Geometry of irreversibility: The film of nonequilibrium states

Gorban, Alexander N. and Karlin, Iliya V. (2002) Geometry of irreversibility: The film of nonequilibrium states. [Preprint]

Full text available as:



A general geometrical framework of nonequilibrium thermodynamics is developed. The notion of macroscopically definable ensembles is developed. The thesis about macroscopically definable ensembles is suggested. This thesis should play the same role in the nonequilibrium thermodynamics, as the Church-Turing thesis in the theory of computability. The primitive macroscopically definable ensembles are described. These are ensembles with macroscopically prepared initial states. The method for computing trajectories of primitive macroscopically definable nonequilibrium ensembles is elaborated. These trajectories are represented as sequences of deformed equilibrium ensembles and simple quadratic models between them. The primitive macroscopically definable ensembles form the manifold in the space of ensembles. We call this manifold the film of nonequilibrium states. The equation for the film and the equation for the ensemble motion on the film are written down. The notion of the invariant film of non-equilibrium states, and the method of its approximate construction transform the problem of nonequilibrium kinetics into a series of problems of equilibrium statistical physics. The developed methods allow us to solve the problem of macro-kinetics even when there are no autonomous equations of macro-kinetics.

Item Type:Preprint
Keywords:Irreversibility; entropy; kinetics; curvature; Church-Turing thesis; macroscopically definable ensembles
Subjects:Computer Science > Dynamical Systems
ID Code:3167
Deposited By:Gorban, Prof Alexander N.
Deposited On:04 Oct 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.

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

Del Rio-Correa, J. L., Garcia-Colin, L. S. Phys. Rev. E, 48 (1993) 891.

Zubarev, D., Morosov, V., Ropke, G. Statistical Mechanics of Nonequilibrium Processes, Akademie Verlag, Berlin, Vol. 1. (1996).

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

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

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

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

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

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

Maxwell's Demon 2: Entropy, Classical and Quantum Information, Computing. 2nd edition. H. S. Leff, A. F. Rex, eds. IOP, Philadelphia, 2003.

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

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, 19 (2002).

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

Copeland, B. J. The Church-Turing Thesis, In: The Stanford Encyclopedia of Philosophy (Fall 2002 Edition)


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.

Karlin, I. V., Gorban, A. N. Succi, S., Boffi, V. Maximum Entropy Principle for Lattice Kinetic Equations, Phys. Rev. Lett., 81, 1 (1998), 6-9.

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.

Karlin, I. V., Ansumali, S., De Angelis, E., Ottinger, H. C., Succi, S. Entropic Lattice Boltzmann Method for Large Scale Turbulence Simulation, E-print,

\noindent On-line:

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.

Gorban, A. N., Karlin, I. V., Zinovyev, A. Yu. Constructive Methods of Invariant Manifolds for Kinetic Problems, Preprint IHES/M/03/50, Institut des Hautes Études Scientifiques in Bures-sur-Yvette (France), 2003.

\noindent On-line:,


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

Ruelle, D. Smooth Dynamics and New Theoretical Ideas in Nonequilibrium Statistical Mechanics, Journal of Statistical Physics, 95 (1-2) (1999), 393-468.

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

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. Equilibrium Encircling. Equations of Chemical Kinetics and Their Thermodynamic Analysis, Nauka, Novosibirsk, 1984. [In Russian]

Gorban, A. N., Bykov, V. I., Yablonskii, G. S. Essays on chemical relaxation, Nauka, Novosibirsk, 1986. [In Russian]

Gorban, A. N., Karlin, I. V., Ilg, P., Ottinger, H. C. Corrections and enhancements of quasi-equilibrium states, J. Non-Newtonian Fluid Mech., 96(1-2)} (2001), 203--219.

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

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

Gorban, A. N., Karlin, I. V. Macroscopic dynamics through coarse-graining: A solvable example, Phys. Rev. E, 65 (2002), 026116(1-5).

Gorban, A. N., Karlin, I. V. Quasi-equilibrium approximations and non-standard expansions in the theory of the Boltzmann kinetic equation, Mathematical Modeling in Biology and Chemistry (New Approaches), R. G. Khlebopros, ed., Nauka, Novosibirsk, 1992, 69-117. [In Russian]

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

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

Beyn, W.-J., Kless, W. Numerical Taylor expansions of invariant manifolds in large dynamical systems, Numerische Mathematik, 80 (1989), 1-38.

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

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

Chapman, S., Cowling, T. G. The Mathematical Theory of Non-uniform Gases, Cambridge University Press, Cambridge, (1970).

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.

Karlin, I. V., Tatarinova, L. L., Gorban, A. N., Ottinger, H. C. Irreversibility in the short memory approximation, {\it Physica A,} to appear. Online:

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

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

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

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

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

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.

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].


Repository Staff Only: item control page