Nonlinear closure relations theory for transport processes in nonequilibrium systems

Physical Review E - Statistical, Nonlinear, and Soft Matter Physics

Volumn 79, Issue 5, 2009, Pages

  Sonnino, Giorgio ^a  

^a UNIVERSITÉ LIBRE DE BRUXELLES   (Belgium)

Author keywords

[No Author keywords available]

Indexed keywords

ABSENCE OF TURBULENCE; CLOSED FORM SOLUTIONS; CLOSURE EQUATIONS; COVARIANT FORMALISM; ENTROPY PRODUCTION; GENERAL COVARIANCE; LOCAL EQUILIBRIUM; MACROSCOPIC THEORY; MAGNETICALLY CONFINED PLASMAS; MATHEMATICAL FORMALISM; NONEQUILIBRIUM SYSTEM; NONLINEAR CORRECTION; NONLINEAR FLUX-FORCE RELATIONS; ONSAGER; POTENTIAL GRADIENTS; SKEW-SYMMETRIC; THERMODYNAMIC FIELDS; TOKAMAK PLASMAS; TRANSPORT COEFFICIENT; TRANSPORT EQUATION; TRANSPORT PROCESS; TRANSPORT RELATIONS; UNIMOLECULAR; UNIVERSAL CRITERION;

ELECTRIC POTENTIAL; FUSION REACTORS; GRAFTING (CHEMICAL); PLASMA CONFINEMENT; SYNTHESIS (CHEMICAL); THERMODYNAMICS; TRANSMISSION CONTROL PROTOCOL;

NONLINEAR EQUATIONS;

EID: 67549093117     PISSN: 15393755     EISSN: 15502376     Source Type: Journal    
DOI: 10.1103/PhysRevE.79.051126     Document Type: Article

References (56)

1. G. Sonnino and P. Peeters Phys. Plasmas 15, 062309 (2008). 10.1063/1.2939377
2. L. Onsager, Phys. Rev. 37, 405 (1931) 10.1103/PhysRev.37.405
3. L. Onsager, Phys. Rev. 38, 2265 (1931). 10.1103/PhysRev.38.2265
4. I. Prigogine, Etude Thermodynamique des Phénomènes Irréversibles (Desoer, Liège, 1947), Vol. 69, p. 49.
5. Here, we adopt the De Groot-Mazur terminology. The state of the system can be described by a number of independent variables. In general, one distinguishes two types of macroscopic variables. The variables of the first type denoted by the symbol a are even functions of the particle velocities. The other variables denoted by the symbol b are odd functions of the particle velocities. Thermodynamic processes involving only variables a (b) are referred to as a-a (b-b) processes. It is possible to show that the Onsager reciprocal relations read as L μν a-a = L νμ a-a, L μν b-b = L νμ b-b, and L μν a-b =- L νμ b-a.
6. P. Glansdorff and I. Prigogine, Physica (Amsterdam) 20, 773 (1954) 10.1016/S0031-8914(54)80190-X
7. P. Glansdorff and I. Prigogine, Thermodynamic Theory of Structure, Stability and Fluctuations (John Wiley & Sons, London, 1971), Vol. 106.
8. R. Balescu, Transport Processes in Plasmas: Neoclassical Transport (Elsevier, Amsterdam, 1988), Vol 2., p. 170.
9. I. Gyarmati, Acta Chim. Acad. Sci. Hung. 43, 353 (1965).
10. J. C. M. Li, J. Chem. Phys. 37, 1592 (1962). 10.1063/1.1733345
11. P. Van Rysselberghe, J. Chem. Phys. 36, 1329 (1962). 10.1063/1.1732735
12. G. Sonnino, Nuovo Cimento Soc. Ital. Fis., B 115, 1057 (2000)
13. G. Sonnino, Thermodynamic Field Theory: An Approach to Thermodynamics of Irreversible Processes, Proceedings of the 9th International Workshop on Instabilities and Nonequilibrium Structures, edited by, Viña del Mar, (Kluwer, Dordrecht, 2001), p. 291
14. G. Sonnino, A Field Theory Approach to Thermodynamics of Irreversible Processes, Thèse d'Habilitation à Diriger des Recherches (Institut Non Linèaire de Nice, Nice, France, 2002);
15. G. Sonnino, Nuovo Cimento Soc. Ital. Fis., B 118, 1115 (2003).
16. G. Sonnino and J. Evslin, Int. J. Quantum Chem. 107, 968 (2007) 10.1002/qua.21134
17. G. Sonnino and J. Evslin, Phys. Lett. A 365, 364 (2007). 10.1016/j.physleta.2007.01.076
18. Th. De Donder, Bull. Cl. Sci., Acad. R. Belg. 23, 244 (1937)
19. I. Prigogine and R. Hansen, Bull. Cl. Sci., Acad. R. Belg. 28, 301 (1942)
20. I. Prigogine, Bull. Cl. Sci., Acad. R. Belg. 32, 30 (1946)
21. J. Meixner, Ann. Phys. 433, 409 (1942) 10.1002/andp.19424330602
22. J. Meixner, Ann. Phys. 435, 244 (1943). 10.1002/andp.19434350403
23. I. Prigogine, Thermodynamics of Irreversible Processes (John Wiley & Sons, New York, 1954), p. 42.
24. F. L. Hinton and R. D. Hazeltine, Rev. Mod. Phys. 48, 239 (1976). 10.1103/RevModPhys.48.239
25. H. Weyl, Zur Infinitesimalgeometrie; Einordnung der projektiven und der konformen Auffassung (Göttinger Nachrichten, 1921), p. 110.
26. G. Sonnino, Int. J. Quantum Chem. 98, 191 (2004). 10.1002/qua.10873
27. G. Ruppeiner, Phys. Rev. Lett. 50, 287 (1983) 10.1103/PhysRevLett.50.287
28. F. Weinhold, J. Chem. Phys. 63, 2488 (1975) 10.1063/1.431636
29. P. Salamon, J. Nulton, and R. S. Berry, J. Chem. Phys. 82, 2433 (1985) 10.1063/1.448337
30. J. Casas-Vázquez and D. Jou, J. Chem. Phys. 83, 4715 (1985). 10.1063/1.448996
31. M. Grmela and H. C. Ottinger, Phys. Rev. E 56, 6620 (1997) 10.1103/PhysRevE.56.6620
32. H. C. Ottinger, Beyond Equilibrium Thermodynamics (Wiley, Hoboken, 2005), p. 422.
33. In some examples of chemical reactions, the only condition of invariance of entropy production may not be sufficient to assure the equivalent character of two descriptions (Jμ, Xμ) and (Jμ′, X′μ). In Ref. we can find the case where it is also necessary to impose additional invariances of the rate of change in the number of moles. This is necessary to avoid certain paradoxes to which Verschaffelt called its attention (cf., also).
34. We may qualify as thermodynamic tensor (taken as a single noun) a set of quantities where only transformation 22 is involved. This is in order to qualify as a tensor, a set of quantities, which satisfies certain laws of transformation when the coordinates undergo a general transformation. Consequently, every tensor is a thermodynamic tensor but the converse is not true.
35. L. P. Eisenhart, Non-Riemannian Geometry (American Mathematical Society, Providence, Rhode Island, 1927), Vol. 8.
36. J. L. Synge and A. Schild, Tensor Calculus (Dover, New York, 1949), p. 294.
37. J. E. Verschaffelt, Bull. Cl. Sci., Acad. R. Belg. 37, 853 (1951).
38. R. O. Davies, Physica 18, 182 (1952). 10.1016/S0031-8914(52)80021-7
39. Of course, R νλκ μ and Rνλ do not coincide with the Riemannian curvature tensor and the Ricci tensor, respectively (see also Appendix).
40. A. Palatini, Rend. Circ. Mat. Palermo 43, 203 (1919). 10.1007/BF03014670
41. S. Weinberg, Gravitation and Cosmology: Principles and Applications of the General Theory of Relativity (John Wiley & Sons, New York, 1972), p. 142.
42. Equation 66 does not apply in two dimensions. Two-dimensional problem may be met in the limit case of a system driven out of equilibrium by two (independent) scalar thermodynamic forces such as, for example, two chemical affinities (and not when we analyze, for example, a system submitted to two vectorial thermodynamic forces in one dimension), where the diffusion of the chemical species is neglected. This ideal example is, however, analyzed in Ref.. Equation 66 should be replaced by R=2 R1212 /g where R1212 and g indicate the 1212 component of the thermodynamic curvature tensor and the determinant of the matrix gμν, respectively (see, for example, Ref.).
43. S. R. De Groot and P. Mazur, Non-Equilibrium Thermodynamics (Dover, New York, 1984).
44. R. Balescu, Transport Processes in Plasmas: Classical Transport (Elsevier, Amsterdam, 1988), Vol 1.
45. For more details, refer to G. Sonnino, Phys. Rev. Lett. (to be published).
46. H. Haken, Synergetics: Introduction and Advanced Topics (Springer-Verlag, Berlin, Heidelberg, 1976), p. 202
47. A. Puhl, M. Malek Mansour, and M. Mareschal, Phys. Rev. A 40, 1999 (1989) 10.1103/PhysRevA.40.1999
48. H. Haken, Physica D 97, 95 (1996) 10.1016/0167-2789(96)00080-2
49. M. Malek Mansour, J. Dethier, and F. Baras, J. Chem. Phys. 114, 9265 (2001). 10.1063/1.1367389
50. D. Kondepudi and I. Prigogine, Modern Thermodynamics: From Heat Engines to Dissipative Structures (John Wiley & Sons, Chichester, New York, 1998), p. 346.
51. D. Fitts, NonEquilibrium Thermodynamics: A Phenomenological Theory of Irreversible Processes in Fluid Systems (McGraw-Hill, New York, 1962), p. 31.
52. C. Vidal, G. Dewel, and P. Borckmans, Au-delà de l'équilibre (Hermann Iditeurs Des Sciences et des Arts, Paris, 1994), p. 341.
53. Notice that Jν,μ - Jμ,ν is a thermodynamic tensor of second order.
54. H. C. Ottinger, Beyond Equilibrium Thermodynamics (Wiley, New York, 2005), p. 464
55. G. Lebon, J. Casas-Vázquez, and D. Jou, Understanding Nonequilibrium Thermodynamics (Springer, New York, 2008), p. 196.
56. R. Balescu, Aspects in Anomalous Transport in Plasmas, Series in Plasma Physics (Institute of Physics, Bristol, 2005), p. 417.