Chemical Instabilities and Broken Symmetry: The Hard Mode Case

  • D. Walgraef
  • G. Dewel
  • P. Borckmans
Part of the Springer Series in Synergetics book series (SSSYN, volume 8)


Chemical instabilities are interesting for the theorist for several reasons. The most important to our purposes are the following. These instabilities occur far out of thermal equilibrium and ordered states may appear through space or time symmetry breaking. In the vicinity of the transition points anomalies in the static and dynamical behaviour of the fluctuations, correlation functions, …, occur. These anomalies are very similar to critical slowing down, divergence of correlation lengths, existence of long range fluctuations, which are well known to play an essential role in the understanding of equilibrium phase transitions. So despite the loss of important properties like detailed balance or fluctuation-dissipation theorems, far from equilibrium chemical instabilities show important similarities with ordinary equilibrium phase transitions [1–3]. Moreover, near the instability the behaviour of the system may be thought to be governed by a few modes only which evolve on long time scales and for these modes detailed balance may be recovered [4]. Great progress has been made these last years in this domain, namely by the use of renormalization group theories to justify scaling properties, universality concepts and to calculate critical indices also by the derivation of exact relations or theorems associated to the breaking of continuous symmetries in the ordered phases [3,7]. As our understanding of the real world lives on analogies, it is then an attractive idea to apply these powerful concepts to get a deeper understanding of non equilibrium phase transitions.


Instability Point Single Mode Laser Renormalization Group Theory Equilibrium Phase Transition Topological Phase Transition 
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  1. 1.
    G. Nicolis, I. Prigogine: Self-Organization in Non-Equilibrium Systems (Wiley, New York 1977)Google Scholar
  2. 2.
    H. Haken: Synergetics, 2ed., Springer Series in Synergetics, Vol.1 (Springer, Berlin, Heidelberg, New York 1978)Google Scholar
  3. 3.
    Proceedings of the XVIIth Solvay Conference on Physics: Order and Fluctuations in Equilibrium and Non-Equilibrium Statistical Mechanics (Wiley, New York 1981, ed. by G. Nicolis, G. Dewel, J. TurnerGoogle Scholar
  4. 4.
    R. Graham: Phys. Rev. A10, 1762 (1974)ADSGoogle Scholar
  5. 5.
    G. Dewel, D. Walgraef, P. Borckmans: Z. Physik B28 ,235 (1977)MathSciNetADSGoogle Scholar
  6. 5a.
    D. Walgraef, G. Dewel, P. Borckmans: Phys. Rev. A2l ,397 (1980)ADSGoogle Scholar
  7. 6.
    D. Walgraef, G. Dewel, P. Borckmans: Adv. in Chem. Phys., submittedGoogle Scholar
  8. 7.
    S.K. Ma: Modern Theory of Critical Phenomena (Benjamin, New York 1976)Google Scholar
  9. 8.
    P.C. Hohenberg, B.I. Halperin: Rev. Mod. Phys. 49 ,435 (1977)ADSCrossRefGoogle Scholar
  10. 9.
    V. Degiorgio, M. Scully: Phys. Rev. A2, 1170 (1970)ADSGoogle Scholar
  11. 10.
    J.W. Turner: “Stationary and Time Dependent Solutions of Master Equations in Several Variables”, in Dynamics of Synergetics Systems ,ed. by H. Haken (Springer, Berlin, Heidelberg, New York 1980)Google Scholar
  12. 11.
    A. Winfree: J. Math. Biol. 1, 73 (1974)MathSciNetMATHCrossRefGoogle Scholar
  13. 12.
    R. Graham: in Fluctuations, Instabilities and Phase Transitions ,ed. by T. Riste (Plenum, New York 1975)Google Scholar
  14. 13.
    P. Schranner, S. Grossmann, P. Richter: Z. Physik B35 ,363 (1979)MathSciNetADSCrossRefGoogle Scholar
  15. 14.
    H. Hentschel: Z. Physik B31 ,401 (1978)ADSGoogle Scholar
  16. 15.
    N.N. Bogolubov: Lectures in Quantum Statistics (Mac Donald, London 1971) Vol.2Google Scholar
  17. 16.
    H. Mashiyama, A. Ito, T. Ohta: Progr. Theor. Phys. 54 ,1050 (1975)ADSCrossRefGoogle Scholar
  18. 17.
    V.L. Pokrovskii: Adv. in Phys. 28 ,597 (1979)ADSGoogle Scholar
  19. 18.
    J.K. Kosterlitz, D. Thouless: J. Phys. C6 ,1181 (1973)ADSGoogle Scholar
  20. 19.
    B.I. Halperin: in Physics of Low Dimension Systems ,Proceedings of the Kyoto Summer Institute, ed. by Y. Nagaoka, S. Hikami (Publication Office, Progress in Theoretical Physics, Kyoto 1979)Google Scholar
  21. 20.
    D. Nelson: To appear in the Proceedings of the 1980 Summer School on Fundamental Problems in Statistical Mechanics, Enschede, NetherlandsGoogle Scholar
  22. 21.
    Y. Kuramoto: Prog. Theor. Physics 56 ,679 (1976)MathSciNetADSCrossRefGoogle Scholar
  23. 22.
    Y. Kuramoto, T. Yamada: Prog. Theor. Phys. 56 ,724 (1976)MathSciNetADSCrossRefGoogle Scholar
  24. 23.
    P. Ortoleva: J. Chem. Phys. 69 ,300 (1978)MathSciNetADSCrossRefGoogle Scholar
  25. 24.
    T. Yamada, Y. Kuramoto: Prog. Theor. Phys. 55 ,2035 (1976)MathSciNetADSCrossRefGoogle Scholar
  26. 24.
    A.T. Winfree: Science 175 ,634 (1972)ADSCrossRefGoogle Scholar
  27. 25.
    M.L. Smoes: “Chemical Waves in the Oscillatory Zhabotinskii System. A Transition from Temporal to Spatio-Temporal Organization”, in Dynamics of Synergetic Systems ,ed. by H. Haken (Springer, Berlin, Heidelberg, New York 1980)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1981

Authors and Affiliations

  • D. Walgraef
    • 1
  • G. Dewel
    • 1
  • P. Borckmans
    • 1
  1. 1.Chimie-Physique IIUniversité Libre de BruxellesBruxellesBelgium

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