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.
Similar content being viewed by others
References
E.W. Montroll, G.H. Weiss, J. Math. Phys. 6, 167 (1965)
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)
E.W. Montroll, B.J. West, in Fluctuation Phenomena, edited by E.W. Montroll, J.L. Lebowitz (North-Holland, Amsterdam, 1987)
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)
M. Lax, Rev. Mod. Phys. 32, 25 (1960)
H. Scher, M. Lax, Phys. Rev. B 7, 4491 (1973)
H. Scher, M. Lax, Phys. Rev. B 7, 4502 (1973)
T. Odagaki, M. Lax, Phys. Rev. B 24, 5284 (1981)
E. Hernandez-García, M.A. Rodríguez, L. Pesquera, M. San Miguel, Phys. Rev. B 42, 10653 (1990)
E. Hernandez-García, M.O. Cáceres, Phys. Rev. A 42, 4503 (1990)
M.O. Cáceres, H. Matsuda, T. Odagaki, D.P. Prato, W. Lamberti, Phys. Rev. B 56, 5897 (1997)
B.D. Hughes, in Random Walks and Random Environments (Clarendon Press, Oxford, 1995), Vols. 1 and 2
C.E. Budde, M.O. Cáceres, Phys. Rev. Lett. 60, 2712 (1988)
M.O. Cáceres, C.E. Budde, M.A. Re, Phys. Rev. E 52, 3462 (1995)
A. Compte, Phys. Rev. E 53, 4191 (1996)
A. Compte, Phys. Rev. E 55, 6821 (1997)
A. Compte, M.O. Cáceres, Phys. Rev. Lett. 81, 3140 (1998)
U. Landman, W.W. Montroll, M.F. Shlesinger, Proc. Natl. Acad. Sci. 74, 430 (1977)
G.H. Weiss, J. Stat. Phys. 15, 157 (1976)
M.O. Cáceres, Phys. Rev. A 33, 647 (1986)
C.B. Briozzo, C.E. Budde, O. Osenda, M.O. Cáceres, J. Stat. Phys. 65, 167 (1991)
M.O. Cáceres, C.E. Budde, Physica A 153, 315 (1988)
C.E. Briozzo, C.E. Budde, M.O. Cáceres, Phys. Rev. A 39, 6010 (1989)
M.O. Cáceres, H. Schnörer, A. Blumen, Phys. Rev. A 42, 4462 (1990)
J.W. Haus, K.W. Kehr, Phys. Rep. 150, 263 (1987)
J.-P. Bouchaud, Phys. Rep. 195, 127 (1990)
E.R. Reyes, M.O. Cáceres, P.A. Pury, Phys. Rev. B 61, 308 (2000)
P.A. Pury, M.O. Cáceres, Phys. Rev. E 66, 021112 (2002)
M.O. Cáceres, Phys. Rev. E 69, 036302 (2004)
M.O. Cáceres, Non-equilibrium Statistical Physics with Application to Disordered Systems (Springer, 2017)
R. Balescu, Statistical Dynamics (Imperial College Press, London, 1997)
R. Balescu, Phys. Rev. E 55, 2465 (1997)
O. Mülken, A. Blumen, Phys. Rep. 502, 37 (2011)
W. Dur, R. Raussendorf, V.M. Kendon, H.-L. Briegel, Phys. Rev. A 66, 052319 (2002)
P. Xue, R. Zhang, H. Qin, X. Zhan, Z.H. Bian, J. Li, B.C. Sanders, Phys. Rev. Lett. 114, 140502 (2015)
A.K. Chattah, M.O. Cáceres, Physica D 168, 258 (2002)
J.P. Keating, N. Linden, J.C.F. Matthews, A. Winter, Phys. Rev. A 76, 012315 (2007)
Y. Yin, D.E. Katsanos, S.N. Evangelou, Phys. Rev. A 77, 022302 (2008)
N.G. van Kampen, J. Stat. Phys. 78, 299 (1995)
R. Alicki, K. Lendi, in Quantum Dynamical Semigroups and Applications, Lecture Notes in Physics (Springer-Verlag, Berlin, 1987), Vol. 286
Y. Jung, E. Barkai, R.J. Silbey, Chem. Phys. 284, 181 (2002)
M.O. Cáceres, M. Nizama, J. Phys. A 43, 455306 (2010)
V.M. Kenkre, E.W. Montroll, M.F. Shlesinger, J. Stat. Phys. 9, 45 (1973)
N.H. Abel, Solution de quelques problèmes à l’aide d’intégrales définies, Werke 1, 10 (1823)
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)
And also using CTRW with multiple paths: M.O. Cáceres, G.L. Insua, J. Phys. A: Math. Gen. 38, 3711 (2005)
V. Ditkine, A. Proudnikov, Transformations intégrales et calcul opérationnel (Editions MIR, Moscou, 1978)
M. Nizama, M.O. Cáceres, J. Phys. A: Math. Theor. 45, 335303 (2012)
M. Nizama, M.O. Cáceres, Physica A 400, 31 (2014)
Author information
Authors and Affiliations
Corresponding author
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.
Rights and permissions
About this article
Cite this article
Cáceres, M.O. On the quantum CTRW approach. Eur. Phys. J. B 90, 74 (2017). https://doi.org/10.1140/epjb/e2017-80009-8
Received:
Revised:
Published:
DOI: https://doi.org/10.1140/epjb/e2017-80009-8