager, L. 1931. Reciprocal relations in irreversible processes. Phys. Rev. 37: 405–426 and 38: 2265–2279.ADSGoogle Scholar
  139. Öttinger, H.C. 2005. Beyond Equilibrium Thermodynamics. Wiley, Hoboken.Google Scholar
  140. Pavon, D., D. Jou and J. Casas-Vazquez. 1980. About the relativistic temperature gradient. Phys. Lett. A 78: 317–318.ADSGoogle Scholar
  141. Peltier, J.C. 1839. Observations sur les multiplicateurs et sur les piles thermo-électriques. Imprimerie E.J. Bailly, Paris.Google Scholar
  142. Prigogine, I. 1947. Etude Thermodynamique des Phénomènes Irréversibles. Desoer, Liège.Google Scholar
  143. Prigogine, I. 1961. Introduction to Thermodynamics of Irreversible Processes. Interscience, New York.Google Scholar
  144. Prigogine, I. 1980. From Time to Becoming. Time and Complexity in the Physical Sciences. Freeman, San Francisco.Google Scholar
  145. Reichl, L.E. 1980. A Modern Course in Statistical Physics. Univ. Texas Press, Austin, Texas.Google Scholar
  146. Rubi, M. and J. Casas-Vazquez. 1980. Thermodynamical aspects of micropolar fluids. J. Non-Equilib. Thermodyn. 5: 155–164.ADSGoogle Scholar
  147. Ruggeri, T. 1993. Recent results on wave propagation in continuous media. In: CISM Courses and Lectures, Vol. 344, pp. 105–154. Springer, Wien, New York.Google Scholar
  148. Seebeck, T.J. 1821. Über den Magnetism der Galvanische Kette. K. Akad. Wiss. Berlin.Google Scholar
  149. Sieniutycz, S. 1984. Variational approach to extended irreversible thermodynamics of heat and mass transfer. J. Non-Equilib. Thermodyn. 9: 61–71.zbMATHADSGoogle Scholar
  150. Sieniutycz, S. and P. Salomon. (eds.). 1992. Extended thermodynamics systems. Taylor and Francis, New York.Google Scholar
  151. Sieniutycz, S. 1994. Conservation Laws in Variational Thermo-Hydrodynamics. Springer, Berlin.Google Scholar
  152. Soret, C. 1879. Sur l’état d’équilibre que prend au niveau de sa concentration une dissolution saline primitivement homogène dont deux parties sont portées à des températures différentes. Arch. Sci. Phys. Nat. Genève 2: 187.Google Scholar
  153. Stewart, J.M. 1977. Non-transient relativistic thermodynamics and kinetic theory. Proc. Roy. Soc. London A 357: 59–75.ADSGoogle Scholar
  154. Stocker, H. and W. Greiner. 1986. High energy heavy ion collisions-probing the equation of state of highly excited hadronic matter. Phys. Rep. 137: 277–392.ADSGoogle Scholar
  155. Stokes, G.C. 1851. On the effect of the internal friction of fluids on the motion of the pendulums. Transactions of the Cambridge Phil. Soc. IX: 8–106.ADSGoogle Scholar
  156. Struchtrup, H. 2005. Macroscopic Transport Equations for Rarefied Gas Flows. Springer, Berlin, New York.Google Scholar
  157. Thomson, W. 1848. On an absolute thermometric scale founded on Carnot’s theory of the motive power of heat and calculated from Regnault’s observations. Phil. Mag. 33: 313–317.Google Scholar
  158. Thomson, W. 1851. On the dynamical theory of heat, with numerical results deduced from Mr Joule’s equivalent of a thermal unit, and M. Regnault’s observations on steam. Transactions of the Royal Society of Edinburgh XX (part II): 261–268 and 289–298.Google Scholar
  159. Thomson, W. 1854. Account of experimental investigations to answer questions originating in the mechanical theory of thermoelectric currents. Edinburgh Roy. Soc. Proc. III: 255.Google Scholar
  160. Truesdell, C. 1966. Six Lectures on Modern Natural Philosophy. Springer, Heidelberg.Google Scholar
  161. Truesdell, C. 1969. Rational Thermodynamics. MacGraw Hill, New York.Google Scholar
  162. Truesdell, C. and W. Noll. 1965. The Non-Linear Field Theories. In: Handbuch der Physik, Bd. III/3. Springer, Berlin.Google Scholar
  163. Turing, A. 1952. The chemical basis of morphogenesis. Philos. Trans. R. Soc. London B 237: 37–72.ADSGoogle Scholar
  164. van Kampen, N.G. 1987. Chapman-Enskog as an application of the method for eliminating fast variables. J. Stat. Phys. 46: 709–727.zbMATHMathSciNetADSGoogle Scholar
  165. Velasco, R.M. and L.S. Garcia-Colin. 1993. The kinetic foundations of non-local nonequilibrium thermodynamics. J. Non-Equilib. Thermodyn. 18: 157–172.zbMATHADSGoogle Scholar
  166. Verhas, J. 1983. On the entropy current. J. Non-Equilib. Thermodyn. 8: 201–206.ADSGoogle Scholar
  167. Vernotte, P. 1958. La véritable équation de la chaleur. Compt. Rend. Acad. Sci. Paris 247: 2103–2107.MathSciNetGoogle Scholar
  168. Vidal, C., G. Dewel and P. Borkmans. 1994. Au-delà de l’Equilibre. Hermann, Paris.Google Scholar
  169. Walgraef, D. 1997. Spatio-Temporal Pattern Formation. Springer, New York.Google Scholar
  170. Wei, J. 1966. Irreversible thermodynamics in Engineering. Ind. Eng. Chem. 58: 55–60.Google Scholar
  171. Weiss, W. 1990. Hierarchie der Erweiterten Thermodynamik. Dissertation Tech. Univ. Berlin.Google Scholar
  172. Wilhelm, H.E. and S.H. Choi. 1975. Nonlinear hyperbolic theory of thermal waves in metals. J. Chem. Phys. 63: 2119–2123.MathSciNetADSGoogle Scholar
  173. Woods, L.C. 1981. The bogus axioms of continuum mechanics. Bull. Inst. Math. Appl. 17: 98–102 and 1982. 18: 64–67.zbMATHMathSciNetGoogle Scholar

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© EDP Sciences and Springer-Verlag Berlin Heidelberg 2014

Authors and Affiliations

  1. 1.Département d’Astrophysique, Géophysique et Océanographie, Bâtiment B5, Liège UniversityLiègeBelgium
  2. 2.Departament de Física, Universitat Autònoma de BarcelonaCataloniaSpain

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