Skip to main content
SpringerLink
Log in

Quantization of Hénon's map with dissipation

  • Published:
Zeitschrift für Physik B Condensed Matter

Abstract

Hénon's map with dissipation is suspended to the nonlinearly kicked damped harmonic oscillator and then quantized. The ensuing master equation between two subsequent kicks is solved exactly in the representation by the Wigner distribution, resulting in a quantized version of Hénon's dissipative map. The semi-classical limit of the map is studied. The leading quantum corrections are shown to be associated with dissipation and can be formulated as a classical map with classical stochastic perturbations. The next-to-leading quantum corrections, arising from the nonlinearity of the kicks, are similar as in the area conserving map and cannot be described within the framwork of classical statistics. The Wigner distribution in the steady state is investigated in the limit of strong dissipation, where Hénon's map is reduced to the logistic map. The insensitivity of the main results against details of the quantization procedure is demonstrated by comparing with the results of a different phenomenological quantization procedure.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Lichtenberg, A.J., Lieberman, M.A.: Regular and stochastic motion. In: Applied Mathematical Sciences. Vol. 38. Berlin, Heidelberg, New York: Springer-Verlag 1983; Universality in chaos. Cvitanović, P. (ed.). Bristol: Adam Hilger 1984

    Google Scholar 

  2. Helleman, R.H.G.: In: Fundamental problems in statistical mechanics. Cohen, E.G.D. (ed.), Vol. 5. Amsterdam: North Holland 1980

    Google Scholar 

  3. Collet, P., Eckmann, J.P.: Iterated maps on the interval as dynamical systems. Basel: Birkhäuser 1980

    Google Scholar 

  4. Hénon, M.: Commun. Math. Phys.50, 69 (1976)

    Google Scholar 

  5. Casati, G., Chirikov, B.V., Izrailev, F.M., Ford, J.: Stochastic behavior in classical and quantum hamiltonian systems. In: Lecture Notes in Physics. Casati, G., Ford, J. (eds.), Vol. 93, p. 334. Berlin, Heidelberg, New York: Springer-Verlag 1979

    Google Scholar 

  6. Hannay, J.H., Berry, M.V.: Physica1D, 267 (1980)

    Google Scholar 

  7. Izrailev, F.M., Shepalyansky, D.L.: Theor. Math. Phys.43, 553 (1980)

    Google Scholar 

  8. Shepalyansky, D.L.: Theor. Math. Phys.49, 925 (1981)

    Google Scholar 

  9. Zaslavsky, G.M.: Phys. Rep.80, 157 (1981)

    Google Scholar 

  10. Hogg, T., Huberman, B.A.: Phys. Rev. Lett.48, 711 (1982); Phys. Rev. A28, 22 (1983)

    Google Scholar 

  11. Shmuel Fishman, Grempel, D.R., Prange, R.E.: Phys. Rev. Lett.49, 509 (1982)

    Google Scholar 

  12. Grempel, D.R., Prange, R.E., Shmuel Fishman: Phys. Rev. A29, 1639 (1984)

    Google Scholar 

  13. Berry, M.V., Balázs, N.L., Tabor, M., Voros, A.: Ann. Phys. (NY)122, 26 (1979)

    Google Scholar 

  14. Korsch, H.J., Berry, M.V.: Physica3D, 627 (1981)

    Google Scholar 

  15. Graham, R.: Phys. Lett.99A, 131 (1983)

    Google Scholar 

  16. Graham, R.: Z. Phys. B — Condensed Matter59, 75 (1985)

    Google Scholar 

  17. Kaplan, J.L., York, J.A.: Functional differential equations and approximation of fixed points. In: Lecture Notes in Mathematics. Peitgen, H.-O., Walther, H.-O. (eds.), Vol. 730, p. 228. Berlin, Heidelberg, New York: Springer-Verlag 1979

    Google Scholar 

  18. Haken, H.: Laser theory. In: Encyclopedia of Physics. Vol. XXV/2c. Berlin, Heidelberg, New York: Springer-Verlag 1970

    Google Scholar 

  19. Haken, H.: Phys. Lett.53A, 77 (1975)

    Google Scholar 

  20. Graham, R.: Phys. Rev. Lett.53, 2020 (1984)

    Google Scholar 

  21. Lorenz, E.N.: J. Atmos. Sci.20, 130 (1963)

    Google Scholar 

  22. Hénon, M., Pomeau, Y.: Turbulence and navier stokes equations. In: Lecture Notes in Mathematical Physics. Temam, R. (ed.), Vol. 565, p. 29. Berlin, Heidelberg, New York: Springer-Verlag 1976

    Google Scholar 

  23. Schwinger, J.: Phys. Rev.75, 1912 (1949)

    Google Scholar 

  24. Louisell, W.H.: Quantum statistical properties of radiation. London: Wiley 1973

    Google Scholar 

  25. Haake, F.: Quantum statistics in optics and solid state physics. In: Springer Trasts in Modern Physics. Vol. 66, p. 98. Berlin, Heidelberg, New York: Springer-Verlag 1973

    Google Scholar 

  26. Caldeira, A.O., Leggett, A.J.: Ann. Phys. (NY)149, 374 (1983)

    Google Scholar 

  27. Grabert, H., Weiss, U., Talkner, P.: Z. Phys. B — Condensed Matter55, 87 (1984)

    Google Scholar 

  28. Haake, F., Reibold, R.: Strong damping and low temperature anomalies for the harmonic oscillator (to be published)

  29. Wigner, E.P.: Phys. Rev.40, 749 (1932)

    Google Scholar 

  30. Tatarskii, V.I.: Sov. Phys. Usp.26, 311 (1983)

    Google Scholar 

  31. Hillery, M., O'Connell, R.F., Scully, M.O., Wigner, E.P.: Phys. Rep.106, 121 (1984)

    Google Scholar 

  32. Risken, H., Schmid, C., Weidlich, W.: Z. Phys.193, 37 (1966)

    Google Scholar 

  33. Abramowitz, M., Stegun, I.A.: Handbook of mathematical functions. New York: Dover 1965

    Google Scholar 

  34. Holmes, P.: Philos. Trans. R. Soc. A292, 419 (1979)

    Google Scholar 

  35. Lozi, R.: J. Phys. (Paris)39C5, 9 (1978); In: Intrinsic stochasticity in plasmas. Laval, G., Gresillon, D. (eds.). Orsay: Edition de Physique 1979

    Google Scholar 

  36. Mayer-Kress, G., Haken, H.: J. Stat. Phys.26, 149 (1981)

    Google Scholar 

  37. Crutchfield, J.P., Nauenberg, M., Rudnick, J.: Phys. Rev. Lett.46, 993 (1981)

    Google Scholar 

  38. Shraiman, B., Wayne, C.E., Martin, P.C.: Phys. Rev. Lett.46, 935 (1981)

    Google Scholar 

  39. Crutchfield, J.P., Farmer, J.D., Huberman, B.A.: Phys. Rep.92, 45 (1982)

    Google Scholar 

  40. Bridges, P., Rowlands, G.: Phys. Lett.63A, 189 (1977)

    Google Scholar 

  41. Daido, H.: Prog. Theor. Phys.63, 1190 (1980)

    Google Scholar 

  42. Tél, T.: Z. Phys. B — Condensed Matter49, 157 (1982); J. Stat. Phys.33, 195 (1983)

    Google Scholar 

  43. Yamaguchi, Y., Mishima, N.: Phys. Lett.104A, 179 (1984)

    Google Scholar 

  44. Grossmann, S., Thomae, S.: Z. Naturforsch.32a, 1353 (1977)

    Google Scholar 

  45. Ott, E., Hanson, J.D.: Phys. Lett.85A, 20 (1981)

    Google Scholar 

  46. Graham, R.: Phys. Rev. A28, 1679 (1983) (Erratum: Phys. Rev. A30, 2805 (1984))

    Google Scholar 

  47. Ben-Mizrachi, A., Procaccia, I., Grassberger, P.: Phys. Rev. A29, 975 (1984)

    Google Scholar 

  48. Agarwal, G.S.: Phys. Rev.178, 2025 (1969); Phys. Rev. A4, 739 (1971)

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Fachbereich Physik, Universität GHS, Postfach 6843, D-4300, Essen, Federal Republic of Germany

    R. Graham & T. Tél

Authors
  1. R. Graham
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. T. Tél
    View author publications

    You can also search for this author in PubMed Google Scholar

Additional information

Dedicated to B. Mühlschlegel on the occasion of his 60th birthday

On leave from and address after 1st of February 1985: Institute for Theoretical Physics, Eötvös University, H-1088 Budapest, Hungary

Rights and permissions

Reprints and permissions

About this article

Cite this article

Graham, R., Tél, T. Quantization of Hénon's map with dissipation. Z. Physik B - Condensed Matter 60, 127–136 (1985). https://doi.org/10.1007/BF01304431

Download citation

  • Received:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF01304431

Keywords

Search

Navigation