Quasi-equilibrium closure hierarchies for the Boltzmann equation

Physica A: Statistical Mechanics and its Applications

Volumn 360, Issue 2, 2006, Pages 325-364

  Gorban, Alexander N ^a,b   Karlin, Iliya V ^c  

^a UNIVERSITY OF LEICESTER   (United Kingdom)

^b INSTITUTE OF COMPUTATIONAL MODELING   (Russian Federation)

^c ETH ZURICH   (Switzerland)

Author keywords

Boltzmann equation; Entropy; Fokker Planck equation; Kinetics; MaxEnt; Model reduction

Indexed keywords

APPROXIMATION THEORY; ENTROPY; HIERARCHICAL SYSTEMS; HYDRODYNAMICS; KINETIC THEORY; MATHEMATICAL MODELS; NONLINEAR SYSTEMS; PROBLEM SOLVING; SCATTERING; THERMODYNAMICS; TRANSPORT PROPERTIES;

BOLTZMANN EQUATION; FOKKER-PLANCK EQUATION; KINETICS; MAXENT; MODEL REDUCTION;

STATISTICAL MECHANICS;

EID: 27744535129     PISSN: 03784371     EISSN: None     Source Type: Journal    
DOI: 10.1016/j.physa.2005.07.016     Document Type: Article

References (54)

1. A.N. Gorban, I.V. Karlin, Quasi-equilibrium approximation and non-standard expansions in the theory of the Boltzmann kinetic equation, in: R. G. Khlebopros (Ed.), Mathematical Modelling in Biology and Chemistry. New Approaches, Nauka, Novosibirsk, 1991, pp. 69-117 (in Russian).
2. A.N. Gorban, I.V. Karlin, Quasi-equilibrium closure hierarchies for the Boltzmann equation, 2003, Preprint online: http://arXiv.org/abs/cond-mat/ 0305599.
3. E.T. Jaynes, Information theory and statistical mechanics, in: K.W. Ford (Ed.), Statistical Physics. Brandeis Lectures, vol. 3, Benjamin, New York, 1963, pp. 160-185.
4. D. Zubarev, V. Morozov, G. Röpke, Statistical mechanics of nonequilibrium processes, vol. 1, Basic, Concepts, Kinetic Theory, Akademie Verlag, Berlin, 1996, Relaxation and Hydrodynamic Processes, vol. 2, Akademie Verlag, Berlin, 1997.
5. A.M. Kogan, and L.I. Rozonoer On the macroscopic description of kinetic processes Dokl. AN SSSR 158 3 1964 566 569
6. A.M. Kogan Derivation of Grad-type equations and study of their properties by the method of entropy maximization Prikl. Math. Mech. 29 1 1965 122 133
7. L.I. Rozonoer, Thermodynamics of nonequilibrium processes far from equilibrium, in: Thermodynamics and Kinetics of Biological Processes, Nauka, Moscow, 1980, pp. 169-186.
8. J. Karkheck, and G. Stell Maximization of entropy, kinetic equations, and irreversible thermodynamics Phys. Rev. A 25 6 1984 3302 3327
9. A.N. Gorban, and I.V. Karlin Thermodynamic parameterization Physica A 190 1992 393 404
10. A.N. Gorban, I.V. Karlin, Uniqueness of thermodynamic projector and kinetic basis of molecular individualism, Physica A 336 (3-4) (2004) 391-432, Preprint online: http://arxiv.org/abs/cond-mat/0309638.
11. A.N. Gorban, and I.V. Karlin Method of invariant manifolds and regularization of acoustic spectra Transport Theory Stat. Phys. 23 1994 559 632
12. S.H. Lam, and D.A. Goussis The CSP method for simplifying kinetics Int. J. Chem. Kinetics 26 1994 461 486
13. P. Constantin, C. Foias, B. Nicolaenko, R. Temam, Integral Manifolds and Inertial Manifolds for Dissipative Partial Differential Equations, Applied Mathematical Sciences, vol. 70, Springer, New York, 1988.
14. D.A. Jones, and E.S. Titi C 1 Approximations of inertial manifolds for dissipative nonlinear equations J. Differential Equations 127 1 1996 54 86
15. A.N. Gorban, I.V. Karlin, A.Y. Zinovyev, Constructive methods of invariant manifolds for kinetic problems, Phys. Rep. 396 (4-6) (2004) 197-403, Preprint online: http://arxiv.org/abs/cond-mat/0311017.
16. A.N. Gorban, I.V. Karlin, Invariant Manifolds for Physical and Chemical Kinetics, Springer Lecturer Notes in Physics, vol. 660, 2005.
17. A.N. Gorban Equilibrium Encircling, Equations of Chemical Kinetics and Their Thermodynamic Analysis 1984 Nauka Novosibirsk
18. G.W. Gibbs Elementary Principles of Statistical Mechanics 1960 Dover New York
19. H. Grad On the kinetic theory of rarefied gases Commun. Pure Appl. Math. 2 4 1949 331 407
20. J.T. Alvarez-Romero, and L.S. García-Colín The foundations of informational statistical thermodynamics revisited Physica A 232 1-2 1996 207 228
21. R.E. Nettleton, and E.S. Freidkin Nonlinear reciprocity and the maximum entropy formalism Physica A 158 2 1989 672 690
22. N.N. Orlov, L.I. Rozonoer, The macrodynamics of open systems and the variational principle of the local potential, J. Franklin Inst. 318 (1984) 283-314, 315-347.
23. N.N. Bugaenko, A.N. Gorban, and I.V. Karlin Universal expansion of the triplet distribution function Teoreticheskaya i Matematicheskaya Fisika 88 3 1991 430 441 (Transl.: Theoret. Math. Phys. (1992) 977-985)
24. C.D. Levermore Moment closure hierarchies for kinetic theories J. Stat. Phys. 83 1996 1021 1065
25. R. Balian, Y. Alhassid, and H. Reinhardt Dissipation in many-body systems: a geometric approach based on information theory Phys. Rep. 131 1 1986 1 146
26. P. Ilg, I.V. Karlin, and H.C. Öttinger Canonical distribution functions in polymer dynamics: I. Dilute solutions of flexible polymers Physica A 315 2002 367 385
27. P. Ilg, I.V. Karlin, M. Kröger, and H.C. Öttinger Canonical distribution functions in polymer dynamics: II Liquid-crystalline polymers Physica A 319 2003 134 150
28. P. Ilg, M. Kröger, Magnetization dynamics rheology, and an effective description of ferromagnetic units in dilute suspension, Phys. Rev. E 66 (2002) 021501. Erratum, Phys. Rev. E 67 (2003), 049901(E).
29. P. Ilg, I.V. Karlin, Combined micro-macro integration scheme from an invariance principle: application to ferrofluid dynamics, J. Non-Newtonian Fluid Mech. 120 (1-3) (2004) 33-40, Preprint online: http://arxiv.org/abs/cond-mat/ 0401383.
30. E. Wigner On the quantum correction for thermodynamic equilibrium Phys. Rev. 40 1932 749 759
31. A.O. Caldeira, and A.J. Leggett Influence of damping on quantum interference: an exactly soluble model Phys. Rev. A 31 1985 1059 1066
32. P. Degond, and C. Ringhofer Quantum moment hydrodynamics and the entropy principle J. Stat. Phys. 112 2003 587 627
33. S. Chapman, and T. Cowling Mathematical Theory of Non-uniform Gases third ed. 1970 Cambridge University Press Cambridge
34. A.N. Gorban, and I.V. Karlin General approach to constructing models of the Boltzmann equation Physica A 206 1994 401 420
35. C. Cercignani The Boltzmann Equation and its Applications 1988 Springer New York
36. A.V. Bobylev The Chapman-Enskog and Grad methods for solving the Boltzmann equation Sov. Phys. Dokl. 27 1 1982 29 31
37. A.N. Gorban, and I.V. Karlin Structure and approximations of the Chapman-Enskog expansion Sov. Phys. JETP 73 1991 637 641
38. A.N. Gorban, I.V. Karlin, P. Ilg, and H.C. Öttinger Corrections and enhancements of quasi-equilibrium states J. Non-Newtonian Fluid Mech. 96 2001 203 219
39. M. Slemrod Renormalization of the Chapman-Enskog expansion: isothermal fluid flow and Rosenau saturation J. Stat. Phys. 91 1-2 1998 285 305
40. I.V. Karlin, A.N. Gorban, Hydrodynamics from Grad's equations: what can we learn from exact solutions? Ann. Phys. (Leipzig) 11 (2002) 783-833, Preprint online: http://arXiv.org/abs/cond-mat/0209560.
41. I.V. Karlin On the relaxation of the chemical reaction rate K.I. Zamaraev G.S. Yablonskii Mathematical Problems of Chemical Kinetics 1989 Nauka Novosibirsk 7 42 (in Russian)
42. A.N. Gorban, and I.V. Karlin Scattering rates versus moments: alternative Grad equations Phys. Rev. E 54 1996 R3109
43. D. Jou, J. Casas-Vázquez, and G. Lebon Extended Irreversible Thermodynamics 1993 Springer Berlin
44. A.N. Gorban, and I.V. Karlin New Methods for Solving the Boltzmann Equations 1994 AMSE Press Tassin, France
45. J.O. Hirschfelder, C.F. Curtiss, and R.B. Bird Molecular Theory of Gases and Liquids 1954 Wiley New York
46. J. Dorfman, and H. van Beijeren B. Berne Statistical Mechanics B 1977 Plenum New York
47. J. Kestin, K. Knierim, E.A. Mason, B. Najafi, S.T. Ro, and M. Waldman Equilibrium and transport properties of the noble gases and their mixtures at low density J. Phys. Chem. Ref. Data 13 1 1984 229 303
48. C. Tsallis Possible generalization of Boltzmann-Gibbs statistics J. Stat. Phys. 52 1988 479 487
49. S. Abe, Y. Okamoto (Eds.), Nonextensive Statistical Mechanics and its Applications, Springer, Heidelberg, 2001.
50. S. Kullback Information Theory and Statistics 1959 Wiley New York
51. A.R. Plastino, H.G. Miller, and A. Plastino Minimum Kullback entropy approach to the Fokker-Planck equation Phys. Rev. E 56 1997 3927 3934
52. I.G. Kevrekidis, C.W. Gear, J.M. Hyman, P.G. Kevrekidis, O. Runborg, and C. Theodoropoulos Equation-free, coarse-grained multiscale computation: enabling microscopic simulators to perform system-level analysis Commun. Math. Sci. 1 4 2003 715 762
53. R. Temam, Infinite-dimensional Dynamical Systems in Mechanics and Physics, Applied Mathematical Sciences, vol 68, Springer, New York, 1988 (second edition, 1997).
54. C. Foias, M.S. Jolly, I.G. Kevrekidis, G.R. Sell, and E.S. Titi On the computation of inertial manifolds Phys. Lett. A 131 7-8 1988 433 436