Advertisement

Fluctuations and Transitions in the Absorptive Optical Bistability

  • J. C. Englund
  • W. C. Schieve
  • W. Zurek
  • R. F. Gragg

Abstract

From a previously suggested stochastic differential equation (S.D.E.), the fluctuations in amplitude, x, both in and out of the steady state are described for the absorptive bistability. The Stratonovic Fokker-Planck (F.P.) equation obtained is utilized to develop a theory of switching between locally stable states of amplitude x. This approach is a generalization of Kramers’ early work and the ideas of nucleation theory to the situation of non-constant diffusion, an interesting characteristic of the optical bistability. The importance of fluctuations (noise) is emphasized in switching. This result is compared to the approximation of Kramers and Landauer-Swanson. Also, comparisons are made to mean first passage estimates, and a recent work of Hanggi, Bulsara and Janda. The important question of the dominance of the low eigenvalue of the F.P. equation is investigated numerically by the development of a variational eigenvalue calculation for the bistability. The one eigenvalue approximation is found to hold for a wide range of y values and it is found that the variational treatment and the above theory agree well. Critical slowing is seen for c = 4, q = 0.01. The numerical algorithm may readily be applied to other one-dimensional bistable models.

Keywords

Stochastic Differential Equation Passage Time Nucleation Theory Time Dependent Phenomenon Time Dependent Transition 
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.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    G. Nicolis, I. Prigogine, “Self-Organization in Non-Equilibrium Systems,” John Wiley, New York (1977).Google Scholar
  2. 2.
    H. Haken, “Synergetics,” Springer-Verlag, Berlin (1977).CrossRefGoogle Scholar
  3. 3.
    Solvay Conference in Physics, Nov. 1978. (Prooceedings to appear.)Google Scholar
  4. 4.
    “Dissipative Structures in the Social and Physical Sciences,” ed. by W. C. Schieve and P. Allen (University of Texas Press, Austin) (to appear).Google Scholar
  5. 5.
    Austin Conference on Dissipative Structures in Chemistry and Physics, March, 1980. (Proceedings in preparation).Google Scholar
  6. 6.
    T. G. Kurtz, J. Chem. Phys. 57, 2976 (1972); Math. Prog. Study 5, 67 (1976); Stoch. Proc. Appl. 6, 223 (1978).ADSCrossRefGoogle Scholar
  7. 7.
    R. Bonifacio, M. Gronchi, L. A. Lugiato, Phys. Rev. A18, 2266 (1978); F. Casagrande and L. A. Lugiato, Nuovo Cimento B48, 287 (1978).ADSGoogle Scholar
  8. 8.
    H. Risken, Statistical Properties of Laser Light, in “Progress in Optics,” Vol. XII, 241, ed. by E. Wolf, North Holland, Amsterdam (1974).Google Scholar
  9. 9.
    R. Bonifacio, L. A. Lugiato, Opt. Comm. 19, 172 (1976).ADSCrossRefGoogle Scholar
  10. 10.
    A. Bulsara, W. C. Schieve, R. F. Gragg, Phys. Lett. 68A, 294 (1978).ADSGoogle Scholar
  11. 11.
    R. F. Gragg, W. C. Schieve, A. R. Bulsara, Phys. Rev. A19, 2052 (1979); J. C. Englund, W. C. Schieve and R. F. Gragg, Int. J. Q. Chem. Symp. 13, 695 (1979).ADSGoogle Scholar
  12. 12.
    A. Schenzle and H. Brand, Opt. Comm. 27, 485 (1978). ( See also A. Schenzle, these proceedings. )ADSCrossRefGoogle Scholar
  13. 13.
    K. Kondo, M. Mabuchi, H. Husegawa, Opt. Comm. 32, 136 (1980).ADSCrossRefGoogle Scholar
  14. 14.
    F. T. Arecchi and A. Politi, Opt. Comm. 29, 361 (1979).ADSCrossRefGoogle Scholar
  15. 15.
    H. A. Kramers, Physica 7, 284 (1940).MathSciNetADSMATHCrossRefGoogle Scholar
  16. 16.
    S. Chandrasekhar, Rev. Mod. Phys. 15, 1 (1943).MathSciNetADSMATHCrossRefGoogle Scholar
  17. 17.
    R. Landauer and J. A. Swanson, Phys. Rev. 121, 1668 (1960).ADSCrossRefGoogle Scholar
  18. 18.
    F. F. Abraham, “Homogeneous Nucleation Theory,” Academic Press, New York (1974) (references therein).Google Scholar
  19. 19.
    K. Binder and D. Stauffer, Adv. Phys. 25, 343 (1976).ADSCrossRefGoogle Scholar
  20. 20.
    W. Zurek and W. C. Schieve, “The Nucleation Paradigm,” (see Ref. 4).Google Scholar
  21. 21.
    C. H. Weiss, First Passage Time Problems in Chemical Physics, in Adv. Chem. Phys. 13, 1 (1966).Google Scholar
  22. 22.
    A. R. Bulsara and W. C. Schieve, Opt. Comm. 26, 384 (1978).ADSCrossRefGoogle Scholar
  23. 23.
    R. Bonifacio and P. Meystre, Opt. Comm. 27, 147 (1978) and Opt. Comm. 29, 131 (1979); F. A. Hopf, P. Meystre, P. D. Drummond and D. F. Wall’s, Opt. Comm. 31, 245 (1976). (See also the papers of P. Meystre, Et al., and F. A. Hopf, Et al., in these proceedings.)ADSCrossRefGoogle Scholar
  24. 24.
    L. Arnold, “Stochastic Differential Equations,” Wiley-Inter-science, New York (1974).MATHGoogle Scholar
  25. 25.
    R. E. Mortensen, J. Stat. Phys. 1, 271 (1969).ADSCrossRefGoogle Scholar
  26. 26.
    R. F. Gragg, W. C. Schieve, J. Englund, “Stochastic Differential Equations in the Optical Bistability,” (in preparation).Google Scholar
  27. 27.
    J. C. Doob, “Stochastic Processes,” John Wiley, New York (1953).MATHGoogle Scholar
  28. 28.
    R. L. Stratonovic, SIAM J. Control 4, 363 (1966).MathSciNetGoogle Scholar
  29. 29.
    A. H. Gray, Jr., and T/ K. Caughy, J. Math, and Phys. 44, 288 (1965).Google Scholar
  30. 30.
    E. Wong and M. Zakai, Ann. Math. Stat. 36, 1560 (1965).MathSciNetMATHCrossRefGoogle Scholar
  31. 31.
    N. Goel and N. Richter-Dyn, “Stochastic Problems in Biology,” John Wiley and Sons, (1973).Google Scholar
  32. 32.
    L. Arnold, W. Horsthemke, R. Lefever, Z. Physik, B29, 367 (1978); W. Horsthemke, and R. Lefever, Phys. Lett. 64A, 19 (1977).ADSGoogle Scholar
  33. 33.
    R. F. Gragg, Ph.D. Thesis, University of Texas, Austin, August, 1980.Google Scholar
  34. 34.
    M. Suzuki, Proceedings of XVII Conf. on Phys, Nov. 1978.Google Scholar
  35. 35.
    R. C. Desai and R. Zwanzig, J. Stat. Phys. 19, 1 (1978).MathSciNetADSCrossRefGoogle Scholar
  36. 36.
    N. G. van Kampen, J. Stat. Phys. 17, 71 (1977).ADSCrossRefGoogle Scholar
  37. 37.
    H. Dekker and N. G. van Kampen, Phys. Lett. 73A, 374 (1979).ADSGoogle Scholar
  38. 38.
    H. Tomita, A. Ito and H. Kidachi, Prog. Theor. Phys. 56, 786 (1976).MathSciNetADSCrossRefGoogle Scholar
  39. 39.
    A. Schenzle and H. Brand, Opt. Comm. 31, 401 (1979).ADSCrossRefGoogle Scholar
  40. 40.
    S. G. Mikhlin, “Variational Methods in Math. Phys.,” transl. by Boddington, Macmillan, (1964).Google Scholar
  41. 41.
    P. Hanggi, A. Bulsara, R. Janda, “Spectrum and Dynamic Response Function of Transmitted Light in the Absorptive Optical Bistability,” Phys. Rev. (to appear).Google Scholar
  42. 42.
    H. Mori, H. Fujisaka and H. Schigematso, Prog. Theor. Phys. 51, 1209 (1974); see also L. S. Garcia-Colin and J. L. del Rio, J. Stat. Phys. 16, 235 (1978) and references therein.CrossRefGoogle Scholar
  43. 43.
    L. van Hove, Phys. Rev. 95, 1374 (1954); S. Ma and G. F. Mazenko, Phys. Rev. B11, 4077 (1975).ADSMATHCrossRefGoogle Scholar

Copyright information

© Plenum Press, New York 1981

Authors and Affiliations

  • J. C. Englund
    • 1
  • W. C. Schieve
    • 1
  • W. Zurek
    • 1
  • R. F. Gragg
    • 1
  1. 1.Center for Studies in Statistical Mechanics and ThermodynamicsUniversity of Texas at AustinAustinUSA

Personalised recommendations