On the quantum CTRW approach

Regular Article
Part of the following topical collections:
  1. Topical issue: Continuous Time Random Walk still trendy: Fifty-year history, current state, and outlook

Abstract

The concept of continuous-time random walk is generalized into the quantum approach using a completely positive map. This approach introduces in a phenomenological way the concept of disorder in the transport problem of a quantum open system. If the waiting-time of the continuous-time renewal approach is exponential we recover a semigroup for a dissipative quantum walk. Two models of non-Markovian evolution have been solved considering different types of waiting-time functions.

References

  1. 1.
    E.W. Montroll, G.H. Weiss, J. Math. Phys. 6, 167 (1965)ADSCrossRefGoogle Scholar
  2. 2.
    G.H. Weiss, in Aspects and Applications of the Random Walk, edited by H.E. Stanley, E. Guyon (Random Material and Processes, North-Holland, Amsterdam, 1994)Google Scholar
  3. 3.
    E.W. Montroll, B.J. West, in Fluctuation Phenomena, edited by E.W. Montroll, J.L. Lebowitz (North-Holland, Amsterdam, 1987)Google Scholar
  4. 4.
    C.R. Cox, in Renewal Process, Monographs on Statistic and Applied Probability, Gen. edited by C.R. Cox, D.V. Hinkley (Chapman and Hall, London, 1962) (reprinted 1982)Google Scholar
  5. 5.
    M. Lax, Rev. Mod. Phys. 32, 25 (1960)ADSCrossRefGoogle Scholar
  6. 6.
    H. Scher, M. Lax, Phys. Rev. B 7, 4491 (1973)ADSMathSciNetCrossRefGoogle Scholar
  7. 7.
    H. Scher, M. Lax, Phys. Rev. B 7, 4502 (1973)ADSCrossRefGoogle Scholar
  8. 8.
    T. Odagaki, M. Lax, Phys. Rev. B 24, 5284 (1981)ADSCrossRefGoogle Scholar
  9. 9.
    E. Hernandez-García, M.A. Rodríguez, L. Pesquera, M. San Miguel, Phys. Rev. B 42, 10653 (1990)ADSCrossRefGoogle Scholar
  10. 10.
    E. Hernandez-García, M.O. Cáceres, Phys. Rev. A 42, 4503 (1990)ADSCrossRefGoogle Scholar
  11. 11.
    M.O. Cáceres, H. Matsuda, T. Odagaki, D.P. Prato, W. Lamberti, Phys. Rev. B 56, 5897 (1997)ADSCrossRefGoogle Scholar
  12. 12.
    B.D. Hughes, in Random Walks and Random Environments (Clarendon Press, Oxford, 1995), Vols. 1 and 2Google Scholar
  13. 13.
    C.E. Budde, M.O. Cáceres, Phys. Rev. Lett. 60, 2712 (1988)ADSMathSciNetCrossRefGoogle Scholar
  14. 14.
    M.O. Cáceres, C.E. Budde, M.A. Re, Phys. Rev. E 52, 3462 (1995)ADSCrossRefGoogle Scholar
  15. 15.
    A. Compte, Phys. Rev. E 53, 4191 (1996)ADSCrossRefGoogle Scholar
  16. 16.
    A. Compte, Phys. Rev. E 55, 6821 (1997)ADSCrossRefGoogle Scholar
  17. 17.
    A. Compte, M.O. Cáceres, Phys. Rev. Lett. 81, 3140 (1998)ADSCrossRefGoogle Scholar
  18. 18.
    U. Landman, W.W. Montroll, M.F. Shlesinger, Proc. Natl. Acad. Sci. 74, 430 (1977)ADSCrossRefGoogle Scholar
  19. 19.
    G.H. Weiss, J. Stat. Phys. 15, 157 (1976)ADSCrossRefGoogle Scholar
  20. 20.
    M.O. Cáceres, Phys. Rev. A 33, 647 (1986)ADSCrossRefGoogle Scholar
  21. 21.
    C.B. Briozzo, C.E. Budde, O. Osenda, M.O. Cáceres, J. Stat. Phys. 65, 167 (1991)ADSCrossRefGoogle Scholar
  22. 22.
    M.O. Cáceres, C.E. Budde, Physica A 153, 315 (1988)ADSMathSciNetCrossRefGoogle Scholar
  23. 23.
    C.E. Briozzo, C.E. Budde, M.O. Cáceres, Phys. Rev. A 39, 6010 (1989)ADSCrossRefGoogle Scholar
  24. 24.
    M.O. Cáceres, H. Schnörer, A. Blumen, Phys. Rev. A 42, 4462 (1990)ADSCrossRefGoogle Scholar
  25. 25.
    J.W. Haus, K.W. Kehr, Phys. Rep. 150, 263 (1987)ADSCrossRefGoogle Scholar
  26. 26.
    J.-P. Bouchaud, Phys. Rep. 195, 127 (1990)ADSMathSciNetCrossRefGoogle Scholar
  27. 27.
    E.R. Reyes, M.O. Cáceres, P.A. Pury, Phys. Rev. B 61, 308 (2000)ADSCrossRefGoogle Scholar
  28. 28.
    P.A. Pury, M.O. Cáceres, Phys. Rev. E 66, 021112 (2002)ADSCrossRefGoogle Scholar
  29. 29.
    M.O. Cáceres, Phys. Rev. E 69, 036302 (2004)ADSCrossRefGoogle Scholar
  30. 30.
    M.O. Cáceres, Non-equilibrium Statistical Physics with Application to Disordered Systems (Springer, 2017)Google Scholar
  31. 31.
    R. Balescu, Statistical Dynamics (Imperial College Press, London, 1997)Google Scholar
  32. 32.
    R. Balescu, Phys. Rev. E 55, 2465 (1997)ADSCrossRefGoogle Scholar
  33. 33.
    O. Mülken, A. Blumen, Phys. Rep. 502, 37 (2011)ADSMathSciNetCrossRefGoogle Scholar
  34. 34.
    W. Dur, R. Raussendorf, V.M. Kendon, H.-L. Briegel, Phys. Rev. A 66, 052319 (2002)ADSCrossRefGoogle Scholar
  35. 35.
    P. Xue, R. Zhang, H. Qin, X. Zhan, Z.H. Bian, J. Li, B.C. Sanders, Phys. Rev. Lett. 114, 140502 (2015)ADSCrossRefGoogle Scholar
  36. 36.
    A.K. Chattah, M.O. Cáceres, Physica D 168, 258 (2002)ADSCrossRefGoogle Scholar
  37. 37.
    J.P. Keating, N. Linden, J.C.F. Matthews, A. Winter, Phys. Rev. A 76, 012315 (2007)ADSCrossRefGoogle Scholar
  38. 38.
    Y. Yin, D.E. Katsanos, S.N. Evangelou, Phys. Rev. A 77, 022302 (2008)ADSMathSciNetCrossRefGoogle Scholar
  39. 39.
    N.G. van Kampen, J. Stat. Phys. 78, 299 (1995)ADSMathSciNetCrossRefGoogle Scholar
  40. 40.
    R. Alicki, K. Lendi, in Quantum Dynamical Semigroups and Applications, Lecture Notes in Physics (Springer-Verlag, Berlin, 1987), Vol. 286Google Scholar
  41. 41.
    Y. Jung, E. Barkai, R.J. Silbey, Chem. Phys. 284, 181 (2002)ADSCrossRefGoogle Scholar
  42. 42.
    M.O. Cáceres, M. Nizama, J. Phys. A 43, 455306 (2010)ADSMathSciNetCrossRefGoogle Scholar
  43. 43.
    V.M. Kenkre, E.W. Montroll, M.F. Shlesinger, J. Stat. Phys. 9, 45 (1973)ADSCrossRefGoogle Scholar
  44. 44.
    N.H. Abel, Solution de quelques problèmes à l’aide d’intégrales définies, Werke 1, 10 (1823)Google Scholar
  45. 45.
    Applications of the Abel’s pdf in fractal dynamics can be seen in: A.A. Budini, M.O. Cáceres, J. Phys. A: Math. Gen. 37, 5959 (2004)Google Scholar
  46. 46.
    And also using CTRW with multiple paths: M.O. Cáceres, G.L. Insua, J. Phys. A: Math. Gen. 38, 3711 (2005)Google Scholar
  47. 47.
    V. Ditkine, A. Proudnikov, Transformations intégrales et calcul opérationnel (Editions MIR, Moscou, 1978)Google Scholar
  48. 48.
    M. Nizama, M.O. Cáceres, J. Phys. A: Math. Theor. 45, 335303 (2012)CrossRefGoogle Scholar
  49. 49.
    M. Nizama, M.O. Cáceres, Physica A 400, 31 (2014)ADSMathSciNetCrossRefGoogle Scholar

Copyright information

© EDP Sciences, SIF, Springer-Verlag Berlin Heidelberg 2017

Authors and Affiliations

  1. 1.Centro Atómico Bariloche, CNEA, Instituto Balseiro, Universidad Nacional de CuyoBarilocheArgentina

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