Journal of Statistical Physics

, Volume 49, Issue 3–4, pp 569–605 | Cite as

Statistical macrodynamics of large dynamical systems. Case of a phase transition in oscillator communities

  • Yoshiki Kuramoto
  • Ikuko Nishikawa


A model dynamical system with a great many degrees of freedom is proposed for which the critical condition for the onset of collective oscillations, the evolution of a suitably defined order parameter, and its fluctuations around steady states can be studied analytically. This is a rotator model appropriate for a large population of limit cycle oscillators. It is assumed that the natural frequencies of the oscillators are distributed and that each oscillator interacts with all the others uniformly. An exact self-consistent equation for the stationary amplitude of the collective oscillation is derived and is extended to a dynamical form. This dynamical extension is carried out near the transition point where the characteristic time scales of the order parameter and of the individual oscillators become well separated from each other. The macroscopic evolution equation thus obtained generally involves a fluctuating term whose irregular temporal variation comes from a deterministic torus motion of a subpopulation. The analysis of this equation reveals order parameter behavior qualitatively different from that in thermodynamic phase transitions, especially in that the critical fluctuations in the present system are extremely small.

Key words

Large dissipative system population of limit cycle oscillators order parameter phase transition via mutual entrainment approximate invariant measure dynamical extension of self-consistent equation critical slowing down anomalous critical fluctuation 


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  1. 1.
    D. A. Linkens,IEEE Trans. Circuits Systems CAS-21:294 (1974); R. Zwanzig, inTopics in Statistical Mechanics and Biophysics, R. A. Piccirelli, ed. (American Institute of Physics, New York, 1976), p. 187.Google Scholar
  2. 2.
    H. Haken,Advanced Synergetics (Springer, Berlin, 1983).Google Scholar
  3. 3.
    P. Bergé, Y. Pomeau, and C. Vidal,Order within Chaos (Wiley, New York, 1986), Chapter VII and Appendix C.Google Scholar
  4. 4.
    Y. Kuramoto,Chemical Oscillations, Waves, and Turbulence (Springer, Berlin, 1984), Chapter 5.Google Scholar
  5. 5.
    A. T. Winfree,J. Theor. Biol. 16:15 (1967).Google Scholar
  6. 6.
    A. T. Winfree,The Geometry of Biological Time (Springer, New York, 1980).Google Scholar
  7. 7.
    Y. Kuramoto, inInternational Symposium on Mathematical Problems in Theoretical Physics, H. Araki, ed. (Springer, New York, 1975), p. 420;Physica 106A:128 (1981); S. Shinomoto and Y. Kuramoto,Prog. Theor. Phys. 75:1105 (1986); H. Sakaguchi and Y. Kuramoto,Prog. Theor. Phys. 76:576 (1986).Google Scholar
  8. 8.
    Y. Aizawa,Prog. Theor. Phys. 56:703 (1976).Google Scholar
  9. 9.
    Y. Kuramoto,Prog. Theor. Phys. 79(Suppl.):223 (1984).Google Scholar
  10. 10.
    Y. Yamaguchi, K. Kometani, and H. Shimizu,J. Stat. Phys. 26:719 (1981).Google Scholar
  11. 11.
    H. Daido,Prog. Theor. Phys. 75:1460 (1986); and preprint.Google Scholar
  12. 12.
    P. Glansdorff and I. Prigogine,Thermodynamic Theory of Structure, Stability and Fluctuations (Wiley, London, 1971).Google Scholar
  13. 13.
    G. Nicolis and I. Prigogine,Self-Organization in Nonequilibrium Systems (Wiley, New York, 1977).Google Scholar
  14. 14.
    S. Wolfram,Rev. Mod. Phys. 55:601 (1983).Google Scholar
  15. 15.
    K. Kaneko,Prog. Theor. Phys. 74:1033 (1985); G.-L. Oppo and R. Kapral,Phys. Rev. A 33:4219 (1986).Google Scholar
  16. 16.
    S. Chapman and T. G. Cowling,The Mathematical Theory of Nonuniform Gases, 3rd ed. (Cambridge University Press, 1970).Google Scholar
  17. 17.
    S. Wolfram,J. Stat. Phys. 45:471 (1986).Google Scholar
  18. 18.
    J. C. Neu,SIAM J. Appl. Math. 37:307 (1979);38:305 (1980).Google Scholar
  19. 19.
    A. H. Cohen, P. J. Holms, and R. H. Rand,J. Math. Biol. 13:345 (1982).Google Scholar
  20. 20.
    G. B. Ermentrout and J. Rinzel,J. Math. Biol. 11:269 (1981); G.B. Ermentrout and N. Kopell,SIAM J. Math. Anal. 15:215 (1984).Google Scholar
  21. 21.
    N. Kopell and G. B. Ermentrout,Commun. Pure Appl. Math. 39:623 (1986).Google Scholar

Copyright information

© Plenum Publishing Corporation 1987

Authors and Affiliations

  • Yoshiki Kuramoto
    • 1
  • Ikuko Nishikawa
    • 1
  1. 1.Department of PhysicsKyoto UniversityKyotoJapan

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