Journal of Statistical Physics

, Volume 20, Issue 5, pp 473–486 | Cite as

Critical fluctuation universality in chemically oscillatory systems: A soluble master equation

  • M. DelleDonne
  • P. Ortoleva
Articles
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Abstract

Multiple time scale arguments are used to show that near a Hopf bifurcation to a chemical oscillation the dynamics of the system reduces to that of a classic soluble limit cycle system. A birth and death master equation is then introduced and the spectrum of the resulting transition operator is shown to be complex. Exact solutions of the master equation are obtained both for the steady and (for a rather general class of systems) “excited” states. Thus a simple basis of universality of critical properties in chemical oscillations is provided.

Key words

Chemical oscillations critical fluctuations master equation critical universality limit cycle chemical instability dissipative structure scaling theory 
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References

  1. 1.
    R. J. Field and R. M. Noyes,J. Chem. Phys. 63:2289 (1975); A. T. Winfree,Science 175:634 (1972); B. Hess and A. Boiteau,Am. Rev. Biochem. 40:237 (1971); B. Chance, E. K. Pye, A. K. Gosh, and B. Hess, eds.Biological and Biochemical Oscillators (Academic Press, 1973).Google Scholar
  2. 2.
    R. Landauer,J. Appl. Phys. 33 (1962); J. W. Woo and R. Landauer,IEEE J. Quant. Elect. 7: 435 (1970).Google Scholar
  3. 3.
    R. D. Hempstead and M. Lax,Phys. Rev. 161:350 (1967).Google Scholar
  4. 4.
    P. Ortoleva, inAdvances in Theoretical Chemistry, H. Eyring, ed. (1978).Google Scholar
  5. 5.
    I. Stakgold,SIAM Soc. Ind. Appl. Math. Rec. 13:289 (1971); D. Sattinger,Topics in Stability and Bifurcation Theory (Lecture Notes in Mathematics, 309) (Springer, Berlin, 1973).Google Scholar
  6. 6.
    M. DelleDonne and P. Ortoleva,J. Chem. Phys. 67:1861 (1977).Google Scholar
  7. 7.
    H.-S. Hahn, A. Nitzan, P. Ortoleva, and J. Ross,Proc. Nat. Acad. Sci. 71:4067 (1974).Google Scholar
  8. 8.
    A. Nitzan, P. Ortoleva, J. Deutch, and J. Ross,J. Chem. Phys. 61:1056 (1974).Google Scholar
  9. 9.
    Y. Kuramoto and T. Tsuzuki,Prag. Theor. Phys. 52:1399 (1974);54:687 (1975).Google Scholar
  10. 10.
    A. Wunderlin and H. Haken,Z. Physik, 393 (1975).Google Scholar
  11. 11.
    A. Nitzan,Phys. Rev. A 17:1513 (1978).Google Scholar
  12. 12.
    A. Nitzan and P. Ortoleva (submitted for publication).Google Scholar
  13. 13.
    D. S. Cohen, J. C. Neu, and R. R. Rosales,SIAM J. Appl. Math. (1978).Google Scholar
  14. 14.
    L. Caesari,Asymptotic Behavior and Stability Problems in Ordinary Differential Equations (Springer, Berlin, 1971).Google Scholar
  15. 15.
    J. Dreitlein and M. Smoes,J. Theor. Biol. 46:569 (1974).Google Scholar
  16. 16.
    N. Koppel and L. Howard,Studies in Applied Math. 52:291 (1973).Google Scholar
  17. 17.
    P. Ortoleva and J. Ross,J. Chem. Phys. 60:5890 (1974).Google Scholar
  18. 18.
    K. G. Wilson and J. Kogut,Physics Reports 12:75 (1974).Google Scholar
  19. 19.
    P. C. Hohenburg and B. I. Halperin,Rev. Mod. Phys. 49:435 (1977).Google Scholar
  20. 20.
    D. A. McQuarrie,J. Appl. Prob. 4:413 (1967).Google Scholar
  21. 21.
    M. Dole and M. Inokuchi,J. Chem. Phys. 39:310 (1963).Google Scholar
  22. 22.
    M. Malek-Mansour and G. Nicolis,J. Stat. Phys. 13:197 (1975); C. W. Gardiner, K. J. McNeil, D. F. Walls, and I. S. Matheson,J. Stat. Phys. 14:307 (1976); M. DelleDonne and P. Ortoleva,J. Stat. Phys. 18:319 (1978).Google Scholar
  23. 23.
    P. Ortoleva, unpublished notes; P. Ortoleva and J. Ross,Adv. Chem. Phys. XXIX:49 (1975).Google Scholar
  24. 24.
    O. Rossler,Z. Naturforsch. 31a:259 (1976); O. Rossler and P. Ortoleva,Lecture Notes in Biomathematics 21 (Springer-Verlag, 1978), p. 51.Google Scholar
  25. 25.
    M. DelleDonne and P. Ortoleva,Z. Naturforsch. 33a:588 (1978).Google Scholar
  26. 26.
    J. M. Deutch, S. Hudson, P. Ortoleva, and J. Ross,J. Chem. Phys. 57:4327 (1972).Google Scholar
  27. 27.
    D. Feinn, P. Ortoleva, W. Schalf, S. Schmidt, and M. Wolff,J. Chem. Phys. 69:27 (1978); R. Lovett, P. Ortoleva, and J. Ross,J. Chem. Phys. 69:947 (1978).Google Scholar
  28. 28.
    P. Ortoleva,J. Chem. Phys. 69:300 (1978).Google Scholar
  29. 29.
    R. Graham, inSpringer Tracts in Modern Physics 66 (1973), p. 1.Google Scholar
  30. 30.
    Rostocker,Physikalische Manuscripte 2:55 (1977).Google Scholar
  31. 31.
    R. Feistel and W. Ebeling,Physica 93A:114 (1978).Google Scholar
  32. 32.
    G. Nicolis,Suppl. Prog. Theor. Phys. 64 (1978).Google Scholar

Copyright information

© Plenum Publishing Corporation 1979

Authors and Affiliations

  • M. DelleDonne
    • 1
  • P. Ortoleva
    • 1
  1. 1.Department of ChemistryIndiana UniversityBloomington

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