Models of Stochastic Behavior in Non-Equilibrium Steady States

  • Robert Graham
Part of the NATO Advanced Study Institutes Series book series (NSSB, volume 73)


Macroscopic systems are frequently described in terms of a set of macroscopic variables staisfying a given system of deterministic differential equations. The fluctuations of macroscopic variables being small, such a deterministic description is, in many cases, sufficiently accurate and already contains a wealth of information: e.g., for systems which are driven by stationary external forces and interacting with time independent reservoirs, the deterministic description allows to discuss the steady state as a fixed point or a limit cycle or a more complicated ‘strange attractor’ in the space of the macroscopic variables. Within the same description it is also possible to discuss the stability of the various steady states, in which the system may be found, and the time dependent relaxation towards such states from a given initial state.


Bifurcation Point Fokker Planck Equation Strange Attractor Detailed Balance Steady State Distribution 


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  1. 1.
    M. Lax, Rev. Mod. Phys. 32: 25 (1960);ADSMATHCrossRefGoogle Scholar
  2. 1a.
    M. Lax, Rev. Mod. Phys. 38:359 (1966)MathSciNetADSCrossRefGoogle Scholar
  3. 1b.
    M. Lax, Rev. Mod. Phys. 38:541 (1966).MathSciNetADSMATHCrossRefGoogle Scholar
  4. 2.
    T. Kirkpatrick, E.G.D. Cohen, J.R. Dorfman, Phys. Rev. Lett. 44: 472 (1980).ADSCrossRefGoogle Scholar
  5. 2a.
    I. Procaccia, D. Ronis, I. Oppenheim, Phys. Rev. A20: 2533 (1979).ADSGoogle Scholar
  6. 3.
    H. Haken, Encyclopedia of Physics 25/2c, Springer, New York 1970.Google Scholar
  7. 4.
    R. Graham, Springer Tracts in Mod. Phys. 66, Springer, New York 1973.Google Scholar
  8. 5.
    H. Haken, Rev. Mod. Phys. 47: 67 (1975).MathSciNetADSCrossRefGoogle Scholar
  9. 6.
    R. Graham, in “Order and Fluctuations in Equilibrium and Non-equilibrium Statistical Mechanics”, ed. Nicolis et al., John Wiley, New York (1981), p. 235.Google Scholar
  10. 7.
    ’ E. N. Lorenz, J. Atmos. Sci. 20: 130 (1963).ADSCrossRefGoogle Scholar
  11. 8.
    D. Ruelle, F. Takens, Commun. Math. Phvs. 20: 167 (1971).MathSciNetADSMATHCrossRefGoogle Scholar
  12. 9.
    R.L. Stratonovich, “Topics in the Theory of Random Noise,” Vol. I, II, Gordon and Breach, New York 1963.Google Scholar
  13. 10.
    H. Risken, Z. Physik 251: 231 (1972).ADSCrossRefGoogle Scholar
  14. 11.
    R. Graham, Z. Physik B40: 149 (1980).ADSGoogle Scholar
  15. 12.
    R. Graham, H. Haken Z. Physik 243: 289 (1971)MathSciNetADSCrossRefGoogle Scholar
  16. 12a.
    R. Graham, H. Haken Z. Physik 245:141 (1971).MathSciNetADSCrossRefGoogle Scholar
  17. 13.
    R. Graham, in Coherence and Quantum Optics, ed. L. Mandel, E. Wolf, Plenum, New York 1973.Google Scholar
  18. 14.
    R. Graham, in “Fluctuations, Instabilities and Phase Transitions,” ed. T. Riste, Plenum, New York 1975.Google Scholar
  19. 15.
    H. Risken, H.D. Vollmer, Z. Physik 201: 323 (1967);ADSCrossRefGoogle Scholar
  20. 15a.
    H. Risken, H.D. Vollmer, Z. Physik 204:240 (1967)ADSCrossRefGoogle Scholar
  21. 16.
    K. Seybold, H. Risken, Z. Physik 267: 323 (1974).ADSCrossRefGoogle Scholar
  22. 17.
    A. Schenzle, H. Brand, Phys. Rev. A4: 1628 (1979).ADSGoogle Scholar
  23. 18.
    R. Graham, Phys. Lett. 80A: 351 (1980).ADSGoogle Scholar
  24. 19.
    S.L. McCall, Phys. Rev. A9: 1515 (1974)ADSGoogle Scholar
  25. 19a.
    H.M. Gibbs, S.L. McCall, T.N.C. Venkatesan, Phys. Rev. Lett. 36: 1135 (1976)ADSCrossRefGoogle Scholar
  26. 20..
    R. Graham, A. Schenzle, in “Proceedings of the International Optical Bistability Conference,” Asheville, N.C. USA, 1980; R. Graham, A. Schenzle, Phys. Rev. A23, to appear.Google Scholar
  27. 21.
    R. Graham, H.J. Scholz, Phys. Rev. A22: 1198 (1980).MathSciNetADSGoogle Scholar
  28. 22.
    L.D. Landau, E.M. Lifshitz, “Fluid Mechanics”, Pergamon, London 1958.Google Scholar
  29. 23.
    L.D. Landau, E.M. Lifshitz, “Statistical Physics”, Pergamon, London 1959.Google Scholar
  30. 24.
    S. Grossmann, J. Chem. Phys. 65: 2007 (1976).Google Scholar
  31. 25.
    C. Gardiner, J. Stat. Phys. 14: 309 (1976).MathSciNetADSCrossRefGoogle Scholar
  32. 26.
    N. van Kampen, in “Topics in Statistical Mechanics and Biophysics,” ed. R. A. Piccirelli, 1976.Google Scholar
  33. 27.
    W. Horsthemke, L. Brenig, Z. Physik B27: 341 (1977).MathSciNetADSGoogle Scholar
  34. 28.
    S.R. DeGroot, P. Mazur, “Non equilibrium thermodynamics,” North Holland, Amsterdam 1962.Google Scholar
  35. 29.
    A. Schenzle, H. Brand, Optics Commun. 23: 151 (1978).Google Scholar
  36. 30.
    A. Schenzle, H. Brand, Optics Commun. 31: 401 (1979).ADSCrossRefGoogle Scholar
  37. 31.
    R. Graham, Phys. Rev. Lett. 31: 1479 (1973)ADSCrossRefGoogle Scholar
  38. 31a.
    R. Graham, Phys. Rev. A10: 1762 (1974).ADSGoogle Scholar
  39. 32.
    W.A. Smith, Phys. Rev. Lett. 32: 1164 (1974).ADSCrossRefGoogle Scholar
  40. 33.
    I.S. Gradsteyn, I.M. Ryshik, “Table of Integrals, Series and Products,” Academic, New York 1965.Google Scholar
  41. 34.
    M. Abramowitz, I. Stegun, “Handbook of Mathematical Functions,” Dover, New York 1965.Google Scholar
  42. 35.
    H. Haken, Phys. Lett. 53A: 77(1975).Google Scholar
  43. 36.
    R. Graham, Phys. Lett. 58A: 440 (1976).ADSGoogle Scholar
  44. 37.
    K.A. Robbins, Proc. Natl. Acad. Sci. USA 73: 4297 (1976).ADSCrossRefGoogle Scholar
  45. 38.
    J.B. McLaughlin, P.C. Martin, Phys. Rev. A12: 186 (1975); O. Lanford in “Turbulence Seminar”, Lecture Notes in Mathematics 615: 113, Springer, New York 1977.ADSGoogle Scholar

Copyright information

© Plenum Press, New York 1981

Authors and Affiliations

  • Robert Graham
    • 1
  1. 1.Fachbereich PhysikUniversity EssenWest Germany

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