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On the quantum CTRW approach

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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.

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References

  1. E.W. Montroll, G.H. Weiss, J. Math. Phys. 6, 167 (1965)

    Article  ADS  Google Scholar 

  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)

  3. E.W. Montroll, B.J. West, in Fluctuation Phenomena, edited by E.W. Montroll, J.L. Lebowitz (North-Holland, Amsterdam, 1987)

  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)

  5. M. Lax, Rev. Mod. Phys. 32, 25 (1960)

    Article  ADS  Google Scholar 

  6. H. Scher, M. Lax, Phys. Rev. B 7, 4491 (1973)

    Article  ADS  MathSciNet  Google Scholar 

  7. H. Scher, M. Lax, Phys. Rev. B 7, 4502 (1973)

    Article  ADS  Google Scholar 

  8. T. Odagaki, M. Lax, Phys. Rev. B 24, 5284 (1981)

    Article  ADS  Google Scholar 

  9. E. Hernandez-García, M.A. Rodríguez, L. Pesquera, M. San Miguel, Phys. Rev. B 42, 10653 (1990)

    Article  ADS  Google Scholar 

  10. E. Hernandez-García, M.O. Cáceres, Phys. Rev. A 42, 4503 (1990)

    Article  ADS  Google Scholar 

  11. M.O. Cáceres, H. Matsuda, T. Odagaki, D.P. Prato, W. Lamberti, Phys. Rev. B 56, 5897 (1997)

    Article  ADS  Google Scholar 

  12. B.D. Hughes, in Random Walks and Random Environments (Clarendon Press, Oxford, 1995), Vols. 1 and 2

  13. C.E. Budde, M.O. Cáceres, Phys. Rev. Lett. 60, 2712 (1988)

    Article  ADS  MathSciNet  Google Scholar 

  14. M.O. Cáceres, C.E. Budde, M.A. Re, Phys. Rev. E 52, 3462 (1995)

    Article  ADS  Google Scholar 

  15. A. Compte, Phys. Rev. E 53, 4191 (1996)

    Article  ADS  Google Scholar 

  16. A. Compte, Phys. Rev. E 55, 6821 (1997)

    Article  ADS  Google Scholar 

  17. A. Compte, M.O. Cáceres, Phys. Rev. Lett. 81, 3140 (1998)

    Article  ADS  Google Scholar 

  18. U. Landman, W.W. Montroll, M.F. Shlesinger, Proc. Natl. Acad. Sci. 74, 430 (1977)

    Article  ADS  Google Scholar 

  19. G.H. Weiss, J. Stat. Phys. 15, 157 (1976)

    Article  ADS  Google Scholar 

  20. M.O. Cáceres, Phys. Rev. A 33, 647 (1986)

    Article  ADS  Google Scholar 

  21. C.B. Briozzo, C.E. Budde, O. Osenda, M.O. Cáceres, J. Stat. Phys. 65, 167 (1991)

    Article  ADS  Google Scholar 

  22. M.O. Cáceres, C.E. Budde, Physica A 153, 315 (1988)

    Article  ADS  MathSciNet  Google Scholar 

  23. C.E. Briozzo, C.E. Budde, M.O. Cáceres, Phys. Rev. A 39, 6010 (1989)

    Article  ADS  Google Scholar 

  24. M.O. Cáceres, H. Schnörer, A. Blumen, Phys. Rev. A 42, 4462 (1990)

    Article  ADS  Google Scholar 

  25. J.W. Haus, K.W. Kehr, Phys. Rep. 150, 263 (1987)

    Article  ADS  Google Scholar 

  26. J.-P. Bouchaud, Phys. Rep. 195, 127 (1990)

    Article  ADS  MathSciNet  Google Scholar 

  27. E.R. Reyes, M.O. Cáceres, P.A. Pury, Phys. Rev. B 61, 308 (2000)

    Article  ADS  Google Scholar 

  28. P.A. Pury, M.O. Cáceres, Phys. Rev. E 66, 021112 (2002)

    Article  ADS  Google Scholar 

  29. M.O. Cáceres, Phys. Rev. E 69, 036302 (2004)

    Article  ADS  Google Scholar 

  30. M.O. Cáceres, Non-equilibrium Statistical Physics with Application to Disordered Systems (Springer, 2017)

  31. R. Balescu, Statistical Dynamics (Imperial College Press, London, 1997)

  32. R. Balescu, Phys. Rev. E 55, 2465 (1997)

    Article  ADS  Google Scholar 

  33. O. Mülken, A. Blumen, Phys. Rep. 502, 37 (2011)

    Article  ADS  MathSciNet  Google Scholar 

  34. W. Dur, R. Raussendorf, V.M. Kendon, H.-L. Briegel, Phys. Rev. A 66, 052319 (2002)

    Article  ADS  Google Scholar 

  35. P. Xue, R. Zhang, H. Qin, X. Zhan, Z.H. Bian, J. Li, B.C. Sanders, Phys. Rev. Lett. 114, 140502 (2015)

    Article  ADS  Google Scholar 

  36. A.K. Chattah, M.O. Cáceres, Physica D 168, 258 (2002)

    Article  ADS  Google Scholar 

  37. J.P. Keating, N. Linden, J.C.F. Matthews, A. Winter, Phys. Rev. A 76, 012315 (2007)

    Article  ADS  Google Scholar 

  38. Y. Yin, D.E. Katsanos, S.N. Evangelou, Phys. Rev. A 77, 022302 (2008)

    Article  ADS  MathSciNet  Google Scholar 

  39. N.G. van Kampen, J. Stat. Phys. 78, 299 (1995)

    Article  ADS  Google Scholar 

  40. R. Alicki, K. Lendi, in Quantum Dynamical Semigroups and Applications, Lecture Notes in Physics (Springer-Verlag, Berlin, 1987), Vol. 286

  41. Y. Jung, E. Barkai, R.J. Silbey, Chem. Phys. 284, 181 (2002)

    Article  ADS  Google Scholar 

  42. M.O. Cáceres, M. Nizama, J. Phys. A 43, 455306 (2010)

    Article  ADS  MathSciNet  Google Scholar 

  43. V.M. Kenkre, E.W. Montroll, M.F. Shlesinger, J. Stat. Phys. 9, 45 (1973)

    Article  ADS  Google Scholar 

  44. N.H. Abel, Solution de quelques problèmes à l’aide d’intégrales définies, Werke 1, 10 (1823)

    Google Scholar 

  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)

  46. And also using CTRW with multiple paths: M.O. Cáceres, G.L. Insua, J. Phys. A: Math. Gen. 38, 3711 (2005)

  47. V. Ditkine, A. Proudnikov, Transformations intégrales et calcul opérationnel (Editions MIR, Moscou, 1978)

  48. M. Nizama, M.O. Cáceres, J. Phys. A: Math. Theor. 45, 335303 (2012)

    Article  Google Scholar 

  49. M. Nizama, M.O. Cáceres, Physica A 400, 31 (2014)

    Article  ADS  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. Centro Atómico Bariloche, CNEA, Instituto Balseiro, Universidad Nacional de Cuyo, and CONICET, 8400, Bariloche, Argentina

    Manuel O. Cáceres

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  1. Manuel O. Cáceres
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Correspondence to Manuel O. Cáceres.

Additional information

Contribution to the Topical Issue “Continuous Time Random Walk Still Trendy: Fifty-year History, Current State and Outlook”, edited by Ryszard Kutner and Jaume Masoliver.

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Cáceres, M.O. On the quantum CTRW approach. Eur. Phys. J. B 90, 74 (2017). https://doi.org/10.1140/epjb/e2017-80009-8

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  • DOI: https://doi.org/10.1140/epjb/e2017-80009-8

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