Lyapunov Instabilities of Extended Systems

  • Hong-liu Yang
  • Günter Radons
Part of the Lecture Notes in Computational Science and Engineering book series (LNCSE, volume 52)


One of the most successful theories in modern science is statistical mechanics, which allows us to understand the macroscopic (thermodynamic) properties of matter from a statistical analysis of the microscopic (mechanical) behavior of the constituent particles. In spite of this, using certain probabilistic assumptions such as Boltzmann’s Stosszahlansatz causes the lack of a firm foundation of this theory, especially for non-equilibrium statistical mechanics. Fortunately, the concept of chaotic dynamics developed in the 20th century [1] is a good candidate for accounting for these difficulties. Instead of the probabilistic assumptions, the dynamical instability of trajectories can make available the necessary fast loss of time correlations, ergodicity, mixing and other dynamical randomness [2]. It is generally expected that dynamical instability is at the basis of macroscopic transport phenomena and that one can find certain connections between them. Some beautiful theories in this direction were already developed in the past decade. Examples are the escape-rate formalism by Gaspard and Nicolis [3, 4] and the Gaussian thermostat method by Nosé, Hoover, Evans, Morriss and others [5, 6], where the Lyapunov exponents were related to certain transport coefficients.


Lyapunov Exponent Extend System Lyapunov Spectrum Coordinate Part Momentum Part 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    1. J.-P. Eckmann and D. Ruelle. Ergodic theory of chaos and strange attractors. Rev. Mod. Phys., 57:617, 1985; E. Ott. Chaos in Dynamical Systems. Cambridge University Press, Cambridge, 1993.CrossRefMathSciNetGoogle Scholar
  2. 2.
    2. N. S. Krylov. Works on the Foundations of Statistical Mechanics. Princeton University Press, Princeton, 1979.Google Scholar
  3. 3.
    3. P. Gaspard. Chaos, Scattering, and Statistical Mechanics. Cambridge University Press, Cambridge, 1998.zbMATHGoogle Scholar
  4. 4.
    4. J. P. Dorfman. An Introduction to Chaos in Nonequilibrium Statistical Mechanics. Cambridge University Press, Cambridge, 1999.zbMATHGoogle Scholar
  5. 5.
    5. D. J. Evans and G. P. Morriss. Statistical Mechanics of Nonequilibrium Liquids. Academic, New York, 1990.Google Scholar
  6. 6.
    6. Wm. G. Hoover. Time Reversibility, Computer Simulation, and Chaos. World Scientific, Singapore, 1999.zbMATHGoogle Scholar
  7. 7.
    7. H. A. Posch and R. Hirschl. Simulation of blliards and of hard-body fluids. In D. Szasz, editor, Hard Ball Systems and the Lorenz Gas, Springer, Berlin, 2000.Google Scholar
  8. 8.
    8. C. Forster, R. Hirschl, H. A. Posch, and Wm. G. Hoover. Perturbed phase-space dynamics of hard-disk fluids. Physics D, 187:294, 2004.zbMATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    9. J.-P. Eckmann and O. Gat. Hydrodynamic Lyapunov modes in translationinvariant systems. J. Stat. Phys., 98:775, 2000.zbMATHCrossRefMathSciNetGoogle Scholar
  10. 10.
    10. S. McNamara and M. Mareschal. Origin of the hydrodynamic Lyapunov modes. Phys. Rev. E, 64:051103, 2001.CrossRefGoogle Scholar
  11. 11.
    11. A. de Wijn and H. van Beijeren. Goldstone modes in Lyapunov spectra of hard sphere systems. Phys. Rev. E, 70:016207, 2004.CrossRefGoogle Scholar
  12. 12.
    12. T. Taniguchi and G. P. Morriss. Stepwise structure of Lyapunov spectra for many-particle systems using a random matrix dynamics. Phys. Rev. E, 65:056202, 2002.CrossRefGoogle Scholar
  13. 13.
    13. T. Taniguchi and G. P. Morriss. Boundary effects in the stepwise structure of the Lyapunov spectra for quasi-one-dimensional systems. Phys. Rev. E, 68:026218, 2003.CrossRefMathSciNetGoogle Scholar
  14. 14.
    14. Wm. G. Hoover, H. A. Posch, C. Forster, C. Dellago and M. Zhou. Lyapunov modes of two-Dimensional many-body systems; soft disks, hard disks, and rotors. J. Stat. Phys., 109:765, 2002.zbMATHCrossRefMathSciNetGoogle Scholar
  15. 15.
    15. H.-L. Yang and G. Radons. Lyapunov instabilities of Lennard-Jones fluids. Phys. Rev. E, 71:036211, 2005, see also arXiv:nlin.CD/0404027.CrossRefGoogle Scholar
  16. 16.
    16. G. Radons and H.-L. Yang. Static and dynamic correlations in many-particle Lyapunov vectors. arXiv:nlin.CD/0404028.Google Scholar
  17. 17.
    17. H.-L. Yang and G. Radons. Universal features of hydrodynamic Lyapunov modes in extended systems with continuous symmetries. Phys. Rev. Lett., 96:074101, 2006.CrossRefGoogle Scholar
  18. 18.
    18. H.-L. Yang and G. Radons. Hydrodynamic Lyapunov modes in coupled map lattices. Phys. Rev. E, 73:016202, 2006.CrossRefGoogle Scholar
  19. 19.
    19. H.-L. Yang and G. Radons. Dynamical behavior of hydrodynamic Lyapunov modes in coupled map lattices. Phys. Rev. E, 73:016208, 2006.CrossRefGoogle Scholar
  20. 20.
    20. G. Benettin, L. Galgani and J. M. Strelcyn. Kolmogorov entropy and numerical experiments. Phys. Rev. A, 14:2338, 1976.CrossRefGoogle Scholar
  21. 21.
    21. I. Shimada and T. Nagashima. A numerical approach to ergodic problem of dissipative dynamical systems. Prog. Theor. Phys., 61:1605, 1979.zbMATHCrossRefMathSciNetGoogle Scholar
  22. 22.
    22. J. Dongarra, J. Du Croz, I. Duff and S. Hammarling. A set of Level 3 Basic Linear Algebra Subprograms. ACM Trans. Math. Soft., 16:1, 1990.zbMATHCrossRefGoogle Scholar
  23. 23.
    23. L. S. Blackford, et al. ScaLAPACK Users' Guide. Society for Industrial and Applied Mathematics, Philadelphia, 1997.zbMATHGoogle Scholar
  24. 24.
    24. G. Radons, G. Rünger, M. Schwind and H.-L. Yang. Parallel algorithms for the determination of Lyapunov characteristics of large nonlinear dynamical systems. Proceedings of PARA04, WORKSHOP ON STATE-OF-THE-ART IN SCIENTIFIC COMPUTING, Lyngby, June 20–23, 2004, Lecture Notes of Computer Science, vol. 3272, 1131, Springer, Berlin, 2005.Google Scholar
  25. 25.
    25. J. P. Boon and S. Yip. Molecular Hydrodynamics. McGraw-Hill, New York, 1980.Google Scholar
  26. 26.
    26. W. Kob and H. C. Andersen. Testing mode-coupling theory for a supercooled binary Lennard-Jones mixture I: The van Hove correlation function. Phys. Rev. E, 51:4626, 1995.CrossRefGoogle Scholar
  27. 27.
    27. F. H. Stillinger. Statistical mechanics of metastable matter: superheated and stretched liquids. Phys. Rev. E, 52:4685, 1995.CrossRefGoogle Scholar
  28. 28.
    28. R. Livi, M. Pettini, S. Ruffo, M. Sparpaglione and A. Vulpiani. Equipartition threshold in nonlinear large Hamiltonian systems: The Fermi-Pasta-Ulam model. Phys. Rev. A, 31:1039, 1985.CrossRefGoogle Scholar
  29. 29.
    29. J.-P. Eckmann, C. Forster, H. A. Posch and E. Zabey. Lyapunov modes in harddisk systems. J. Stat. Phys., 118:795, 2005.CrossRefMathSciNetGoogle Scholar
  30. 30.
    30. D. Escande, H. Kantz, R. Livi and S. Ruffo. Self-consistent check of the validity of Gibbs calculus using dynamical variables. J. Stat. Phys., 76:605, 1994; M. Antoni and S. Ruffo. Clustering and relaxation in Hamiltonian long-range dynamics. Phys. Rev. E, 52:2361, 1995.zbMATHCrossRefMathSciNetGoogle Scholar
  31. 31.
    31. Y. Kuramoto and T. Tsuzuki. Persistent propagation of concentration waves in dissipative media far from thermal equilibrium. Prog. Theor. Phys., 55:356, 1976; G. I. Sivashinsky. Acta Astron., 4:1177, 1977.CrossRefGoogle Scholar
  32. 32.
    32. C. Forster and H. A. Posch. Lyapunov modes in soft-disk fluids. New J. Phys., 7:32, 2005, see also arXiv:nlin.CD/0409019CrossRefGoogle Scholar

Copyright information

© Springer 2006

Authors and Affiliations

  • Hong-liu Yang
    • 1
  • Günter Radons
    • 1
  1. 1.Institut für PhysikTechnische Universität ChemnitzChemnitzGermany

Personalised recommendations