Skip to main content
SpringerLink
Log in

References

  • Chapter
  • First Online:
Book cover Invariant Manifolds for Physical and Chemical Kinetics

Part of the book series: Lecture Notes in Physics ((LNP,volume 660))

  • 495 Accesses

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

REFERENCES

  1. N.G.Van Kampen,Elimination of fast variables, Physics Reports, 124 (1985), 69–160.

    Google Scholar 

  2. Bogolyubov N.N., Dynamic theory problems in statistical physics, Gostekhizdat, Moscow, Leningrad, 1946.

    Google Scholar 

  3. Lyapunov A.M., The general problem of the stability of motion, Taylor & Francis, London, 1992.

    Google Scholar 

  4. Kolmogorov, A.N., On conservation of conditionally periodic motions under small perturbations of the Hamiltonian. Dokl. Akad. Nauk SSSR, 98 (1954), 527–530.

    Google Scholar 

  5. Arnold, V.I., Proof of a theorem of A.N. Kolmogorov on the invariance of quasi–periodic motions under small perturbations of the Hamiltonian. (English translation) Russian Math Surveys, 18 (1963), 9–36.

    Google Scholar 

  6. Moser, J., Convergent series expansions for quasi–periodic motions, Math. Ann., 169 (1967), 136–176.

    Google Scholar 

  7. Moser, J., On invariant manifolds of vector fields and symmetric partial differential equations, Differential Anal., Bombay Colloq. (1964), 227–236.

    Google Scholar 

  8. Sacker, R.J., A new approach to the perturbation theory of invariant surfaces, Comm. Pure. Appl. Math., 18 (1965), 717–732.

    Google Scholar 

  9. Gorban, A.N., Karlin, I.V., Thermodynamic parameterization, Physica A, 190 (1992), 393–404.

    Google Scholar 

  10. Gorban, A.N., Karlin, I.V., 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.

    Google Scholar 

  11. Gorban, A.N., Karlin, I.V., Method of invariant manifolds and regularization of acoustic spectra, Transport Theory and Stat. Phys., 23 (1994), 559–632.

    Google Scholar 

  12. Gorban, A.N., Karlin, I.V., Zinovyev, A.Yu., Constructive methods of invariant manifolds for kinetic problems, Phys. Reports, 396, 4–6 (2004), 197–403. Preprint online: http://arxiv.org/abs/cond-mat/0311017.

    Google Scholar 

  13. Roberts, A.J., Low–dimensional modelling of dynamical systems applied to some dissipative fluid mechanics, in: Nonlinear dynamics from lasers to butterflies, World Scientific, Lecture Notes in Complex Systems, 1, (2003), Rowena Ball and Nail Akhmediev, eds, 257–313.

    Google Scholar 

  14. Gorban, A.N., Karlin, I.V., The constructing of invariant manifolds for the Boltzmann equation, Adv.Model. and Analysis C, 33(3) (1992), 39–54.

    Google Scholar 

  15. Ehrenfest, P., Ehrenfest-Afanasyeva, T., in: Mechanics Enziklopädie der Mathematischen Wissenschaften, Vol.4., Leipzig, 1911. (Reprinted in: Ehrenfest, P., Collected Scientific Papers, North–Holland, Amsterdam, 1959, pp. 213–300.

    Google Scholar 

  16. Hilbert, D., Begrúndung der kinetischen Gastheorie, Mathematische Annalen, 72 (1912), 562–577.

    Google Scholar 

  17. Karlin, I.V., Dukek, G., Nonnenmacher, T.F., Invariance principle for extension of hydrodynamics: Nonlinear viscosity

    Google Scholar 

  18. Santos, A., Nonlinear viscosity and velocity distribution function in a simple longitudinal flow, Phys. Rev. E 62(2000), 6597–6607.

    Google Scholar 

  19. Santos, A., Comments on nonlinear viscosity and Grad’s moment method, Phys. Rev. E 67 (2003), 053201.

    Google Scholar 

  20. Garz o, V., Santos, A., Kinetic theory of gases in shear flows. nonlinear transport, Book series: Fundamental Theories of Physics, Vol. 131, Kluwer, Dordrecht, 2003.

    Google Scholar 

  21. Karlin, I.V., Gorban, A.N., Dukek, G., Nonnenmacher, T.F. Dynamic correction to moment approximations, Phys. Rev. E., 57 (1998), 1668-1672

    Google Scholar 

  22. Gorban, A.N., Karlin, I.V., Method of invariant manifold for chemical kinetics, Chem. Eng. Sci., 58, 21 (2003), 4751–4768. Preprint online: http://arxiv.org/abs/cond-mat/0207231.

    Google Scholar 

  23. Gorban, A.N., Karlin, I.V., Zmievskii, V.B., Dymova S.V., Reduced description in reaction kinetics, Physica A, 275(3–4) (2000), 361–379.

    Google Scholar 

  24. Karlin, I.V., Zmievskii, V.B., Invariant closure for the Fokker–Planck equation, 1998. Preprint online: http://arxiv.org/abs/adap-org/9801004.

    Google Scholar 

  25. Foias, C., Jolly, M.S., Kevrekidis, I.G., Sell, G.R., Titi, E.S., On the computation of inertial manifolds, Physics Letters A, 131, 7–8 (1988), 433–436.

    Google Scholar 

  26. Gorban, A.N., Karlin, I.V., Zmievskii, V.B., Nonnenmacher, T.F., Relaxational trajectories: global approximations, Physica A, 231 (1996), 648–672.

    Google Scholar 

  27. Gorban, A.N., Karlin, I.V., Zmievskii, V.B., Two–step approximation of space–independent relaxation, Transp.Theory Stat. Phys., 28(3) (1999), 271–296.

    Google Scholar 

  28. Guckenheimer, J., Vladimirsky, A., A fast method for approximating invariant manifolds, SIAM Journal on Applied Dynamical Systems, 3, 3 (2004), 232–260.

    Google Scholar 

  29. Gorban, A.N., Karlin, I.V., Ilg, P., and Öttinger, H.C., Corrections and enhancements of quasi–equilibrium states, J.Non–Newtonian Fluid Mech. 96 (2001), 203–219.

    Google Scholar 

  30. Gorban, A.N., Karlin, I.V., Öttinger, H.C., and Tatarinova, L.L., Ehrenfest’s argument extended to a formalism of nonequilibrium thermodynamics, Phys. Rev. E 63 (2001), 066124.

    Google Scholar 

  31. Gorban, A.N., Karlin, I.V., Reconstruction lemma and fluctuation–dissipation theorem, Revista Mexicana de Fisica 48, Supl. 1 (2002), 238–242.

    Google Scholar 

  32. Gorban, A.N., Karlin, I.V., Macroscopic dynamics through coarse–graining: A solvable example, Phys. Rev. E, 56 (2002), 026116.

    Google Scholar 

  33. Gorban, A.N., Karlin, I.V., Geometry of irreversibility, in: Recent Developments in Mathematical and Experimental Physics, Volume C: Hydrodynamics and Dynamical Systems, Ed. F. Uribe, Kluwer, Dordrecht, 2002, 19–43.

    Google Scholar 

  34. Karlin, I.V., Tatarinova, L.L., Gorban, A.N., Öttinger, H.C., Irreversibility in the short memory approximation, Physica A, 327, 3–4 (2003), 399–424. Preprint online: http://arXiv.org/abs/cond-mat/0305419 v1 18 May 2003.

    Google Scholar 

  35. Karlin, I.V., Ricksen, A., Succi, S., Dissipative quantum dynamics from Wigner distributions, in: Quantum Limits to the Second Law: First International Conference on Quantum Limits to the Second Law, San Diego, California (USA), 29–31 July 2002, AIP Conference Proceedings, 643, 19–24.

    Google Scholar 

  36. Wigner, E., On the quantum correction for thermodynamic equilibrium, Phys. Rev., 40 (1932), 749–759.

    Article  Google Scholar 

  37. Caldeira, A.O., Leggett, A.J. Influence of damping on quantum interference: An exactly soluble model, Phys. Rev. A, 31 (1985), 1059–1066.

    Google Scholar 

  38. Filinov, V.S., Wigner approach to quantum statistical mechanics and quantum generalization molecular dynamics method. 1, Mol. Phys., 88 (1996), 1517–1528; 2, ibidem 1529-1539.

    Google Scholar 

  39. Calzetta, E.A., Hu, B.L., Correlation entropy of an interacting quantum field and H-theorem for the O(N) model, Phys. Rev. D, 68 (2003), 065027.

    Google Scholar 

  40. Gorban, A.N., Karlin, I.V., Short–wave limit of hydrodynamics: a soluble example, Phys. Rev. Lett. 77 (1996), 282–285.

    Google Scholar 

  41. Karlin, I.V., Exact summation of the Chapman-Enskog expansion from moment equations, J. Physics A: Math. Gen., 33 (2000), 8037–8046.

    Google Scholar 

  42. Karlin, I.V., Gorban, A.N., 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.

    Google Scholar 

  43. Gorban, A.N., Karlin, I.V., Structure and approximations of the Chapman–Enskog expansion, Sov. Phys. JETP 73 (1991), 637–641.

    Google Scholar 

  44. Gorban, A.N., Karlin, I.V., Structure and approximations of the Chapman Enskog expansion for linearized Grad equations, Transport Theory and Stat. Phys. 21 (1992), 101-117

    Google Scholar 

  45. Karlin, I.V., Simplest nonlinear regularization, Transport Theory and Stat. Phys., 21 (1992), 291–293.

    Google Scholar 

  46. Fenichel, N., Persistence and smooothness of invariant manifolds for flows, Indiana Univ. Math. J., 21 (1971), 193–226.

    Google Scholar 

  47. Hirsch, M.W., Pugh, C. Shub, M.Invariant manifolds, Lecture Notes in Mathematics, V. 583, Springer, NY, 1977.

    Google Scholar 

  48. Jones, D.A., Stuart, A.M., Titi, E.S., Persistence of invariant sets for dissipative evolution equations, Journal of Mathematical Analysis and Applications, 219, 2 (1998), 479–502.

    Google Scholar 

  49. De la Llave, R., Invariant manifolds associated to invariant subspaces without invariant complements: a graph transform approach, Mathematical Physics Electronic Journal, 9 (2003). http://www.ma.utexas.edu/mpej/MPEJ.html

    Google Scholar 

  50. Poincaré, H.: Les méthodes nouvelles de la mécanique céleste. Vols. 1–3. Gauthier–Villars, Paris, 1892/1893/1899.

    Google Scholar 

  51. Beyn, W.-J., W. Kless Numerical Taylor expansions of invariant manifolds in large dynamical systems, Numerische Mathematik 80 (1998), 1–38.

    Google Scholar 

  52. Kazantzis, N., Singular PDEs and the problem of finding invariant manifolds for nonlinear dynamical systems, Physics Letters, A 272 (4) (2000), 257–263.

    Google Scholar 

  53. Shirkov, D.V., Kovalev, V.F., Bogoliubov renormalization group and symmetry of solution in mathematical physics, Physics Reports, 352 (2001), 219–249. Preprint online: http://arxiv.org/abs/hep-th/0001210.

    Google Scholar 

  54. Zinn-Justin, J., Quantum field theory and critical phenomena, Clarendon Press, Oxford, 1989.

    Google Scholar 

  55. Pashko O., Oono, Y., The Boltzmann equation is a renormalization group equation, Int. J. Mod. Phys. B, 14 (2000), 555–561.

    Google Scholar 

  56. Kunihiro T., A geometrical formulation of the renormalization group method for global analysis, Prog. Theor. Phys. 94 (1995), 503–514; Erratum: ibid. 95 (1996), 835. Preprint online: http://arxiv.org/abs/hep-th/9505166.

    Google Scholar 

  57. Ei, S.-I., Fujii, K., Kunihiro, T., Renormalization–group method for reduction of evolution equations; invariant manifolds and envelopes, Annals Phys. 280 (2000), 236–298. Preprint online: http://arxiv.org/abs/hep-th/9905088.

    Google Scholar 

  58. Hatta Y., Kunihiro T. Renormalization group method applied to kinetic equations: roles of initial values and time, Annals Phys. 298 (2002), 24–57. Preprint online: http://arxiv.org/abs/hep-th/0108159.

    Google Scholar 

  59. Degenhard A., Rodrigues-Laguna J. Towards the evaluation of the relevant degrees of freedom in nonlinear partial differential equations,J.Stat. Phys., 106, No. 516 (2002), 1093–1119.

    Google Scholar 

  60. Forter, D., Nelson D.R., Stephen, M.J., Long–time tails and the large–eddy behavior of a randomly stirred fluid, Phys. Rev. Lett. 36 (1976), 867–870.

    Google Scholar 

  61. Forster, D., Nelson D.R., Stephen, M.J., Large–distance and long–time properties of a randomly stirred fluid, Phys. Rev. A 16 (1977), 732–749.

    Google Scholar 

  62. Adzhemyan, L.Ts., Antonov, N.V., Kompaniets, M.V., Vasil’ev, A.N., Renormalization–group approach to the stochastic Navier Stokes equation: Two–loop approximation, International Journal of Modern Physics B, 17, 10 (2003), 2137–2170.

    Google Scholar 

  63. Chen, H., Succi, S., Orszag, S.,Analysis of subgrid scale turbulence using Boltzmann Bhatnagar-Gross-Krook kinetic equation, Phys. Rev. E, 59, R2527–R2530.

    Google Scholar 

  64. Chen, H., Kandasamy, S., Orszag, S., Shock, R., Succi, S., Yakhot, V., Extended Boltzmann Kinetic Equation for Turbulent Flows, Science, 301, 633–636.

    Google Scholar 

  65. Degond, P., Lemou, M., Turbulence Models for Incompressible Fluids Derived from Kinetic Theory, Journal of Mathematical Fluid Mechanics, 4, 3 (2002), 257–284.

    Google Scholar 

  66. Ansumali, S., Karlin, I.V., Succi, S., Kinetic theory of turbulence modeling: Smallness parameter, scaling and microscopic derivation of Smagorinsky model, Physica A, (2004), to appear. Preprint online: http://arxiv.org/ abs/cond-mat/0310618.

    Google Scholar 

  67. Smagorinsky, J., General Circulation Experiments with the Primitive Equations: I. The Basic Equations, Mon. Weather Rev., 91 (1963), 99–164.

    Google Scholar 

  68. Bricmont, J., Gawedzki, K., Kupiainen, A., KAM theorem and quantum field theory. Commun. Math. Phys. 201 (1999), 699–727. E-print mp_arc 98–526, online: http://mpej.unige.ch/mp_arc/c/98/98-517.ps.gz.

    Google Scholar 

  69. Gorban, A.N., Karlin, I.V., Methods of nonlinear kinetics, in: Encyclopedia of Life Support Systems, Encyclopedia of Mathematical Sciences, EOLSS Publishers, Oxford, 2004, http://www.eolss.net/. Preprint online: http://arXiv.org/abs/cond-mat/0306062.

    Google Scholar 

  70. Chapman, S., Cowling, T., Mathematical theory of non-uniform gases, Third edition, Cambridge University Press, Cambridge, 1970.

    Google Scholar 

  71. Galkin V.S., Kogan M.N., Makashev N.K., Chapman-Enskog generalized method, Dokl. Akademii Nauk SSSR, 220 (1975), 304–307.

    Google Scholar 

  72. Bobylev, A.V., The Chapman–Enskog and Grad methods for solving the Boltzmann equation, Sov. Phys. Dokl., 27 (1982), No. 1, 29–31.

    Google Scholar 

  73. Bobylev, A.V., Exact-solutions of the nonlinear Boltzmann-equation and the theory of relaxation of a Maxwellian gas, Theor. Math. Phys., 60 (1984), 820–841.

    Google Scholar 

  74. Bobylev, A.V., Quasi-stationary hydrodynamics for the Boltzmann equation, J. Stat. Phys. 80, (1995), 1063–1083.

    Google Scholar 

  75. Ernst, M.H., Nonlinear Model-Boltzmann equations and exact solutions, Physics Reports, 78 (1981), 1–171.

    Google Scholar 

  76. García-Col ín, L.S., Green, M.S., Chaos, F., The Chapman-Enskog solution of the generalized Boltzmann equation Physica, 32, 2 (1966), 450–478.

    Google Scholar 

  77. Bowen, J.R., Acrivos, A., Oppenheim, A.K., Singular perturbation refinement to quasi–steady state approximation in chemical Kinetics. Chemical Engineering Science, 18 (1963), 177–188.

    Google Scholar 

  78. Segel, L.A., Slemrod, M., The quasi–steady–state assumption: A case study in perturbation. SIAM Rev., 31 (1989), 446–477.

    Google Scholar 

  79. Fraser, S.J., The steady state and equilibrium approximations: A geometrical picture. J. Chem. Phys., 88(8) (1988), 4732–4738.

    Google Scholar 

  80. Roussel, M.R., S.J. Fraser, Geometry of the steady–state approximation: Perturbation and accelerated convergence methods,J. Chem. Phys., 93 (1990), 1072–1081.

    Google Scholar 

  81. Yablonskii, G.S., Bykov, V.I., Gorban, A.N., Elokhin, V.I., Kinetic models of catalytic reactions. Comprehensive Chemical Kinetics, Vol. 32, Compton R. G. ed., Elsevier, Amsterdam (1991).

    Google Scholar 

  82. Vasil’eva A.B., Butuzov V.F., Kalachev L.V., The boundary function method for singular perturbation problems, SIAM (1995).

    Google Scholar 

  83. Strygin V.V., Sobolev V.A., Spliting of motion by means of integral manifolds. Nauka, Moscow (1988).

    Google Scholar 

  84. Roos, H.G., Stynes, M., Tobiska, L., numerical methods for singularly perturbed differential equations: Convection–diffusion and flow problems, Springer Verlag, 1996.

    Google Scholar 

  85. Mishchenko, E.F., Kolesov, Y.S., Kolesov, A.U., Rozov, N.Kh., Asymptotic methods in singularly perturbed systems, Consultants Bureau, 1994.

    Google Scholar 

  86. Novozhilov, I.V., Fractional analysis: Methods of motion decomposition, Birkhäuser, Boston, 1997.

    Google Scholar 

  87. Milik, A., Singular perturbation on the Web, 1997. http://www.ima.umn.edu/ milik/singdir.html#geo:sing.

    Google Scholar 

  88. Gear, C.W., Numerical initial value problems in ordinary differential equations, Prentice–Hall, Englewood Cliffs, NJ (1971).

    Google Scholar 

  89. Rabitz, H., Kramer, M., Dacol, D., Sensitivity analysis in chemical kinetics, Ann. Rev. Phys. Chem., 34, 419–461 (1983).

    Google Scholar 

  90. Lam, S.H., Goussis, D.A., The CSP Method for Simplifying Kinetics, International Journal of Chemical Kinetics, 26 (1994), 461–486.

    Google Scholar 

  91. Valorani, M., Goussis, D.A., Explicit time-scale splitting algorithm for stiff problems: Auto-ignition of gaseous mixtures behind a steady shock, Journal of Computational Physics, 169 (2001), 44–79.

    Google Scholar 

  92. Valorani, M., Najm, H.N., Goussis, D.A., CSP analysis of a transient flame-vortex interaction: time scales and manifolds, Combustion and Flame 134 (2003), 35–53.

    Google Scholar 

  93. Maas, U., Pope, S.B., Simplifying chemical kinetics: intrinsic low– dimensional manifolds in composition space, Combustion and Flame, 88 (1992), 239–264.

    Google Scholar 

  94. Kaper, H.G., Kaper, T.J., Asymptotic analysis of two reduction methods for systems of chemical reactions, Physica D, 165 (2002), 66–93.

    Google Scholar 

  95. Zagaris, A.,H.G. Kaper, T.J. Kaper Analysis of the computational singular perturbation reduction method for chemical kinetics, Journal of Nonlinear Science,14, 1 (2004), 59–91. Preprint on-line: http://arxiv.org/abs/ math.DS/0305355.

    Google Scholar 

  96. Debussche A., Temam, R., Inertial manifolds and slow manifolds. Appl. Math. Lett., 4, 4 (1991), 73–76.

    Google Scholar 

  97. Foias, C., Prodi, G., Sur le comportement global des solutions non stationnaires des equations de Navier-Stokes en dimension deux, Rend. Sem. Mat. Univ. Padova. 39 (1967), 1–34.

    Google Scholar 

  98. Ladyzhenskaya, O.A., A dynamical system generated by Navier-Stokes equations, J. of Soviet Mathematics, 3 (1975), 458–479.

    Google Scholar 

  99. Chueshov, I.D., Theory of functionals that uniquely determine the asymptotic dynamics of infinite-dimentional dissipative systems, Russian Math. Surveys., 53, 4 (1998), 731–776.

    Google Scholar 

  100. Chueshov, I.D., Introduction to the theory of infinite-dimensional dissipative systems, The Electronic Library of Mathematics, 2002, http://rattler. cameron.edu/EMIS/monographs/Chueshov/. [Translated from Russian edition, ACTA Scientific Publishing House, Kharkov, Ukraine, 1999].

    Google Scholar 

  101. Dellnitz, M., Junge, O., Set oriented numerical methods for dynamical systems, in: B. Fiedler, G. Iooss and N. Kopell (eds.): Handbook of Dynamical Systems II: Towards Applications, World Scientific, 2002, 221–264. http://math-www.upb.de/agdellnitz/papers/handbook.pdf.

    Google Scholar 

  102. Dellnitz, M., Hohmann, A. The computation of unstable manifolds using subdivision and continuation, in H.W. Broer et al. (eds.), Progress in Nonlinear Differential Equations and Their Applications 19:449–459, Birkhäuser, Basel / Switzerland, 1996.

    Google Scholar 

  103. Broer, H.W., Osinga, H.M., Vegter, G. Algorithms for computing normally hyperbolic invariant manifolds, Z. angew. Math. Phys. 48 (1997), 480–524.

    Google Scholar 

  104. Garay, B.M., Estimates in discretizing normally hyperbolic compact invariant manifolds of Ordinary Differential Equations, Computers and Mathematics with Applications, 42 (2001), 1103–1122.

    Google Scholar 

  105. Gorban, A.N., Karlin, I.V., Zinovyev, A.Yu., Invariant grids for reaction kinetics, Physica A, 333 (2004), 106–154. Preprint online: http://www.ihes.fr/ PREPRINTS/P03/Resu/resu-P03–42.html.

    Google Scholar 

  106. Zmievskii, V.B., Karlin, I.V., Deville, M., The universal limit in dynamics of dilute polymeric solutions, Physica A, 275(1–2) (2000), 152–177.

    Google Scholar 

  107. Theodoropoulos, C., Qian, Y.H., Kevrekidis, I.G., Coarse stability and bifurcation analysis using time-steppers: a reaction-diffusion example, Proc. Nat. Acad. Sci., 97 (2000), 9840–9843.

    Google Scholar 

  108. Kevrekidis, I.G., Gear, C.W., Hyman, J.M., Kevrekidis, P.G., Runborg, O., Theodoropoulos, C., Equation-free, coarse-grained multiscale computation: enabling microscopic simulators to perform system-level analysis, Comm. Math. Sci., 14 (2003), 715–762.

    Google Scholar 

  109. Ilg P., Karlin, I.V., Validity of macroscopic description in dilute polymeric solutions, Phys. Rev. E 62 (2000), 1441–1443.

    Google Scholar 

  110. Ilg, P., De Angelis, E., Karlin, I.V., Casciola, C.M., Succi, S., Polymer dynamics in wall turbulent flow, Europhys. Lett., 58 (2002), 616–622.

    Google Scholar 

  111. Boltzmann, L., Lectures on gas theory, University of California Press, 1964.

    Google Scholar 

  112. Cercignani, C., The Boltzmann equation and its applications, Springer, New York, 1988.

    Google Scholar 

  113. Cercignani, C., Illner, R., Pulvirent, M., The mathematical theory of dilute gases, Springer, New York, 1994.

    Google Scholar 

  114. Stueckelberg E.C.G., Theoreme H et unitarite de S, Helv. Phys. Acta 25, 5 (1952), 577–580.

    Google Scholar 

  115. Gorban, A.N., Equilibrium encircling. Equations of chemical kinetics and their thermodynamic analysis, Nauka, Novosibirsk, 1984.

    Google Scholar 

  116. Bhatnagar, P.L., Gross, E.P., Krook, M., A model for collision processes in gases. I. Small amplitude processes in charged and neutral one-component systems, Phys. Rev., 94, 3 (1954), 511–525.

    Article  MATH  Google Scholar 

  117. Gorban, A.N., Karlin, I.V., General approach to constructing models of the Boltzmann equation, Physica A, 206 (1994), 401–420.

    Google Scholar 

  118. Lebowitz, J., Frisch, H., Helfand, E., Non–equilibrium distribution functions in a fluid, Physics of Fluids, 3 (1960), 325.

    Google Scholar 

  119. DiPerna, R.J., Lions, P.L., On the Cauchy problem for Boltzmann equation: Global existence and weak stability, Ann. Math, 130 (1989), 321–366.

    MathSciNet  MATH  Google Scholar 

  120. Enskog, D., Kinetische theorie der Vorange in massig verdunnten Gasen. I Allgemeiner Teil, Almqvist and Wiksell, Uppsala, 1917.

    Google Scholar 

  121. Pöschel, Th., Brilliantov, N. V. Kinetic integrals in the kinetic theory of dissipative gases, In: T. Pöschel, N. Brilliantov (eds.) “Granular Gas Dynamics”, Lecture Notes in Physics, Vol. 624, Springer, Berlin, 2003, 131–162.

    Google Scholar 

  122. Broadwell, J.E., Study of shear flow by the discrete velocity method, J. Fluid Mech. 19 (1964), 401–414.

    Google Scholar 

  123. Broadwell, J.E., Shock structure in a simple discerte velocity gas, Phys.Fluids, 7 (1964), 1243–1247.

    Google Scholar 

  124. Palczewski, A., Schneider, J., Bobylev, A.V., A consistency result for a discrete-velocity model of the Boltzmann equation, SIAM Journal on Numerical Analysis, 34, 5 (1997), 1865–1883.

    Google Scholar 

  125. Zwanzig, R., Ensemble method in the theory of irreversibility. J. Chem. Phys., 33, 5 (1960), 1338–1341.

    Google Scholar 

  126. Robertson, B., Equations of motion in nonequilibrium statistical mechanics, Phys. Rev., 144 (1966), 151–161.

    Google Scholar 

  127. Bird, G.A., Molecular gas dynamics and the direct simulation of gas flows, Clarendon Press, Oxford, 1994.

    Google Scholar 

  128. Oran, E.S., Oh, C.K., Cybyk, B.Z., Direct simulation Monte Carlo: recent advances and applications, Annu Rev. Fluid Mech., 30 (1998), 403–441.

    Google Scholar 

  129. Gatignol, R., Theorie cinetique des gaz a repartition discrete de vitesses. Lecture notes in physics, V. 36, Springer, Berlin, etc, 1975.

    Google Scholar 

  130. Frisch, U., Hasslacher, B., Pomeau, Y., Lattice–gas automata for the Navier–Stokes equation, Phys. Rev. Lett., 56 (1986), 1505–1509.

    Google Scholar 

  131. Mcnamara, Gr., Zanetti, G., Use of the Boltzmann-equation to simulate lattice-gas automata, Phys. Rev. Lett., 61 (1988), 2332–2335.

    Article  Google Scholar 

  132. Higuera, F., Succi, S., Benzi, R., Lattice gas – dynamics with enhanced collisions, Europhys. Lett., 9 (1989), 345–349.

    Google Scholar 

  133. Benzi, R., Succi, S., Vergassola, M., The lattice Boltzmann-equation - theory and applications Physics Reports, 222, 3 (1992), 145–197.

    Google Scholar 

  134. Chen, S., Doolen, G.D., Lattice Boltzmann method for fluid flows, Annu. Rev. Fluid. Mech. 30 (1998), 329–364.

    Google Scholar 

  135. Succi, S., The lattice Boltzmann equation for fluid dynamics and beyond, Clarendon Press, Oxford, 2001.

    Google Scholar 

  136. Succi, S., Karlin, I.V., Chen H., Role of the H theorem in lattice Boltzmann hydrodynamic simulations, Rev. Mod. Phys., 74 (2002), 1203–1220.

    Google Scholar 

  137. Karlin, I.V., Gorban, A.N., Succi, S., Boffi, V., Maximum entropy principle for lattice kinetic equations, Phys. Rev. Lett., 81 (1998), 6–9.

    Google Scholar 

  138. Karlin, I.V., Ferrante, A., Öttinger, H.C., Perfect entropy functions of the Lattice Boltzmann method, Europhys. Lett., 47 (1999), 182–188.

    Google Scholar 

  139. Ansumali, S., Karlin, I.V., Stabilization of the Lattice Boltzmann method by the H theorem: A numerical test, Phys. Rev. E, 62 (6), (2000), 7999–8003.

    Google Scholar 

  140. Ansumali, S., Karlin, I.V., Entropy function approach to the lattice Boltzmann method, J. Stat. Phys., 107 (1/2) (2002), 291–308.

    Google Scholar 

  141. Ansumali, S., Karlin, I.V.,öttinger, H.C., Minimal entropic kinetic models for hydrodynamics, Europhys. Lett., 63 (2003), 798–804.

    Google Scholar 

  142. Ansumali, S., Karlin, I.V., Kinetic Boundary condition for the lattice Boltzmann method, Phys. Rev. E, 66 (2002), 026311.

    Google Scholar 

  143. Ansumali, S., Chikatamarla, S.S., Frouzakis, C.E., Boulouchos, K., Entropic lattice Boltzmann simulation of the flow past square cylinder, Int. J. Mod. Phys. C, 15 (2004), 435–445.

    Google Scholar 

  144. Shan, X., He, X., Discretization of the velocity space in the solution of the Boltzmann equation, Phys. Rev. Lett., 80 (1998), 65–67.

    Google Scholar 

  145. Van Beijeren, H, Ernst, M.H., Modified Enskog equation, Physica A, 68, 3 (1973), 437–456.

    Google Scholar 

  146. Marsden, J.E., Weinstein, A., The Hamiltonian structure of the Maxwell-Vlasov equations, Physica D, 4 (1982), 394–406.

    Google Scholar 

  147. Braun W, Hepp K, Vlasov dynamics and its fluctuations in 1-N limit of interacting classical particles, Comm. Math. Phys., 56, 2 (1977), 101–113.

    Google Scholar 

  148. Van Kampen, N.G., Stochastic processes in physics and chemistry, North–Holland, Amsterdam 1981.

    Google Scholar 

  149. Risken, H., The Fokker–Planck equation, Springer, Berlin, 1984.

    Google Scholar 

  150. Hänggi P., Thomas H., Stochastic Processes: Time Evolution, Symmetries and Linear Response, Physics Reports, 88 (1982), 207–319.

    Google Scholar 

  151. Bird, R.B., Curtiss, C.F., Armstrong, R.C., Hassager, O., Dynamics of Polymer Liquids, 2nd edn., Wiley, New York, 1987.

    Google Scholar 

  152. Doi, M., Edwards, S.F., The theory of polymer dynamics, Clarendon Press, Oxford, 1986.

    Google Scholar 

  153. Öttinger, H.C., Stochastic processes in polymeric fluids, Springer, Berlin, 1996.

    Google Scholar 

  154. Grmela, M., Öttinger, H.C., Dynamics and thermodynamics of complex fluids. I. Development of a general formalism, Phys. Rev. E 56 (1997), 6620–6632.

    Google Scholar 

  155. Öttinger, H.C., Grmela, M., Dynamics and thermodynamics of complex fluids. II. Illustrations of a general formalism, Phys. Rev. E, 56 (1997), 6633–6655.

    Google Scholar 

  156. Kullback, S., Information theory and statistics, Wiley, New York, 1959.

    Google Scholar 

  157. Plastino, A.R., Miller, H.G., Plastino, A., Minimum Kullback entropy approach to the Fokker-Planck equation, Physical Review E 56 (1997). 3927–3934.

    Google Scholar 

  158. Gorban, A.N., Karlin, I.V., Family of additive entropy functions out of thermodynamic limit, Phys. Rev. E, 67 (2003), 016104. Preprint online: http:// arxiv.org/abs/cond-mat/0205511.

    Google Scholar 

  159. Gorban, A.N., Karlin, I.V.,öttinger H.C., The additive generalization of the Boltzmann entropy, Phys. Rev. E, 67, 067104 (2003). Preprint online: http:// arxiv.org/abs/cond-mat/0209319.

    Google Scholar 

  160. Gorban, P., Monotonically equivalent entropies and solution of additivity equation, Physica A, 328 (2003), 380-390. Preprint online: http:// arxiv.org/pdf/cond-mat/0304131.

    Google Scholar 

  161. Tsallis, C., Possible generalization of Boltzmann-Gibbs statistics. J. Stat. Phys., 52 (1988), 479–487.

    Google Scholar 

  162. Abe, S., Okamoto, Y. (Eds.), Nonextensive statistical mechanics and its applications, Springer, Heidelberg, 2001.

    Google Scholar 

  163. Dukek, G., Karlin, I.V., Nonnenmacher, T.F., Dissipative brackets as a tool for kinetic modeling, Physica A, 239(4) (1997), 493–508.

    Google Scholar 

  164. Orlov, N.N., Rozonoer, L.I., The macrodynamics of open systems and the variational principle of the local potential, J. Franklin Inst., 318 (1984), 283–314 and 315–347.

    Google Scholar 

  165. Volpert, A.I., Hudjaev, S.I., Analysis in classes of discontinuous functions and the equations of mathematical physics. Dordrecht: Nijhoff, 1985.

    Google Scholar 

  166. Ansumali S., Karlin, I.V., Single relaxation time model for entropic Lattice Boltzmann methods, Phys. Rev. E, 65 (2002), 056312.

    Google Scholar 

  167. Bykov, V.I., Yablonskii, G.S., Akramov, T.A., The rate of the free energy decrease in the course of the complex chemical reaction. Dokl. Akad. Nauk USSR, 234, 3 (1977) 621–634.

    Google Scholar 

  168. Struchtrup, H., Weiss, W., Maximum of the local entropy production becomes minimal in stationary processes, Phys. Rev. Lett., 80 (1998), 5048–5051.

    Google Scholar 

  169. Grmela, M., Karlin, I.V., Zmievski, V.B., Boundary layer minimum entropy principles: A case study, Phys. Rev. E, 66 (2002), 011201.

    Google Scholar 

  170. Dimitrov, V.I., Simple kinetics, Nauka, Novosibirsk, 1982.

    Google Scholar 

  171. Prigogine, I., Thermodynamics of irreversible processes, Interscience, New York, 1961.

    Google Scholar 

  172. Lifshitz, E.M., Pitaevskii L.P., Physical kinetics (Landau L.D. and Lifshitz E.M. Course of Theoretical Physics, V. 10), Pergamon Press, Oxford, 1968.

    Google Scholar 

  173. Constantin, P., Foias, C., Nicolaenko, B., Temam, R., Integral manifolds and inertial manifolds for dissipative partial differential equations, Applied Math. Sci., 1988, Vol. 70 (Springer Verlag, New York).

    Google Scholar 

  174. Robinson, J.C., A concise proof of the “geometric” construction of inertial manifolds, Phy. Lett. A, 200 (1995), 415–417.

    Google Scholar 

  175. Ryashko, L.B., Shnol, E.E., On exponentially attracting invariant manifolds of ODEs, Nonlinearity, 16 (2003), 147–160.

    Google Scholar 

  176. Walter, W., An elementary proof of the Cauchy–Kovalevsky Theorem, Amer. Math. Month- ly 92 (1985), 115–126.

    Google Scholar 

  177. Evans, L.C., Partial differential equations, AMS, Providence, RI, USA, 1998.

    Google Scholar 

  178. Dubinskii, Ju.A., Analytic pseudo–differential operators and their applications. Kluwer Academic Publishers, Book Series: Mathematics And its Applications Soviet Series: Volume 68, 1991.

    Google Scholar 

  179. Levermore, C.D., Oliver, M., Analyticity of solutions for a generalized Euler equation, J. Differential Equations 133 (1997), 321–339.

    Google Scholar 

  180. Oliver, M., Titi, E.S., On the domain of analyticity for solutions of second order analytic nonlinear differential equations, J. Differential Equations 174 (2001), 55–74.

    Google Scholar 

  181. Arnold, V.I., Geometrical methods in the theory of differential equations, Springer– Verlag, New York–Berlin, 1983.

    Google Scholar 

  182. Arnold, V.I., Vogtmann, K., Weinstein, A., Mathematical methods of classical mechanics, Springer Verlag, 1989.

    Google Scholar 

  183. Bogoliubov, N.N., Mitropolskii, Yu.A., Asymptotic Methods in the Theory of Nonlinear Oscillations, Fizmatgiz, Moscow, 1958 (in Russian).

    Google Scholar 

  184. Kazantzis, N., Kravaris, C., Nonlinear observer design using Lyapunov’s auxiliary theorem, Systems Control Lett., 34 (1998), 241–247.

    Google Scholar 

  185. Krener, A.J., Xiao, M., Nonlinear observer design in the Siegel domain, SIAM J. Control Optim. Vol. 41, 3 (2002), 932–953.

    Google Scholar 

  186. Kazantzis, N., Good, Th., Invariant manifolds and the calculation of the long–term asymptotic response of nonlinear processes using singular PDEs, Computers and Chemical Engineering 26 (2002), 999–1012.

    Google Scholar 

  187. Onsager, L., Reciprocal relations in irreversible processes. I. Phys. Rev. 37 (1931), 405–426; II. Phys. Rev. 38 (1931), 2265–2279.

    Google Scholar 

  188. Nettleton, R.E., Freidkin, E.S., Nonlinear reciprocity and the maximum entropy formalism, Physica A, 158, 2 (1989), 672–690.

    Google Scholar 

  189. Grmela, M., Reciprocity relations in thermodynamics, Physica A, 309, 3–4 (2002), 304–328.

    Google Scholar 

  190. Berdichevsky, V.L., Structure of equations of macrophysics, Phys. Rev. E, 68, 6 (2003), 066126.

    Google Scholar 

  191. Wehrl, A., General properties of entropy, Rev. Mod. Phys. 50, 2 (1978), 221–260.

    Article  Google Scholar 

  192. Schlögl, F., Stochastic measures in nonequilibrium thermodynamics, Phys. Rep. 62, 4 (July 1980), 267–380.

    Google Scholar 

  193. Jaynes E.T., Information theory and statistical mechanics, in: Statistical Physics. Brandeis Lectures, V.3, K. W. Ford, ed., New York: Benjamin, 1963, pp. 160–185.

    Google Scholar 

  194. Grabert, H. Projection operator techniques in nonequilibrium statistical mechanics, Springer Verlag, Berlin, 1982.

    Google Scholar 

  195. Zubarev, D., Morozov, V., Röpke, G. Statistical mechanics of nonequilibrium processes, V.1, Basic concepts, kinetic theory, Akademie Verlag, Berlin, 1996, V.2, Relaxation and hydrodynamic processes, Akademie Verlag, Berlin, 1997.

    Google Scholar 

  196. Evans, M.W., Grigolini, P., Pastori Parravicini, G. (Eds.), Memory function approaches to stochastic problems in condensed matter, Advances in Chemical Physics, V. 62, J. Wiley & Sons, New York etc., 1985.

    Google Scholar 

  197. Uhlenbeck, G.E., in: Fundamental problems in statistical mechanics II, edited by E.G.D. Cohen, North Holland, Amsterdam, 1968.

    Google Scholar 

  198. Glimm, J., Jaffe, A., Quantum Physics: A Functional Integral Point of View, Springer, NY, 1981.

    Google Scholar 

  199. Parisi, G., Statistical Field Theory, Addison-Wesley, Reading, Massachusetts, 1988.

    Google Scholar 

  200. Grad, H.Principles of the kinetic theory of gases, in: S. Flügge, ed., Handbuch der Physics, Band 12, Springer, Berlin, 205–294.

    Google Scholar 

  201. Grad, H., On the kinetic theory of rarefied gases, Comm. Pure and Appl. Math. 24, (1949), 331–407.

    Google Scholar 

  202. Hauge, E.H., Exact and Chapman-Enskog Solutions of the Boltzmann Equation for the Lorentz Model Phys. Fluids 13 (1970), 1201–1208.

    Google Scholar 

  203. Titulaer, U.M., A systematic solution procedure for the Fokker-Planck equation of a Brownian particle in the high–friction case, Physica A, 91, 3–4 (1978), 321–344.

    Google Scholar 

  204. Widder, M.E., Titulaer, U.M., Two kinetic models for the growth of small droplets from gas mixtures, Physica A, 167, 3 (1990), 663–675.

    Google Scholar 

  205. Karlin, I.V., Dukek, G., Nonnenmacher, T.F., Gradient expansions in kinetic theory of phonons, Phys. Rev. B 55 (1997), 6324–6329.

    Google Scholar 

  206. Narayanamurti, V., Dynes, R.C., Phys. Rev. B 12 (1975), 1731–1738.

    Google Scholar 

  207. Narayanamurti, V., Dynes, R.C., Andres, K., Propagation of sound and second sound using heat pulses Phys. Rev. B 11 (1975), 2500–2524.

    Google Scholar 

  208. Guyer, R.A., Krumhansl, J.A., Dispersion relation for 2nd sound in solids, Phys. Rev. 133 (1964), A1411–A1417.

    Google Scholar 

  209. Guyer, R.A., Krumhansl, J.A., Solution of linearized phonon Boltzmann equation, Phys. Rev. 148 (1966), 766–778.

    Google Scholar 

  210. Guyer, R.A., Krumhansl, J.A., Thermal conductivity 2nd sound and phonon hydrodynamic phenomena in nonmetallic crystals, Phys. Rev. 148 (1966), 778–788.

    Google Scholar 

  211. H.Beck, in: Dynamical Properties of Solids, Vol. 2, G.K.Horton and A.A.Maradudin, eds., North-Holland, Amsterdam, 1975, p. 207.

    Google Scholar 

  212. Dreyer, W., Struchtrup, H., Heat pulse experiments revisited, Continuum Mech. Thermodyn. 5 (1993), 3-50.

    Google Scholar 

  213. Ranninger, J., Heat-Pulse Propagation in Ionic Lattices, Phys. Rev. B 5 (1972), 3315–3321.

    Google Scholar 

  214. Paszkiewicz, T., Exact and approximate generalized diffusion equation for the Lorentz gas, Physica A, 123 (1984), 161–174.

    Google Scholar 

  215. Jasiukiewicz Cz., Paszkiewicz, T., The explicit time-dependence of moments of the distribution function for the Lorentz gas with planar symmetry in k-space, Physica A, 145 (1987), 239–254.

    Google Scholar 

  216. Jasiukiewicz Cz., Paszkiewicz, T., Wo’zny, J., Crossover from kinetic to diffusive behavior for a class of generalized models of the Lorentz gas, Physica A, 158 (1989), 864–893.

    Google Scholar 

  217. Jasiukiewicz Cz., Paszkiewicz, T., Relaxation of initial spatially unhomogeneous states of phonon gases scattered by point mass defects embedded in isotropic media, Z. Phys. B, 77 (1989), 209–218.

    Google Scholar 

  218. F.Uribe and E.Piña, Comment on “Invariance principle for extension of hydrodynamics: Nonlinear viscosity”, Phys. Rev. E 57 (1998), 3672–3673.

    Google Scholar 

  219. Karlin, I.V., Exact summation of the Chapman–Enskog expansion from moment equations, J. Phys. A: Math.Gen. 33 (2000), 8037–8046.

    Google Scholar 

  220. Slemrod M., Constitutive relations for monatomic gases based on a generalized rational approximation to the sum of the Chapman–Enskog expansion, Arch. Rat. Mech. Anal, 150 (1) (1999), 1–22.

    Google Scholar 

  221. Slemrod M., Renormalization of the Chapman–Enskog expansion: Isothermal fluid flow and Rosenau saturation J. Stat. Phys, 91, 1–2 (1998), 285–305.

    Google Scholar 

  222. Gibbs, G.W., Elementary Principles of Statistical Mechanics, Dover, 1960.

    Google Scholar 

  223. Kogan, A.M., Rozonoer, L.I., On the macroscopic description of kinetic processes, Dokl. AN SSSR 158 (3) (1964), 566–569.

    Google Scholar 

  224. Kogan, A.M., Derivation of Grad–type equations and study of their properties by the method of entropy maximization, Prikl. Math. Mech. 29 (1) (1965), 122–133.

    Google Scholar 

  225. Rozonoer, L.I., Thermodynamics of nonequilibrium processes far from equilibrium, in: Thermodynamics and Kinetics of Biological Processes (Nauka, Moscow, 1980), 169–186.

    Google Scholar 

  226. Karkheck, J., Stell, G., Maximization of entropy, kinetic equations, and irreversible thermodynamics Phys. Rev. A 25, 6 (1984), 3302–3327.

    Google Scholar 

  227. Alvarez-Romero, J.T., García-Colín, L.S., The foundations of informational statistical thermodynamics revisited, Physica A, 232, 1–2 (1996), 207–228.

    Google Scholar 

  228. Eu, B.C., Kinetic theory and irreversible thermodynamics, Wiley, New York, 1992.

    Google Scholar 

  229. Bugaenko, N.N., Gorban, A.N., Karlin, I.V., Universal Expansion of the Triplet Distribution Function, Teoreticheskaya i Matematicheskaya Fisika, 88, 3 (1991), 430–441 (Transl.: Theoret. Math. Phys. (1992) 977–985).

    Google Scholar 

  230. Levermore C.D., Moment Closure Hierarchies for Kinetic Theories, J. Stat. Phys. 83 (1996), 1021–1065.

    Google Scholar 

  231. Balian, R., Alhassid, Y., Reinhardt, H., Dissipation in many–body systems: A geometric approach based on information theory, Physics Reports 131, 1 (1986), 1–146.

    Google Scholar 

  232. Degond, P., Ringhofer, C., Quantum moment hydrodynamics and the entropy principle, J. Stat. Phys., 112 (2003), 587–627.

    Google Scholar 

  233. Gorban, A.N., Karlin, I.V., Quasi–equilibrium approximation and non-standard expansions in the theory of the Boltzmann kinetic equation, in: “Mathematical Modelling in Biology and Chemistry. New Approaches”, ed. R. G. Khlebopros, Nauka, Novosibirsk, P. 69–117 (1991).[in Russian]

    Google Scholar 

  234. Gorban, A.N., Karlin, I.V., Quasi–equilibrium closure hierarchies for the Boltzmann equation [Translation of the first part of the paper, Preprint, 2003, Preprint online: http://arXiv.org/abs/cond-mat/0305599.

    Google Scholar 

  235. Jou, D., Casas-Vázquez, J., Lebon, G., Extended irreversible thermodynamics, Springer, Berlin, 1993.

    Google Scholar 

  236. Müller, I., Ruggeri, T., Extended Thermodynamics, Springer, NY, 1993.

    Google Scholar 

  237. Gorban, A., Karlin, I., New methods for solving the Boltzmann equations, AMSE Press, Tassin, France, 1994.

    Google Scholar 

  238. Hirschfelder, J.O., Curtiss C.F., Bird, R.B., Molecular theory of gases and liquids, J. Wiley, NY, 1954.

    Google Scholar 

  239. Dorfman, J., van Beijeren, H., in: Statistical Mechanics B, B. Berne, ed., Plenum, NY, 1977.

    Google Scholar 

  240. Résibois, P., De Leener, M., Classical kinetic theory of fluids, Wiley, NY, 1977.

    Google Scholar 

  241. Ford, G., Foch, J., in: Studies in Statistical Mechanics, G. Uhlenbeck and J. de Boer, eds., V. 5, North Holland, Amsterdam, 1970.

    Google Scholar 

  242. Van Rysselberge, P., Reaction rates and affinities, J. Chem. Phys., 29, 3 (1958), 640–642.

    Google Scholar 

  243. Feinberg, M., Chemical kinetics of a sertain class, Arch. Rat. Mech. Anal., 46, 1 (1972), 1–41.

    Google Scholar 

  244. Bykov, V.I., Gorban, A.N., Yablonskii, G.S., Description of nonisothermal reactions in terms of Marcelin – de Donder kinetics and its generalizations, React. Kinet. Catal. Lett., 20, 3–4 (1982), 261–265.

    Google Scholar 

  245. De Donder, T., Van Rysselberghe, P., Thermodynamic theory of affinity. A book of principles. Stanford: University Press, 1936.

    Google Scholar 

  246. Karlin, I.V., On the relaxation of the chemical reaction rate, in: Mathematical Problems of Chemical Kinetics, eds. K.I. Zamaraev and G.S. Yablonskii, Nauka, Novosibirsk, 1989, 7–42. [In Russian].

    Google Scholar 

  247. Karlin, I.V., The problem of reduced description in kinetic theory of chemically reacting gas, Modeling, Measurement and Control C, 34(4) (1993), 1–34.

    Google Scholar 

  248. Gorban, A.N., Karlin, I.V., Scattering rates versus moments: Alternative Grad equations, Phys. Rev. E 54 (1996), R3109.

    Google Scholar 

  249. Treves, F., Introduction to pseudodifferential and Fourier integral operators, Plenum, NY, (1982).

    Google Scholar 

  250. Shubin, M.A., Pseudodifferential operators and spectral theory, Nauka, Moscow, (1978).

    Google Scholar 

  251. Dedeurwaerdere, T., Casas-Vázquez, J., Jou, D., Lebon, G., Foundations and applications of a mesoscopic thermodynamic theory of fast phenomena Phys. Rev. E, 53, 1 (1996), 498–506.

    Google Scholar 

  252. Rodríguez, R.F., García-Colín, L.S., López de Haro, M., Jou, D., Pérez-García, C., The underlying thermodynamic aspects of generalized hydrodynamics, Phys. Lett. A, 107, 1 (1985), 17–20.

    Google Scholar 

  253. Struchtrup, H., Torrilhon M., Regularization of Grad’s 13 Moment Equations: Derivation and Linear Analysis, Phys. Fluids, 15 (2003), 2668–2680.

    Google Scholar 

  254. Ilg, P., Karlin, I.V.,öttinger, H.C., Canonical distribution functions in polymer dynamics: I. Dilute solutions of flexible polymers, Physica A, 315 (2002), 367–385.

    Google Scholar 

  255. Krook, M. Wu, T.T., Formation of Mazwellian tails, Phys. Rev. Lett, 36 (1976), 1107-109.

    Google Scholar 

  256. Krook, M. Wu, T.T., Exact solutions of Boltzmann-equation, Phys. Fluids, 20 (1977), 1589-1595.

    Google Scholar 

  257. Ernst, M.H., Hendriks, E.M., Exactly solvable nonlinear Boltzmann-equation Phys. Lett. A 70 (1979), 183–185.

    Google Scholar 

  258. Hendriks, E.M., Ernst, M.H., The Boltzmann-equation for very hard particles Physica A, 120 (1983), 545–565.

    Google Scholar 

  259. Carleman, T., Sur la théorie de léquation intégro-différentielle de Boltzmann, Acta. Math. 60 (1933), 91–146.

    Google Scholar 

  260. Arkeryd, L., Boltzmann-equation. 1. Existence, Arch. Rat. Mech. Anal. 45 (1972), 1; Boltzmann-equation. 2. Full initial value-problem, ibid. 45 (1972), 17.

    Google Scholar 

  261. Truesdell, C., Muncaster, R., Fundamentals of Maxwell’s Kinetic Theory of a Simple Monatomic Gas, Academic Press, NY, 1980.

    Google Scholar 

  262. Bobylev, A.V., Exact solutions to Boltzmann equations, Dokl. Akad. Nauk SSSR 225 (1975), 1296–1299; One class of invariant solutions to Boltzmann-equation, ibid. 231 (1976), 571–574.

    Google Scholar 

  263. Bobylev, A.V., Cercignani, C. Self-similar solutions of the Boltzmann equation and their applications, J. Stat. Phys., 106 (2002), 1039–1071.

    Google Scholar 

  264. Tjon, J.A., Approach to Maxwellian distribution, Phys. Lett. A, 70 (1979), 369–371.

    Google Scholar 

  265. Cornille, H., Nonlinear Kac model - spatially homogeneous solutions and the Tjon effect, J. Stat. Phys., 39 (1985), 181–213.

    Google Scholar 

  266. Ilg, P., Karlin, I.V., Kröger, M.,öttinger H.C., Canonical distribution functions in polymer dynamics: II Liquid–crystalline polymers, Physica A, 319 (2003), 134–150.

    Google Scholar 

  267. Ilg, P, Kröger, M., 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).

    Google Scholar 

  268. Ilg, P., Karlin, I.V., Combined micro–macro integration scheme from an invariance principle: application to ferrofluid dynamics, J. Non–Newtonian Fluid Mech, 2004, to appear. Ppeprint online: http://arxiv.org/abs/cond-mat/0401383.

    Google Scholar 

  269. Courant, R., Friedrichs, K.O., Lewy, H., On the partial difference equations of mathematical physics., IBM Journal (March 1967), 215–234.

    Google Scholar 

  270. Ames, W.F., Numerical Methods for Partial Differential Equations, 2nd ed. (New York: Academic Press), 1977.

    Google Scholar 

  271. Richtmyer, R.D., and Morton, K.W., Difference methods for initial value problems, 2nd ed., Wiley–Interscience, New York, 1967.

    Google Scholar 

  272. Gorban, A.N., Zinovyev, A.Yu., Visualization of data by method of elastic maps and its applications in genomics, economics and sociology. Institut des Hautes Etudes Scientifiques, Preprint. IHES M/01/36. (2001) . Online: http:// www.ihes.fr/PREPRINTS/M01/Resu/resu-M01-36.html.

    Google Scholar 

  273. Jolliffe, I.T., Principal component analysis, Springer–Verlag, 1986.

    Google Scholar 

  274. Callen, H.B., Thermodynamics and an introduction to thermostatistics, Wiley, New York, 1985.

    Google Scholar 

  275. Use of Legendre transforms in chemical thermodynamics (IUPAC Technical Report), Prepared for publication by R.A. Alberty. Pure Appl.Chem., 73, 8 (2001), pp. 1349–1380. Online: http://www.iupac.org/publications/pac/2001/ pdf/7308x1349.pdf.

    Google Scholar 

  276. Aizenberg, L., Carleman’s formulas in complex analysis: Theory and applications, (Mathematics and its applications; V. 244), Kluwer, 1993.

    Google Scholar 

  277. Gorban, A.N., Rossiev, A.A., Wunsch, D.C.II, Neural network modeling of data with gaps: method of principal curves, Carleman’s formula, and other, The talk was given at the USA–NIS Neurocomputing opportunities workshop, Washington DC, July 1999 (Associated with IJCNN’99). Preprint online: http://arXiv.org/abs/cond-mat/0305508.

    Google Scholar 

  278. Gorban, A.N., Rossiev, A.A., Neural network iterative method of principal curves for data with gaps, Journal of Computer and System Sciences International, 38, 5 (1999), 825–831.

    Google Scholar 

  279. Dergachev, V.A., Gorban, A.N., Rossiev, A.A., Karimova, L.M., Kuandykov, E.B., Makarenko, N.G., Steier, P., The filling of gaps in geophysical time series by artificial neural networks, Radiocarbon, 43, 2A (2001), 365–371.

    Google Scholar 

  280. Gorban A., Rossiev A., Makarenko N., Kuandykov Y., Dergachev V., Recovering data gaps through neural network methods, International Journal of Geomagnetism and Aeronomy, 3, 2 (2002), 191–197.

    Google Scholar 

  281. Lewis, R.M., A unifying principle in statistical mechanics, J. Math. Phys., 8 (1967), 1448–1460.

    Google Scholar 

  282. Chorin, A.J., Hald O.H., Kupferman, R., Optimal prediction with memory, Physica D 166 (2002) 239–257.

    Google Scholar 

  283. Hoover, W.G., Time reversibility, computer simulation, and chaos, Advansed series in nonlinear dynamics, V. 13, World Scientific, Singapore, 1999.

    Google Scholar 

  284. Sone, Y., Kinetic theory and fluid dynamics, Birkhäuser, Boston, 2002.

    Google Scholar 

  285. McKean, H.P. Jr., J. Math. Phys. 8, 547 (1967).

    Google Scholar 

  286. Montroll, E.W., Lebowitz, J.L. (Eds.), Studies in Statistical Mechanics, V.IX, North-Holland, 1981.

    Google Scholar 

  287. Del Río-Correa, J.L., García-Colín, L.S., Increase-in-entropy law, Phys. Rev. E 48 (1993), 819–828.

    Google Scholar 

  288. Leontovich, M.A., An Introduction to thermodynamics, GITTL Publ., Moscow, 1950 (in Russian).

    Google Scholar 

  289. Lebowitz, J.L., Bergmann, P.G., New approach to nonequlibrium processes, Phys. Rev., 99 (1955), 578–587.

    Google Scholar 

  290. Lebowitz, J.L., Bergmann, P.G., Irreversible Gibbsian Ensembles, Annals of Physics, 1:1, 1957.

    Google Scholar 

  291. Lebowitz, J.L., Stationary Nonequilibrium Gibbsian Ensembles, Phys. Rev., 114 (1959), 1192–1202.

    Google Scholar 

  292. Lebowitz, J.L., Botzmann’s entropy and time’s arrow, Physics Today, 46 9 (1993), 32–38.

    Google Scholar 

  293. Leff, H.S., Rex, A.F. (Eds.), Maxwell’s Demon 2: Entropy, Classical and Quantum Information, Computing, 2nd edition, IOP, Philadelphia, 2003.

    Google Scholar 

  294. Von Baeyer, H.C., Maxwell’s Demon: Why Warmth Disperses and Time Passes, Random House, 1998.

    Google Scholar 

  295. Pour-El, M.B., Richards, J.I., Computability in Analysis and Physics, Springer Verlag, NY, 1989.

    Google Scholar 

  296. Copeland, B.J., The Church-Turing Thesis, In: The Stanford Encyclopedia of Philosophy, E.N. Zalta (Ed.) (Fall 2002 Edition), On-line: http:// plato.stanford.edu/archives/fall2002/entries/church-turing/.

    Google Scholar 

  297. Feynman, R., The Character of Physical Law, Cox and Wyman, London, 1965. Lecture No. 5.

    Google Scholar 

  298. Gorban, A.N., Karlin, I.V., Geometry of irreversibility: Film of nonequilibrium states, The lecture given on the V Russian National Seminar “Modeling of Nonequilibrium systems”, Krasnoyarsk, Oct. 18–20, 2002, Printed by Krasnoyarsk State Technical University Press, 2002. [In Russian].

    Google Scholar 

  299. Gorban, A.N., Bykov, V.I., Yablonskii, G.S., Essays on chemical relaxation, Novosibirsk: Nauka, 1986.

    Google Scholar 

  300. Verbitskii, V.I., Gorban, A.N., Utjubaev, G.Sh., Shokin, Yu.I., Moore effect in interval spaces, Dokl. AN SSSR. 304, 1 (1989), 17–21.

    Google Scholar 

  301. Bykov, V.I., Verbitskii, V.I., Gorban, A.N., On one estimation of solution of Cauchy problem with uncertainty in initial data and rigt part, Izv. vuzov, Ser. mat., N. 12 (1991), 5–8.

    Google Scholar 

  302. Verbitskii, V.I., Gorban, A.N., Simultaneously dissipative operators and their applications, Sib. Mat. Jurnal, 33, 1 (1992), 26–31.

    Google Scholar 

  303. Gorban, A.N., Shokin, Yu.I., Verbitskii, V.I., Simultaneously dissipative operators and the infinitesimal Moore effect in interval spaces, Preprint (1997). Preprint online: http://arXiv.org/abs/physics/9702021.

    Google Scholar 

  304. Gorban, A.N., Bykov, V.I., Yablonskii, G.S., Thermodynamic function analogue for reactions proceeding without interaction of various substances, Chemical Engineering Science, 41, 11 (1986), 2739–2745.

    Google Scholar 

  305. Gorban, A.N., Verbitskii, V.I., Thermodynamic restriction and quasithermondynamic conditions in reaction kinetics, in: Mathematical problems of chamical kinetics, K.I. Zamaraev, G.S. Yoblonskii (eds.), Nauka, Novosibirsk, 1989, 43–83

    Google Scholar 

  306. Grassberger, P., On the Hausdorff Dimension of Fractal Attractors, J. Stat. Phys. 26 (1981), 173–179.

    Google Scholar 

  307. Grassberger, P. and Procaccia, I., Measuring the Strangeness of Strange Attractors, Physica D 9 (1983), 189–208.

    Google Scholar 

  308. Frederickson, P., Kaplan, J.L., Yorke, E.D., Yorke, J.A., The Lyapunov dimension of strange attractors. J. Differ. Equations 49 (1983), 185–207.

    Google Scholar 

  309. Ledrappier F., Young, L.-S., The metric entropy of diffeomorphisms: I. Characterization of measures satisfying Pesin’s formula; II. Relations between entropy, exponents and dimensions, Annals of Mathematics, 122 (1985), 509–539, 540–574.

    Google Scholar 

  310. Hentschel, H.G.E., Procaccia, I., The infinite number of generalized dimensions of fractals and strange attractors, Physica D: Nonlinear Phenomena, 8 3 (1983), 435–444.

    Google Scholar 

  311. Ilyashenko, Yu.S., On dimension of attractors of k–contracting systems in an infinite dimensional space, Vest. Mosk. Univ. Ser. 1 Mat. Mekh., No. 3 (1983), 52–58.

    Google Scholar 

  312. Nicolis, C., Nicolis, G., Is there a climate attractor?, Nature, 311 (1984), 529–532.

    Google Scholar 

  313. Constantin, P., Foias, C., Temam, R., Attractors representing turbulent flows, Memoirs of the American Mathematical Society, V. 53, No. 314, Providence, 1985.

    Google Scholar 

  314. Dubois, T., Jauberteau, F., Temam, R., Dynamic multilevel methods and the numerical simulation of turbulence, Cambridge Univ. Press, Cambridge, 1999.

    Google Scholar 

  315. Landau, L.D., Lifshitz, E.M., Fluid Mechanics (Landau L.D. and Lifshitz E.M. Course of Theoretical Physics, V.6), Pergamon Press, Oxford, 1993.

    Google Scholar 

  316. Foias, C., Manley O.P., Temam, R., An estimate of the Hausdorff dimension of the attractor for homogeneous decaying turbulence, Physis Letters A. 122 3#x02013;4 (1987), 140-144.

    Google Scholar 

  317. Foias, C., Sell, G.R., Temam R., Inertial manifolds for dissipative nonlinear evolution equations, Journal of Differential Equations, 73 (1988), 309–353.

    Google Scholar 

  318. Temam R., Infinite–dimensional dynamical systems in mechanics and physics, Applied Math. Sci., Vol 68, Springer Verlag, New York, 1988 (Second edition, 1997).

    Google Scholar 

  319. Málek, J., Prazák, D., Finite fractal dimension of the global attractor for a class of non-Newtonian fluids, Applied Mathematics Letters, 13, 1 (2000), 105–110.

    Google Scholar 

  320. Doering, C.R., Wang, X., Attractor dimension estimates for two-dimensional shear flows, Physica D: Nonlinear Phenomena Volume 123, Issues 1–4, 15 November 1998, Pages 206–222

    Google Scholar 

  321. Doering, C.R., Gibbon, J.D., Holm, D.D., Nicolaenko, B., Finite dimensionality in the complex Ginzburg–Landau equation, Contemporary Mathemathics, 99, 1989, 117–141.

    Google Scholar 

  322. Ghidaglia, J.M., Héron, B., Dimension of the attractors associated to the Ginzburg–Landau partial differential equation, Physica D, 28, 3 (1987), 282–304.

    Google Scholar 

  323. Nicolaenko, B. Scheurer, B., Temam, R., Some global dynamical properties of the Kuramoto–Sivashinsky equations: Nonlinear stability and attractors, Physica D, 16, 2 (1985), 155–183.

    Google Scholar 

  324. Duan, J., Ervin, V.J., Dynamics of a Nonlocal Kuramoto–Sivashinsky Equation, J., Diff. Equ., 143 (1998), 243–266.

    Google Scholar 

  325. Nicolaenko, B. Scheurer, B., Temam, R., Some global dynamical properties of a class of pattern formation equations, Comm. Partial Diff. Equ., 14 (1989), 245–297.

    Google Scholar 

  326. Debussche, A., Dettori, L., On the Cahn–Hilliard equation with a logarithmic free energy, Nonlinear Anal, 24 (1995), 1491–1514.

    Article  MathSciNet  MATH  Google Scholar 

  327. Li, D., Zhong, Ch., Global Attractor for the Cahn–Hilliard System with Fast Growing Nonlinearity, Journal of Differential Equations, 149 (1998), 191–210.

    Google Scholar 

  328. Miranville, A., Zelik, S., Robust exponential attractors for singularly perturbed phase–field type equations, Electronic J., of Diff. Eqns., 2002 (2002), No. 63, 1–28

    Google Scholar 

  329. Miranville, A., Piétrus, A., Rakotoson, J.M., Dynamical aspect of a generalized Cahn–Hilliard equation based on a microforce balance, Asymptotic Anal., 16 (1998), 315–345.

    MathSciNet  Google Scholar 

  330. Grinfeld, M., Novick-Cohen, A., The viscous Cahn-Hilliard equation: Morse decomposition and structure of the global attractor, Trans. Amer. Math. Soc., 351 (1999), 2375–2406.

    Google Scholar 

  331. Cahn, J.W., Hilliard, J.E., Free energy of a nonuniform systems. I. Interfacial free energy, J. Chem. Phys., 28 (1958), 258–267.

    Google Scholar 

  332. Allen, S., Cahn, J.W., A microscopic theory for antiphase boundary motion and its application to antiphase domain coarsening, Acta Metall., 27 (1979), 1084–1095.

    Google Scholar 

  333. Babin, A.V., Vishik, M.I., Attractors of evolutionary equations (Studies in mathematics and its application, V.~25), Elsevier, NY, 1992.

    Google Scholar 

  334. Vishik, M.I., Asymptotic behaviour of solutions of evolutionary equations, Cambridge University Press, 1993.

    Google Scholar 

  335. Chepyzhov, V.V., Vishik M.I., Attractors for equations of mathematical physics, AMS Colloquium Publications, V. 49, American Mathematical Society, Providence, 2002.

    Google Scholar 

  336. Efendiev, M., Miranville, A., The dimension of the global attractor for dissipative reaction-diffusion systems, Applied Mathematics Letters, 16, 3 (2003), 351–355.

    Google Scholar 

  337. Haraux, A, Two remarks on dissipative hyperbolic problems, in: Nonlinear partial differential equations and their applications (H. Brezis, J.L., Lions eds), Research Notes Maths., Vpl 112, Pitman, Boston, 1985, 161-179.

    Google Scholar 

  338. Ghidaglia, J.M., Temam, R., Attractors for damped nonlinear hyperbolic equations, J. Math. Pures Appl., 66 (1987), 273–319.

    Google Scholar 

  339. Ladyzhenskaya, O.A., On finding the minimal global attractors for the Navier–Stokes equation and other PDEs, Uspekhi Mat. Nauk, 42 (1987), 25–60; Engl. transl. Russian Math Surveys, 42 (1987).

    Google Scholar 

  340. Constantin, P., Kevrekidis, I., Titi, E.S., Remarks on a Smoluchowski equation, Discrete and Continuous Dynamical Systems, 11 (2004), 101–112.

    Google Scholar 

  341. Foias, C., Sell, G.R., Titi, E.S., Exponential tracking and approximation of inertial manifolds for dissipative nonlinear equations Journal of Dynamics and Differential Equations, 1 (1989), 199–244.

    Google Scholar 

  342. Jones, D.A., Titi, E.S., C1 Approximations of inertial manifolds for dissipative nonlinear equations, Journal of Differential Equations, 127, 1 (1996), 54–86.

    Google Scholar 

  343. Robinson, J.C., Computing inertial manifolds, Discrete and Continuous Dynamical Systems, 8, 4 (2002), 815-833.

    Google Scholar 

  344. Christofides, P.D., Nonlinear and robust control of partial differential equation systems: Methods and applications to transport–reaction processes, Birkhäuser, Boston, 2001.

    Google Scholar 

  345. Chepyzhov, V.V., Ilyin, A.A., A note on the fractal dimension of attractors of dissipative dynamical systems, Nonlinear Analysis, 44 (2001), 811–819.

    Google Scholar 

  346. Marion, M., Temam, R., Nonlinear Galerkin methods, SIAM J. Numer. Anal., 26 (1989), 1139–1157.

    Google Scholar 

  347. Jones, C., Kaper, T., Kopell, N., Tracking invariant manifolds up to exponentially small errors, SIAM J. Math., Anal. 27 (1996), 558–577.

    Google Scholar 

  348. Yinnian He, Mattheij, R.M.M., Stability and convergence for the reform postprocessing Galerkin method, Nonlinear Anal. Real World Appl., 4 (2000), 517–533.

    Google Scholar 

  349. Garsia-Archilla, B., Novo, J., Titi E.S., Postprocessing the Galerkin method: a novel approach to approximate inertial manifolds, SIAM J. Numer. Anal., 35 (1998), 941–972.

    Google Scholar 

  350. Margolin, L.G., Titi, E.S., Wynne, S., The postprocessing Galerkin and nonlinear Galerkin methods – a truncation analysis point of view, SIAM, Journal of Numerical Analysis, 41, 2 (2003), 695–714.

    Google Scholar 

  351. Novo, J., Titi, E.S., Wynne, S., Efficient methods using high accuracy approximate inertial manifolds, Numerische Mathematik, 87 (2001), 523–554.

    Google Scholar 

  352. Fenichel, N., Geometric singular perturbation theory for ordinary differential equations, J. Diff Eq., 31 (1979), 59–93

    Google Scholar 

  353. Jones, C.K.R.T., Geometric singular perturbation theory, in: Dynamical Systems (Montecatini Terme, 1904), L. Arnold (ed.), Lecture Notes in Mathematics, 1609, Springer-Verlag. Berlin, 1994, 44–118.

    Google Scholar 

  354. Bird, R.B., Wiest, J.M., Constitutive equations for polymeric liquids, Annu. Rev. Fluid Mech. 27 (1995), 169.

    Google Scholar 

  355. Warner, H.R., Kinetic theory and rheology of dilute suspensions of finitely extendible dumbbells, Ind. Eng. Chem. Fundamentals 11 (1972), 379.

    Google Scholar 

  356. Oseen, C.W., Ark. f. Mat. Astr. og Fys. 6 No. 29 (1910) 1.

    Google Scholar 

  357. Burgers, J.M., Verhandelingen Koninkl. Ned. Akad. Wetenschap. 16 Sect. 1, Chap. (1938), 113.

    Google Scholar 

  358. Rotne, J., Prager, S., Variational treatment of hydrodynamic interaction, J. Chem. Phys., 50 (1969) 4831.

    Google Scholar 

  359. Yamakawa, H., Transport properties of polymer chain in dilute solution: Hydrodynamic interaction, J. Chem. Phys. 53 (1970), 436.

    Google Scholar 

  360. Noll, W., A mathematical theory of the mechanical behavior of continuous media, Arch. Ratl. Mech. Anal., 2 (1958), 197.

    Google Scholar 

  361. Astarita, G., Marrucci, G., Principles of non-Newtonian fluid mechanics, McGraw–Hill, London, 1974.

    Google Scholar 

  362. Oldroyd, J.G., Non–Newtonian effects in steady motion of some idealized elastico–viscous liquids, Proc. Roy. Soc. A245 (1958), 278.

    Google Scholar 

  363. Herrchen M., Öttinger, H.C., A detailed comparison of various FENE dumbbell models, J. Non–Newtonian Fluid Mech. 68 (1997), 17

    Google Scholar 

  364. Kröger, M., Simple models for complex nonequilibrium fluids, Physics Reports, 390, 6 (2004), 453–551.

    Google Scholar 

  365. Bird, R.B., Dotson, R.B., Jonson, N.J., Polymer solution rheology based on a finitely extensible bead-spring chain model, J. Non-Newtonian Fluid Mech.7 (1980), 213-235; Corrigendum 8 (1981), 193

    Google Scholar 

  366. Char B.W. et al., Maple V Language Reference Manual, Springer–Verlag, New York, 1991.

    Google Scholar 

  367. Kato, T., Perturbation theory for linear operators, Springer–Verlag, Berlin, 1976.

    Google Scholar 

  368. Thiffeault, J.-L., Finite extension of polymers in turbulent flow, Physics Letters A 308, 5–6 (2003), 445–450.

    Google Scholar 

  369. Gorban, A.N., Gorban, P.A., Karlin I.V., Legendre integrators, post-processing and quasiequilibrium, J. Non–Newtonian Fluid Mech., 120 (2004), 149–167. Preprint on-line: http://arxiv.org/pdf/cond-mat/0308488.

    Google Scholar 

  370. Perkins, Th.T., Smith, D.E., Chu, S., Single polymer dynamics in an elongational flow, Science, 276, 5321 (1997), 2016-2021.

    Google Scholar 

  371. De Gennes, P.G., Molecular individualism, Science, 276, 5321 (1997), 1999-2000.

    Google Scholar 

  372. Smith, D.E., Babcock, H.P., Chu, S., Single-polymer dynamics in steady shear flow. Science 283 (1999), 1724–1727.

    Google Scholar 

  373. Arnold, V.I., Varchenko, A.N., Gussein-Zade, S.M., Singularities of differnetiable maps, Brickhäuser, Boston, 1985-1988. 2 vol.

    Google Scholar 

  374. Langer, J.S., Bar-on, M., Miller, H.D., New computational method in the theory of spinodal decomposition, Phys. Rev. A, 11, 4 (1975), 1417–1429.

    Google Scholar 

  375. Grant, M., San Miguel, M., Vinals, J., Gunton, J.D., Theory for the early stages of phase separation: The long-range-force limit, Phys. Rev. B, 31, 5 (1985), 3027–3039.

    Google Scholar 

  376. Kumaran, V., Fredrickson, G.H., Early stage spinodal decomposition in viscoelastic fluids, J. Chem. Phys., 105, 18 (1996), 8304–8313.

    Google Scholar 

  377. Darwin, Ch., On the origin of species by means of natural selection, or preservation of favoured races in the struggle for life: A Facsimile of the First Edition, Harvard, 1964. http://www.literature.org/authors/darwin-charles/the-origin-of-species/

    Google Scholar 

  378. Haldane, J.B.S., The Causes of Evolution, Princeton Science Library, Princeton University Press, 1990.

    Google Scholar 

  379. Mayr, E. Animal Species and Evolution. Cambridge MA: Harvard University Press 1963

    Google Scholar 

  380. Ewens, W.J., Mathematical Population Genetics. Springer-Verlag, Berlin, 1979.

    Google Scholar 

  381. Rozonoer, L.I., Sedyh, E.I., On the mechanisms of of evolution of self-reproduction systems, 1, Automation and Remote Control, 40, 2 (1979), 243–251; 2, ibid., 40, 3 (1979), 419–429; 3, ibid, 40, 5 (1979), 741–749.

    Google Scholar 

  382. Gorban, A.N., Dynamical systems with inheritance, in: Some problems of community dynamics, R.G. Khlebopros (ed.), Institute of Physics RAS, Siberian Branch, Krasnoyarsk, 1980 [in Russian].

    Google Scholar 

  383. Semevsky, F.N., Semenov S.M., Mathematical modeling of ecological processes, Gidrometeoizdat, Leningrad, 1982 [in Russian].

    Google Scholar 

  384. Gorban, A.N., Khlebopros, R.G., Demon of Darwin: Idea of optimality and natural selection, Nauka (FizMatGiz), Moscow, 1988 [in Russian].

    Google Scholar 

  385. Zakharov, V.E., L’vov, V.S., Starobinets, S.S., Turbulence of spin-waves beyond threshold of their parametric-excitation, Uspekhi Fizicheskikh Nauk 114, 4 (1974), 609–654; English translation Sov. Phys. - Usp. 17, 6 (1975), 896–919.

    Google Scholar 

  386. Zakharov, V.E., L’vov, V.S., Falkovich, G.E., Kolmogorov spectra of turbulence, vol. 1 Wave Turbulence. Springer, Berlin, 1992.

    Google Scholar 

  387. L’vov, V.S., Wave turbulence under parametric excitation applications to magnets, Springer, Berlin, Heidelberg, 1994.

    Google Scholar 

  388. Ezersky A.B., Rabinovich M.I., Nonlinear-wave competition and anisotropic spectra of spatiotemporal chaos of Faraday ripples, Europhysics Letters 13, 3 (1990), 243–249.

    Google Scholar 

  389. Krawiecki, A, Sukiennicki, A., Marginal synchronization of spin-wave amplitudes in a model for chaos in parallel pumping, Physica Status Solidi B–Basic Research 236, 2 (2003), 511–514.

    Google Scholar 

  390. Vorobev, V.M., Selection of normal variables for unstable conservative media, Zhurnal Tekhnicheskoi Fiziki, 62, 8 (1992), 172–175.

    Google Scholar 

  391. Seminozhenko, V.P., Kinetics of interacting quasiparticles in strong external fields. Phys. Reports, 91, 3 (1982), 103–182.

    Google Scholar 

  392. Haken, H., Synergetics, an introduction. Nonequilibrium phase transitions and self–organization in physics, chemistry and biology, Springer, Berlin, Heidelberg, New York, 1978.

    Google Scholar 

  393. Bourbaki, N., Elements of mathematics - Integration I, Springer, Berlin, Heidelberg, New York, 2003.

    Google Scholar 

  394. Smale, S., Structurally stable systems are not dense, Amer. J. Math., 88 (1966), 491–496.

    Google Scholar 

  395. Birkhoff, G.D., Dynamical systems, AMS Colloquium Publications, Providence, 1927. Online: http://www.ams.org/online _bks/coll9/

    Google Scholar 

  396. Hasselblatt, B., Katok, A. (Eds.), Handbook of Dynamical Systems, Vol. 1A, Elsevier, 2002.

    Google Scholar 

  397. Katok, A., Hasselblat, B., Introduction to the Modern Theory of Dynamical Systems, Encyclopedia of Math. and its Applications, Vol. 54, Cambridge University Press, 1995.

    Google Scholar 

  398. Levin, L.A., Randomness conservation inequalities; Information and independence in mathematical theories, Information and Control, 61, 1 (1984), 15–37.

    Google Scholar 

  399. Gause, G.F., The struggle for existence, Williams and Wilkins, Baltimore, 1934. Online: http://www.ggause.com/Contgau.htm.

    Google Scholar 

  400. Volterra, V., Lecons sur la théorie mathematique de la lutte pour la vie, Gauthier-Villars, Paris, 1931.

    Google Scholar 

  401. Gromov, M., A dynamical model for synchronisation and for inheritance in micro-evolution: a survey of papers of A.Gorban, The talk given in the IHES seminar, “Initiation to functional genomics: biological, mathematical and algorithmical aspects”, Institut Henri Poincar’e, November 16, 2000.

    Google Scholar 

  402. Kuzenkov, O.A., Weak solutions of the Cauchy problem in the set of Radon probability measures, Differential Equations, 36, 11 (2000), 1676–1684.

    Google Scholar 

  403. Kuzenkov, O.A., A dynamical system on the set of Radon probability measures, Differential Equations, 31, 4 (1995), 549–554.

    Google Scholar 

  404. Diekmann, O. A. beginner’s guide to adaptive dynamics, in: Mathematical modelling of population dynamics, Banach Center Publications, V.~63, Institute of Mathematics Polish Academy of Sciences, Warszawa, 2004, 47–86.

    Google Scholar 

  405. Blekhman, I.I., Synchronization in science and technology. ASME Press, N.Y., 1988.

    Google Scholar 

  406. Pikovsky, A., Rosenblum, M., Kurths, J., Synchronization: A Universal Concept in Nonlinear Science, Cambridge University Press, 2002.

    Google Scholar 

  407. Josić, K., Synchronization of chaotic systems and invariant manifolds, Nonlinearity 13 (2000) 1321–1336.

    Google Scholar 

  408. Mosekilde, E., Maistrenko, Yu., Postnov, D., Chaotic synchronization: Applications to living systems, World Scientific, Singapore, 2002.

    Google Scholar 

  409. Cooper, S., Minimally disturbed, multi-cycle, and reproducible synchrony using a eukaryotic “baby machine”, Bioessays 24 (2002), 499–501.

    Google Scholar 

  410. Lielens, G., Halin, P., Jaumin, I., Keunings,R., Legat, V., New closure approximations for the kinetic theory of finitely extensible dumbbells, J.Non–Newtonian Fluid Mech. 76 (1998), 249–279.

    Google Scholar 

  411. Ilg, P., Karlin, I.V., Öttinger, H.C., Generating moment equations in the Doi model of liquid–crystalline polymers, Phys. Rev. E, 60 (1999), 5783–5787.

    Google Scholar 

  412. Phan–Thien, N., Goh, C.G., Atkinson, J.D., The motion of a dumbbell molecule in a torsional flow is unstable at high Weissenberg number, J. Non–Newtonian Fluid Mech. 18, 1 (1985), 1–17.

    Google Scholar 

  413. Goh, C.G., Phan–Thien, N., Atkinson, J.D., On the stability of a dumbbell molecule in a continuous squeezing flow, Journal of Non–Newtonian Fluid Mechanics, 18, 1 (1985), 19–23.

    Google Scholar 

  414. Karlin, I.V., Ilg, P.,öttinger, H.C., Invariance principle to decide between micro and macro computation, in: Recent Developments in Mathematical and Experimental Physics, Volume C: Hydrodynamics and Dynamical Systems, Ed. F. Uribe, Kluwer, Dordrecht, 2002, 45–52.

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Multidisciplinary Research Institute 213 Department of Mathematics, University of Leicester, University Road, LE1 7RH, Leicester, United Kingdom

    Alexander N. Gorban

  2. Institute of Computational Modeling, Russian Academy of Sciences, 660036, Krasnoyarsk, Russia

    Alexander N. Gorban

  3. Institut für Energietechnik, Clausiusstrasse 33 ETH Zentrum, CLT A 6, 8092, Zürich, Switzerland

    Ilya V. Karlin

  4. Institute of Computational Modeling, Russian Academy of Sciences, 660036, Krasnoyarsk, Russia

    Ilya V. Karlin

Authors
  1. Alexander N. Gorban
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Ilya V. Karlin
    View author publications

    You can also search for this author in PubMed Google Scholar

Rights and permissions

Reprints and permissions

About this chapter

Cite this chapter

N. Gorban, A., V. Karlin, I. References. In: Invariant Manifolds for Physical and Chemical Kinetics. Lecture Notes in Physics, vol 660. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-31531-5_17

Download citation

  • DOI: https://doi.org/10.1007/978-3-540-31531-5_17

  • Published:

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-22684-0

  • Online ISBN: 978-3-540-31531-5

  • eBook Packages: Physics and AstronomyPhysics and Astronomy (R0)

Publish with us

Policies and ethics

Search

Navigation