Instabilities and Fluctuations

  • Vittorio Degiorgio
Part of the NATO Advanced Study Institutes Series book series (NSSB, volume 77)


Self-organizing systems present generally a critical value of the control parameter λc “below -which the system is in the disordered phase (below threshold) and above which the system goes into the ordered phase. The buildup of the ordered phase is triggered by a spontaneous fluctuation. From a mathematical point of view, one notes that the macroscopic dynamic equations for self-organizing systems are intrinsically homogeneous, i.e. ρ = 0 must be a solution (ρ is the order parameter). Therefore if the system is initially at ρ = 0, it stays there forever and no self-organization takes place unless a spontaneous fluctuation is occurring. The fluctuation may be originated by thermal noise, by quantum noise (e.g., from the spontaneous emission of light), or by external disturbances.


Pump Power Laser Field Random Force Threshold Region Coexistence Curve 
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  1. 1.
    H. Haken, “Synergetics”, Springer Verlag, Berlin (1980)MATHGoogle Scholar
  2. 2.
    G. Nicolis and I. Prigogine,”Self Organization in Non Equilibrium Systems”, Wiley-Interscience, New York (1977)Google Scholar
  3. 3.
    R.L. Stratonovich, “Topics in the Theory of Random Noise”, Gordon and Breach, New York (1963) vol.1.Google Scholar
  4. 4.
    “Noise and Stochastic Processes”, edited by N. Wax, Dover, New York (1964)Google Scholar
  5. 5.
    L.P. Kadanoff et al., Rev. Mod. Phys. 39: 395 (1967)ADSCrossRefGoogle Scholar
  6. 6.
    Besides the general reference 5, see: J. Als-Nielsen and R.J. Birgeneau, Am.J.Phys. 45: 554 (1977)Google Scholar
  7. 7.
    C. Ishimoto and T. Tanaka, Phys.Rev.Lett. 39: 474 (1977)ADSCrossRefGoogle Scholar
  8. 8.
    M. Corti and V. Degiorgio, Phys.Rev.Lett. 45: 1045 (1980)ADSCrossRefGoogle Scholar
  9. 9.
    V. Degiorgio and M. Giglio, unpublished.Google Scholar
  10. 10.
    M. Sargent, M.O. Scully and W. Lamb, “Laser Physics”, Addison-Wesley, Reading (1974; see also Ref. 1 and the papers mentioned therein.Google Scholar
  11. 11.
    F.T. Arecchi and V. Degiorgio in: “Laser Handbook”, edited by F.T. Arecchi and E.O. Schulz-Dubois, North-Holland, Amsterdam, vol. 1: 191 (1972). This review paper contains references about the experimental works on laser statistics by Armstrong and Smith, Freed and Haus, Mandel and coworkers, Pike and coworkers, and others.Google Scholar
  12. 12.
    V. Degiorgio and M.O. Scully, Phys.Rev. A 2: 1170 (1970)ADSCrossRefGoogle Scholar
  13. 12a.
    R. Graham and H. Haken, Z. Physik 237: 31 (1970)MathSciNetADSCrossRefGoogle Scholar
  14. 13.
    M. Corti and V. Degiorgio, Phys.Rev. A 14: 1475 (1976)ADSCrossRefGoogle Scholar
  15. 14.
    L.A. Lugiato, Lett.Nuovo Cimento 23: 609 (1978)ADSCrossRefGoogle Scholar
  16. 15.
    R. Roy, R. Short, J. Durnin and L. Mandel, Phys.Rev.Lett, 45: 1486 (1980)ADSCrossRefGoogle Scholar
  17. 16.
    H. Haken, Phys.Lett. 53A: 77 (1977). The existence of an instability in the laser equations was discussed by H. Risken and K. Nummedal (J.Appl.Phys. 39: 4662 (1968)) independently from Lorenz work.MathSciNetADSGoogle Scholar
  18. 17.
    V. Degiorgio, Phys.Rev. A 20: 2193 (1979)ADSCrossRefGoogle Scholar
  19. 18.
    L.W. Casperson, IEEE J.Quantum Electron. 14: 756 (1978)ADSCrossRefGoogle Scholar
  20. 18.
    L.W. Casperson Phys. Rev. A 23: 248 (1981)ADSCrossRefGoogle Scholar
  21. 19.
    H.M. Gibbs, F.A. Hopf, D.L. Kaplan and R.L. Shoemaker, Phys. Rev.Lett. 46: 474 (1981)ADSCrossRefGoogle Scholar
  22. 20.
    A complete set of references about optical bistability, including the works by Gibbs, McCall, Bonifacio, Lugiato and others, can be found in the March 1981 issue of the IEEE J. of Quantum Electron. Google Scholar
  23. 21.
    K. Ikeda, H. Daido and O. Akimoto, Phys.Rev.Lett. 45 : 709 (1980)ADSCrossRefGoogle Scholar
  24. 22.
    F.T. Arecchi and V. Degiorgio, Phys. Rev. A 3: 1108 (1971)ADSCrossRefGoogle Scholar
  25. 23.
    F. Haake, Phys. Rev. Lett, 41: 1685 (1978)ADSCrossRefGoogle Scholar
  26. 23a.
    M. Suzuki, Phys. Lett. 67A: 339 (1978)ADSGoogle Scholar
  27. 23b.
    F. De Pasquale and P. Tombesi, Phys. Lett. 72A: 7 (1979);ADSGoogle Scholar
  28. 23d.
    F.T. Arecchi and A. Politi, Phys. Rev. Lett.45: 1219 (1980)ADSCrossRefGoogle Scholar
  29. 23e.
    L.A. Pokrovsky, Physica 105A: 105 (l98l)Google Scholar
  30. 24.
    V. Degiorgio, Opt.Commun. 2: 362 (1971).ADSCrossRefGoogle Scholar
  31. 24.
    For the theory of superfluorescence, see R. Bonifacio, P. Schwendimann and F. Haake, Phys. Rev. A 4: 302 (1971)ADSCrossRefGoogle Scholar
  32. 25.
    K. Kawasaki, M.C. Yalabik and J.D. Gunton, Phys.Rev. A 17: 455 (1978)ADSCrossRefGoogle Scholar
  33. 26.
    V.M. Zaitsev and M.I. Shliomis, Sov.Phys. JETP 32: 866 (1971)ADSGoogle Scholar
  34. 27.
    G. Ahlers, M.C. Cross, P.C. Hohenberg and S. Safran, preprintGoogle Scholar
  35. 28.
    R. Graham, Phys. Rev. A 10: 1762 (1974)ADSCrossRefGoogle Scholar
  36. 29.
    H.N. Lekkerkerker and J.P. Boon, in: “Fluctuations, Instabilities, and Phase Transitions” edited by T. Riste, Plenum, New York, p.205 (1975).Google Scholar

Copyright information

© Plenum Press, New York 1982

Authors and Affiliations

  • Vittorio Degiorgio
    • 1
  1. 1.Istituto di Fisica ApplicataUniversità di PaviaPaviaItaly

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