Constructive Methods of Invariant Manifolds for Kinetic Problems

Gorban, Prof Alexander N. and Karlin, Dr. Iliya V. and Zinovyev, Dr. Andrei Yu. (2003) Constructive Methods of Invariant Manifolds for Kinetic Problems. [Book Chapter]

Full text available as:



We present the Constructive Methods of Invariant Manifolds for model reduction in physical and chemical kinetics, developed during last two decades. The physical problem of reduced description is studied in a most general form as a problem of constructing the slow invariant manifold. The invariance conditions are formulated as the differential equation for a manifold immersed in the phase space (the invariance equation). The equation of motion for immersed manifolds is obtained (the film extension of the dynamics). Invariant manifolds are fixed points for this equation, and slow invariant manifolds are Lyapunov stable fixed points, thus slowness is presented as stability. A collection of methods for construction of slow invariant manifolds is presented, in particular, the Newton method subject to incomplete linearization is the analogue of KAM methods for dissipative systems. The systematic use of thermodynamics structures and of the quasi--chemical representation allow to construct approximations which are in concordance with physical restrictions. We systematically consider a discrete analogue of the slow (stable) positively invariant manifolds for dissipative systems, invariant grids. Dynamic and static postprocessing procedures give us the opportunity to estimate the accuracy of obtained approximations, and to improve this accuracy significantly. The following examples of applications are presented: Nonperturbative deviation of physically consistent hydrodynamics from the Boltzmann equation and from the reversible dynamics, for Knudsen numbers Kn~1; construction of the moment equations for nonequilibrium media and their dynamical correction (instead of extension of list of variables) to gain more accuracy in description of highly nonequilibrium flows; determination of molecules dimension (as diameters of equivalent hard spheres) from experimental viscosity data; invariant grids for a two-dimensional catalytic reaction and a four-dimensional oxidation reaction (six species, two balances); universal continuous media description of dilute polymeric solution; the limits of macroscopic description for polymer molecules, etc.

Item Type:Book Chapter
Additional Information:Big review, 237 pages, 241 references, 20 figures.
Keywords:Model Reduction; Invariant Manifold; Entropy; Kinetics; Boltzmann Equation; Fokker--Planck Equation; Postprocessing
Subjects:Computer Science > Dynamical Systems
ID Code:3087
Deposited By:Gorban, Prof Alexander N.
Deposited On:08 Aug 2003
Last Modified:11 Mar 2011 08:55

References in Article

Select the SEEK icon to attempt to find the referenced article. If it does not appear to be in cogprints you will be forwarded to the paracite service. Poorly formated references will probably not work.

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

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.

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.

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

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

Karlin, I. V., Dukek, G., Nonnenmacher, T. F., Invariance principle for extension of hydrodynamics: Nonlinear viscosity. Phys. Rev. E, 55(2) (1997), 1573-1576.

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.

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

Gorban, A. N., Karlin, I. V., Method of invariant manifold for chemical kinetics, Chem. Eng. Sci., to appear. Preprint online:, 9 Jul 2002.

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.

Karlin, I. V., Zmievskii, V. B., Invariant closure for the Fokker-Planck equation, 1998. Preprint online: 4 Feb 1998.

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.

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

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.

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

Gorban, A. N., Karlin, I. V., Ottinger, H. C. , and Tatarinova, L. L., Ehrenfest's argument extended to a formalism of nonequilibrium thermodynamics, Phys. Rev. E, 63 (2001), 066124(1-6).

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

Gorban, A. N., Karlin, I. V., Macroscopic dynamics through coarse-graining: A solvable example Phys. Rev. E, 56 (2002), 026116(1-5).

bibitemGKGeoNeo 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.

Karlin, I. V., Tatarinova, L. L., Gorban, A. N., Ottinger, H. C., Irreversibility in the short memory approximation, Physica A, to appear. Preprint online: v1 18 May 2003.

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.

Gorban, A. N., Karlin, I. V., Short-Wave Limit of Hydrodynamics: A Soluble Example, Phys. Rev. Lett. 77 (1996), 282-285.

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. Online:

Gorban, A. N., Karlin, I. V., Structure and Approximations of the Chapman-Enskog Expansion, Sov. Phys. JETP 73 (1991), 637-641.

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.

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

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

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.

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

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.

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

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

Shirkov, D. V., Kovalev, V. F., Bogoliubov Renormalization Group and Symmetry of Solution in Mathematical Physics, Phys. Rept. 352 (2001). 219-249. Online:

Zinn-Justin, J., Quantum Field Theory and Critical Phenomena (Clarendon Press, Oxford, 1989).

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

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. Online:

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. Online:

Hatta Y., Kunihiro T. Renormalization Group Method Applied to Kinetic Equations: roles of initial values and time, Annals Phys. 298 (2002), 24-57. Online:

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.

Forster, 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.

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.

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.

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:

Gorban, A. N., Karlin, I. V., Methods of nonlinear kinetics, Contribution to the ``Encyclopedia of Life Support Systems (EOLSS Publishers, Oxford, to appear). Online:

Chapman, S., Cowling, T., Mathematical Theory of Non-Uniform Gases, Third edition (Cambridge University Press, Cambridge, 1970).

Hilbert, D. Begründung der kinetischen Gastheorie, Mathematische Annalen, 72 (1912), 562-577.

Bobylev, A. V., The Chapman-Enskog and Grad Methods for Solving the Boltzmann Equation, Sov. Phys. Dokl., 27 (1982), No. 1, 29-31.

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.

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

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

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

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).

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

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

Roos, H. G., Stynes, M., Tobiska, L., Numerical Methods for Singularly Perturbed Differential Equations: Convection-Diffusion and Flow Problems, Springer Verlag, 1996.

Mishchenko, E. F., Kolesov, Y. S., Kolesov, A. U., Rozov, N. Kh., Asymptotic Methods in Singularly Perturbed Systems, Consultants Bureau, 1994.

Novozhilov, I. V., Fractional Analysis: Methods of Motion Decomposition, Birkhauser, 1997.

Milik, A., Singular Perturbation on the Web, 1997. milik/singdir.html#geo:sing

Gear, C. W., Numerical Initial Value Problems in Ordinary Differential Equations. Prentice-Hall, Englewood Cliffs, NJ (1971).

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

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

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

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

Zagaris, A., Kaper, H. G., Kaper, T. J. Analysis of the CSP Reduction Method for Chemical Kinetics, Preprint Argonne National Laboratory ANL/MCS-P1050-0503, May 2003. Online:

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

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.

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

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.

Chueshov, I. D., Introduction to the Theory of Infinite-Dimensional Dissipative Systems, The Electronic Library of Mathematics, 2002, [Translated from Russian edition, ACTA Scientific Publishing House, Kharkov, Ukraine, 1999]

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, .

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, Birkhauser Verlag, Basel / Switzerland, 1996.

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

Garay, B. M., Estimates in Discretizing Normally Hyperbolic Compact Invariant Manifolds of Ordinary Differential Equations, Computers and Mathematics with Applications, 42 (2001), 1103-1122.

Gorban, A. N., Karlin, I. V., Zinovyev, A. Yu., Invariant grids for reaction kinetics, Preprint, Institut des Hautes Étude Scientifiques, 2003. Online:

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

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.

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

Cercignani, C., The Boltzmann Equation and its Applications, Springer, New York, 1988.

Cercignani, C., Illner, R., Pulvirent, M., The Mathematical theory of Dilute Gases, Springer, New York, 1994.

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.

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

Lebowitz, J., Frisch, H., Helfand, E., Non-equilibrium Distribution Functions in a Fluid. Physics of Fluids, 3 (1960), 325.

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

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

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

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

Bird G.A. Molecular Gas Dynamics and the Direct Simulation of gas Flows, Clarendon Press, Oxford, 1994.

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

Frisch U., Hasslacher B. and Pomeau Y. (1986). Lattice-gas automata for the Navier-Stokes equation, Phys. Rev. Lett., 56, 1505-1509. [Introduction of the Lattice Gas models for hydrodynamics]

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

Succi S., The Lattice Boltzmann Equation for Fluid Dynamics and Beyond. Clarendon Press, Oxford, 2001.

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

Van Kampen, N. G., Stochastic Processes in Physics and Chemistry, North-Holland, Amsterdam 1981.

Risken, H., The Fokker-Planck Equation, Springer, Berlin, 1984.

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

Doi, M., Edwards, S. F., The Theory of Polymer Dynamics, Clarendon Press, Oxford, 1986.

Ottinger, H. C., Stochastic Processes in Polymeric Fluids, Springer, Berlin, 1996.

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

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

Gorban, A. N., Karlin, I. V., Family of additive entropy functions out of thermodynamic limit, Phys. Rev. E, 67 (2003), 016104. Online:

Gorban, A. N., Karlin, I. V., Ottinger H. C., The additive generalization of the Boltzmann entropy, Phys. Rev. E, 67, 067104 (2003). Online:

Gorban, P., Monotonically equivalent entropies and solution of additivity equation, arxiv:cond-mat/0304131, Physica A, (2003), to appear. Online:

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

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

Gorban, A. N., Equilibrium Encircling. Equations of Chemical Kinetics and Their Thermodynamic Analysis, Nauka, Novosibirsk, 1984.

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

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.

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

Ansumali S., Karlin, I. V. Single relaxation time model for entropic Lattice Boltzmann methods. Phys. Rev. E, 65 (2002), 056312(1-9).

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

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.

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

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

Dimitrov, V.I., Prostaya kinetika [Simple Kinetics]. Novosibirsk: Nauka, 1982.

Prigogine, I., Thermodynamics of Irreversible Processes, Interscience, New York, 1961.

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

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).

Robinson, J. C., A concise proof of the ``geometric construction of inertial manifolds, Phy. Lett. A, 200 (1995), 415-417.

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

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

Evans, L. C. Partial Differential Equations, AMS, Providence, RI, USA, 1998.

Dubinskii, Ju. A. Analytic Pseudo-Differential Operators and their Applications. Kluwer Academic Publishers, Book Series: Mathematics And its Applications Soviet Series: Volume 68, 1991.

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

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.

Lyapunov A. M., The general Problem of the Stability of Motion, Taylor & Francis, 1992.

Poincar'e, H.: Les m'ethodes nouvelles de la m'ecanique c'eleste. Vols. 1-3. Paris: Gauthier-Villars. (1892/1893/1899).

Arnold, V. I., Geometrical Methods in the Theory of Differential Equations, Springer- Verlag, New York-Berlin, 1983.

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

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

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.

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

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

Schlogl, F., Stochastic measures in nonequilibrium thermodynamics, Phys. Rep. 62, 4 (July 1980), 267-380.

Jaynes E. T., Information theory and statistical mechanics, in: Statistical Physics. Brandeis Lectures, V. 3, 160-185 (1963).

Grabert, H. Projection Operator Techniques in Nonequilibrium Statistical Mechanics (Springer Verlag, Berlin, 1982).

Zubarev, D., Morozov, V., Ropke, 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).

Ottinger, H. C. Derivation of Two-Generator Framework of Nonequilibrium Thermodynamics for Quantum Systems, Phys. Rev. E 62 (2000), 4720-4724.

Uhlenbeck, G. E. in: Fundamental Problems in Statistical Mechanics II, edited by E. G. D. Cohen, (North Holland, Amsterdam, 1968).

Grad, H. On the kinetic theory of rarefied gases, Comm. Pure and Appl. Math. 2(4) (1949), 331-407.

Hauge, E. H. Phys. Fluids 13 (1970), 1201.

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 (May 1978), 321-344.

Widder, M. E., Titulaer, U. M. Two kinetic models for the growth of small droplets from gas mixtures, Physica A: Statistical and Theoretical Physics, 167, 3 (1990), 663-675.

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

bibitemK3Karlin, I. V., Exact summation of the Chapman-Enskog expansion from moment equations, J. Phys. A: Math. Gen. 33 (2000), 8037-8046.

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.

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

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

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

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.

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

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

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).

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

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

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]

Gorban, A. N., Karlin, I. V., Quasi-Equilibrium Closure Hierarchies for The Boltzmann Equation [Translation of the first part of the paper citeMBCh]. Preprint, 2003, Online:

Jou, D., Casas-V'azquez, J., Lebon, G., Extended Irreversible Thermodynamics, Springer, Berlin, 1993.

A. Gorban and I. Karlin, New Methods for Solving the Boltzmann Equations, AMSE Press, Tassin, France, 1994.

Hirschfelder, J. O., Curtiss C. F., Bird, R. B. Molecular Theory of Gases and Liquids, (J. Wiley, NY, 1954).

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

R'esibois, P., De Leener, M., Classical Kinetic Theory of Fluids, Wiley, NY, 1977.

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

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

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

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.

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

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].

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.

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

Treves, F., Introduction to Pseudodifferential and Fourier Integral Operators, Plenum, NY, (1982).

Shubin, M. A., Pseudodifferential Operators and Spectral Theory, Nauka, Moscow, (1978).

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.

Rodr'iguez, R. F., Garc'ia-Col'in, 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.

Struchtrup, H., Torrilhon M., Regularization of Grad's 13 Moment Equations: Derivation and Linear Analysis, Phys. Fluids, accepted (2003).

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

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

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

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

Richtmyer, R.D., and Morton, K.W., Difference Methods for Initial Value Problems, 2nd ed. (New York: Wiley-Interscience), 1967.

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:

Jolliffe, I. T. Principal Component Analysis. Springer-Verlag, 1986.

Callen, H. B. Thermodynamics and an Introduction to Thermostatistics (Wiley, New York), 1985.

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:

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

Aizenberg, L., Carleman's Formulas in Complex Analysis: Theory and Applications. (Mathematics and Its Applications; V. 244) Kluwer, 1993.

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). Online:

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

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.

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.

Ehrenfest, P., Collected Scientific Papers, North-Holland, Amsterdam, 1959, pp. 213-300.

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

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

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

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.

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.

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

Gorban, A. N., Shokin, Yu. I., Verbitskii, V. I. Simultaneously dissipative operators and the infinitesimal Moore effect in interval spaces, Preprint (1997). Online:

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.

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

Grassberger, P. and Procaccia, I., Measuring the Strange


Repository Staff Only: item control page