Role of thermodynamics in extensions of mesoscopic dynamical theories

1. L. Euler, Principes généraux du mouvement des fluides, Académie Royale des Sciences et des Belles-Lettres de Berlin, Mémoires., vol. 11, 1755 (English translation available in Physica D 237, 1825-1839, 2008).Search in Google Scholar

2. C. Truesdell, Rational thermodynamics. New York, NY, USA: Springer, 2nd ed., 1984.10.1007/978-1-4612-5206-1Search in Google Scholar

3. R. B. Bird, O. Hassager, R. C. Armstrong, and C. F. Curtiss, Dynamics of Polymeric Fluids, vol. 1. New York, NY, USA: Wiley, 1987.Search in Google Scholar

4. R. B. Bird, O. Hassager, R. C. Armstrong, and C. F. Curtiss, Dynamics of Polymeric Fluids, vol. 2. New York, NY, USA: Wiley, 1987.Search in Google Scholar

5. C. Cattaneo, Sulla conduzione del calore, Atti del Seminario Matematico e Fisico della Universita di Modena, vol. 3, pp. 83–101, 1948.Search in Google Scholar

6. R. Peierls, Zur kinetischen Theorie der Wärmeleitung in Kristallen, Ann. Phys., vol. 395, pp. 1055–1101, 1929.Search in Google Scholar

7. J. W. Gibbs, Collected Works. New York, NY, USA: Longmans Green and Co., 1984.Search in Google Scholar

8. E. T. Jaynes, Foundations of probability theory and statistical mechanics, in Delaware Seminar in the Foundation of Physics, New York, NY, USA: Springer, 1967.10.1007/978-3-642-86102-4_6Search in Google Scholar

9. R. Hermann, Geometry, Physics and Systems. New York, NY, USA: Marcel Dekker, 1984.Search in Google Scholar

10. V. I. Arnold, Mathematical methods of classical mechanics. New York, NY, USA: Springer, 1989.10.1007/978-1-4757-2063-1Search in Google Scholar

11. V. I. Arnold, Sur la géometrie différentielle des groupes de lie de dimension infini et ses applications dans l’hydrodynamique des fluides parfaits, ann, Inst. Fourier, vol. 16, p. 319, 1966.10.5802/aif.233Search in Google Scholar

12. M. Grmela, Thermodynamical lift of the nonlinear onsager-casimir vector field, in Proceedings of the Workshop on Hamiltonian Systems, Transformation Groups and Spectral Transform Methods (J. E. Harnad, J. Marsden, ed.), pp. 199–207, CRM, Univ. de Montreal, 1990.Search in Google Scholar

13. M. Grmela, Multiscale equilibrium and nonequilibrium thermodynamics in chemical engineering, Advances in Chemical Engineering, vol. 39, pp. 76–128, 2010.10.1016/S0065-2377(10)39002-8Search in Google Scholar

14. M. Grmela, Contact geometry of mesoscopic thermodynamics and dynamics, Entropy, vol. 16, pp. 1652–1686, 2014.Search in Google Scholar

15. I. E. Dzyaloshinskii and G. E. Volovick, Poisson brackets in condense matter physics, Ann. Phys., vol. 125, pp. 67–97, 1980.10.1016/0003-4916(80)90119-0Search in Google Scholar

16. M. Grmela, Particle and bracket formulations of kinetic equations, Contemporary Math., vol. 28, pp. 125–132, 1984.10.1090/conm/028/751978Search in Google Scholar

17. A. N. Kaufman, Dissipative hamiltonian systems: A unifying principle, Phys. Letters A, vol. 100, p. 419, 1984.10.1016/0375-9601(84)90634-0Search in Google Scholar

18. P. J. Morrison, Bracket formulation for irreversible classical fields, Phys. Letters A, vol. 100, p. 423, 1984.10.1016/0375-9601(84)90635-2Search in Google Scholar

19. M. M. Grmela and H. Öttinger, Dynamics and thermodynamics of complex fluids: General formulation, Phys.Rev.E, vol. 56, pp. 6620–6633, 1997.Search in Google Scholar

20. H. C. Öttinger and M. Grmela, Dynamics and thermodynamics of complex fluids: Illustration of the general formalism, Phys.Rev.E, vol. 56, pp. 6633–6650, 1997.Search in Google Scholar

21. M. Grmela, Fluctuations in extended mass-action-law dynamics, Physica D, vol. 241, pp. 976–986, 2012.10.1016/j.physd.2012.02.008Search in Google Scholar

22. A. Clebsch, Über die integration der hydrodynamische gleichungen, J. Reine Angew. Math, vol. 56, pp. 1–10, 1895.Search in Google Scholar

23. M. Grmela, Role of thermodynamics in multiscale physics, Computers and Math. Appl., vol. 65, pp. 1457–1470, 2013.Search in Google Scholar

24. D. Jou, J. Casas-Vàzquez, and G. Lebon, Extended Irreversible Thermodynamics. Berlin: Springer, 4th ed., 2010.10.1007/978-90-481-3074-0Search in Google Scholar

25. M. Grmela, A. Ammar, and F. C. G. Maîtrejean, A mesoscopic rheological model of moderately concentrated colloids, J. Non-Newtonian Fluid Mech., vol. 212, pp. 1–12, 2014.10.1016/j.jnnfm.2014.07.005Search in Google Scholar

26. R. J. DiPerna and P. L. Lions, Global solutions of Boltzmann’s equation and the entropy inequality, Arch. Rational Mech. Anal., vol. 114, pp. 47–55, 1991.10.1007/BF00375684Search in Google Scholar

27. S. K. Godunov, Symmetric form of the magnetohydrodynamic equation, Chislennye Metody Mekhaniki Sploshnoi Sredy, vol. 3, no. 1, pp. 26–34, 1972.Search in Google Scholar

28. K. O. Friedrichs, Conservation equations and the laws of motion in classical physics, Commun. Pure Appl. Math., vol. 31, pp. 123 – 131, 1978.10.1002/cpa.3160310107Search in Google Scholar

29. B. D. Coleman and W. Noll, The thermodynamics of elastic materials with heat conduction and viscosity, Arch. Ration. Mech. Anal., vol. 13, pp. 167–178, 1963.10.1007/BF01262690Search in Google Scholar

30. H. Grad, Principles of Kinetic Theory of Gases, vol. 12. Springer-Verlag: Encyclopedia of Physics, 1958.10.1007/978-3-642-45892-7_3Search in Google Scholar

31. R. T. Müller, I., Rational Extended Thermodynamics. New York, NY, USA: Springer, 1998.10.1007/978-1-4612-2210-1Search in Google Scholar

32. T. Ruggeri, The entropy principle: from continuum mechanics to hyperbolic systems of balance laws, Bollettino dell’Unione Matematica Italiana,, vol. Serie 8, Vol. 8-B, no. 1, pp. 1–20, 2005.Search in Google Scholar

33. V. A. Cimmelli, D. Jou, T. Ruggeri, and P. Ván, Entropy principle and recent results in non-equilibrium theories, Entropy, vol. 16, pp. 1756–1807, 2014.Search in Google Scholar

34. H. Struchtrup, Macroscopic transport equations for rarefied gas flows. Berlin, Germany: Springer, 2005.10.1007/3-540-32386-4Search in Google Scholar

35. G. M. Kremer, An introduction to the Boltzmann equation and transport processes in gases. Berlin, Germany: Springer, 2010.10.1007/978-3-642-11696-4Search in Google Scholar

36. P. Ván, A. Berezovski, and J. Engelbrecht, Internal variables and dynamic degrees of freedom, J. Non-Equilib. Thermod., vol. 33, pp. 235–254, 2008.10.1515/JNETDY.2008.010Search in Google Scholar

37. A. Berezovski, J. Engelbrecht, and M. A. Maugin, Generalized thermo-mechanics with dual internal variables, Archives Appl. Mech., vol. 81, pp. 229–240, 2011.10.1007/s00419-010-0412-0Search in Google Scholar

38. J. Casas-Vázquez and D. Jou, Temperature in non-equilibrium states: A review of open problems and current proposals, Rep. Prog. Phys., vol. 66, pp. 1937–2023, 2003.Search in Google Scholar

39. M. Grmela, Weakly nonlocal hydrodynamics, Phys. Rev. E, vol. 47, p. 351, 1993.10.1103/PhysRevE.47.351Search in Google Scholar

40. M. Grmela, Extensions of classical hydrodynamics, J. Stat. Phys., vol. 132, pp. 581–602, 2008.10.1007/s10955-008-9558-3Search in Google Scholar

41. V. A. Cimmelli, An extension of liu procedure in weakly nonlocal thermodynamics, J. Mat. Phys., vol. 48, p. 113510, 2007.Search in Google Scholar

42. P. Ván, Weakly nonlocal irreversible thermodynamics, Ann. Physik, vol. 12, pp. 142–169, 2003.10.1002/andp.200310002Search in Google Scholar

43. E. Johannessen and D. Bedeaux, The nonequilibrium van der waals square gradient model. (ii) local equilibrium of the Gibbs surface, Physica A, vol. 330, pp. 354–372, 2003.10.1016/j.physa.2003.09.054Search in Google Scholar

44. J. W. Cahn and J. E. Hilliard, Free energy of a nonuniform system. interfacial free energy, J. Chem. Phys., vol. 28, p. 258, 1958.10.1063/1.1744102Search in Google Scholar

45. M. Grmela, G. Lebon, and C. Dubois, Multiscale thermodynamics and mechanics of heat, Phys. Rev. E, vol. 83, 2011.10.1103/PhysRevE.83.061134Search in Google Scholar

46. T. Ruggeri, Can constitutive relations be represented by non-local equations?, Quart. Appl. Math., vol. 70, pp. 597–611, 2012.10.1090/S0033-569X-2012-01314-3Search in Google Scholar

47. M. Grmela and J. Teichmann, Lagrangian formulation of the Maxwell-Cattaneo hydrodynamics, Int. J. Engng Sci., vol. 21, pp. 297–313, 1983.10.1016/0020-7225(83)90115-5Search in Google Scholar

48. T. Arima, S. Taniguchi, T. Ruggeri, and M. Sugiyama, Extended thermodynamics of dense gases, Cont. Mech. Thermodyn., vol. 24, pp. 271–292, 2012.10.1007/s00161-011-0213-xSearch in Google Scholar

49. Z. Y. Guo and G. W. Hou, Thermal wawe based on thermomass model, Journal of Heat Transfer, vol. 132, p. 072403, 2010.Search in Google Scholar

50. V. A. Cimmelli and W. Kosinski, Nonequilibrium semiempirical temperature in materials with thermal relaxation, Arch. Mech., vol. 43, p. 753, 1991.Search in Google Scholar

51. B. L. Holian and P. M. Mareschal, Heat-flow equation motivated by the ideal-gas shock wave, Phys. Rev. E, vol. 82, p. 026707, 2010.Search in Google Scholar

52. L. Onsager and S. Machlup, Fluctuations and irreversible processes, Phys Rev., vol. 91, pp. 1505–1512, 1953.Search in Google Scholar