Advertisement

Communications in Mathematical Physics

, Volume 101, Issue 2, pp 173–185 | Cite as

Large deviations from classical paths. Hamiltonian flows as classical limits of quantum flows

  • Ph. Blanchard
  • M. Sirugue
Article

Abstract

We prove that in the limit→0, the probability for the paths of the stochastic jump process associated to the quantum time evolution to be in a tublet around the classical trajectory is of order 1−exp{−A/ℏ}. We give some applications of this result to the study of the classical limit of Wigner functions.

Keywords

Neural Network Statistical Physic Complex System Time Evolution Nonlinear Dynamics 
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.
    Albeverio, S., Blanchard, Ph., Combe, Ph., Høegh-Krohn, R., Sirugue, M.: Local relativistic invariant flow for quantum fields. Commun. Math. Phys.90, 329–351 (1983)Google Scholar
  2. 2.
    Combe, Ph., Guerra, F., Rodriguez, R., Sirugue, M., Sirugue-Collin, M.: Quantum dynamical time evolutions as stochastic flows in phase space. Proceeding of the VIIth conference of I.A.M.P., Boulder—Colorado (August 1983) Physica124A, 561–574 (1984)Google Scholar
  3. 3.
    Donsker, M. D., Varadhan, S. R. S.: Asymptotic evaluation of certain markov expectations for large time. Commun. Pure Appl. Math. I,28, 1–47 (1975) II,29, 279–301 (1976); III,29, 389–461 (1976)Google Scholar
  4. 4.
    Ventsel', A. D.: Rough limit theorem on large deviations for Markov stochastic processes I. Theory Probab. Appl.21, 227–242 (1976)Google Scholar
  5. 5.
    Ventsel', A. D.: Rough limit theorem on large deviations for Markov stochastic processes II. Theory Probab. Appl.21, 499–512 (1976)Google Scholar
  6. 6.
    Ventsel', A. D.: Rough limit theorem on large deviations for Markov stochastic processes III. Theory Probab. Appl.24, 675–692 (1979)Google Scholar
  7. 7.
    Azencott, R.: Grandes déviations et applications. Cours de probabilité de Saint-Flour. Lecture Notes in Mathematics, Vol.774, Berlin, Heidelberg, New York: Springer 1978Google Scholar
  8. 8.
    Jona-Lasinio, G., Martinelli, F., Scoppola, E.: New approach to the semi-classical limit of quantum mechanics I. Multiple tunnelings in one dimension. Commun. Math. Phys.80, 223–254 (1981)Google Scholar
  9. 9.
    Jona-Lasinio, G., Martinelli, F., Scoppola E.: The semi-classical limit of quantum mechanics: a qualitative theory via stochastic mechanics. Phys. Rep.77, 313–327 (1981)Google Scholar
  10. 10.
    Azencott, R. Doss, H.: L'équation de Schrödinger quand→0: une approche probabiliste. Stochastic aspects of classical and quantum systems proceedings Marseille. In: Lecture Notes in Mathematics, vol. 1109, pp. 1–17, Berlin, Heidelberg, New York: Springer 1985Google Scholar
  11. 11.
    Simon, B.: Instantons, double wells and large deviations. Bull. A.M.S. March 1983Google Scholar
  12. 12.
    De Angelis, G. F., Jona-Lasinio, G., Sirugue, M.: Probabilistic solutions of Pauli type equations. J. Phys. A.16, 2433–2444 (1983)Google Scholar
  13. 13.
    Gihman, I. I., Skorohod, A. V.: Stochastic differential equations. Berlin, Heidelberg, New York: Springer 1972Google Scholar
  14. 14.
    Hepp, K.: The classical limit for quantum mechanical correlation functions. Commun. Math. Phys.35, 265–277 (1974)Google Scholar
  15. 15.
    Fano, U.: Description of states in quantum mechanics by density matrix and operator techniques. Rev. Mod. Phys.29, 74–93 (1957)Google Scholar
  16. 16.
    Weyl, H.: Gruppentheorie und Quantenmechanik, Leipzig: Hirzel, 1931Google Scholar
  17. 17.
    Gihman, I. I., Skohorod, A. V.: The theory of stochastic processes I, II and III. Berlin, Heidelberg, New York: Springer 1974Google Scholar
  18. 18.
    Bertrand, J., Gaveau, G.: Transformations canoniques et renormalisation pour certaines équations d'évolution. J. Funct. Anal.50, 81–99 (1983)Google Scholar
  19. 19.
    Moyal, J. E.: Quantum mechanics as a statistical theory. Proc. Camb. Phil. Soc.45, 99–124 (1949)Google Scholar
  20. 20.
    Bertrand, J., Rideau, G.: Stochastic jump processes in the phase space representation of quantum mechanics. Proceedings of the VIth conference of I.A.M.P., Berlin (1981). Lecture Notes in Physics, Vol.153, Berlin, Heidelberg, New York: Springer 1982; Stochastic processes and evolution of quantum observables. Preprint Paris VII, March 1983Google Scholar
  21. 21.
    Grossmann, A., Seiler, R.: Heat equation on phase space and the classical limit of quantum mechanical expectation values. Commun. Math. Phys.48, 195–197 (1976)Google Scholar
  22. 22.
    Sirugue, M., Sirugue-Collin, M., Truman A.: Semi-classical approximation and microcanonical ensemble. Ann. Inst. Henri Poincaré41, 429–444 (1984)Google Scholar

Copyright information

© Springer-Verlag 1985

Authors and Affiliations

  • Ph. Blanchard
    • 1
  • M. Sirugue
    • 2
  1. 1.Theoretische PhysikUniversität BielefeldBielefeld 1Federal Republic of Germany
  2. 2.Zentrum für interdisziplinäre ForschungUniversität BielefeldBielefeld 1Federal Republic of Germany

Personalised recommendations