Skip Navigation Links
Discontinuity, Nonlinearity, and Complexity

Dimitry Volchenkov (editor), Dumitru Baleanu (editor)

Dimitry Volchenkov(editor)

Mathematics & Statistics, Texas Tech University, 1108 Memorial Circle, Lubbock, TX 79409, USA

Email: dr.volchenkov@gmail.com

Dumitru Baleanu (editor)

Cankaya University, Ankara, Turkey; Institute of Space Sciences, Magurele-Bucharest, Romania

Email: dumitru.baleanu@gmail.com


Topology of Delocalization in the Nonlinear Anderson Model and Anomalous Diffusion on Finite Clusters

Discontinuity, Nonlinearity, and Complexity 4(2) (2016) 151--162 | DOI:10.5890/DNC.2016.06.003

A.V. Milovanov$^{1}$,$^{2}$,$^{4}$; A. Iomin$^{3}$,$^{4}$

$^{1}$ ENEA National Laboratory, Centro Ricerche Frascati, I-00044 Frascati, Rome, Italy

$^{2}$ Space Research Institute, Russian Academy of Sciences, 117997 Moscow, Russia

$^{3}$ Department of Physics and Solid State Institute, Technion, Haifa 32000, Israel

$^{4}$ Max-Planck-Institut f¨ur Physik komplexer Systeme, 01187 Dresden, Germany

Download Full Text PDF

 

Abstract

This study is concernedwith destruction of Anderson localization by a nonlinearity of the power-law type. We suggest using a nonlinear Schr¨odinger model with random potential on a lattice that quadratic nonlinearity plays a dynamically very distinguished role in that it is the only type of power nonlinearity permitting an abrupt localization-delocalization transition with unlimited spreading already at the delocalization border. For super-quadratic nonlinearity the borderline spreading corresponds to diffusion processes on finite clusters. We have proposed an analytical method to predict and explain such transport processes. Our method uses a topological approximation of the nonlinearAnderson model and, if the exponent of the power nonlinearity is either integer or half-integer, will yield the wanted value of the transport exponent via a triangulation procedure in an Euclidean mapping space. A kinetic picture of the transport arising from these investigations uses a fractional extension of the diffusion equation to fractional derivatives over the time, signifying non-Markovian dynamics with algebraically decaying time correlations.

Acknowledgments

A.V.M. and A.I. thank theMax-Planck-Institute for the Physics of Complex Systems for hospitality and financial support. This work was supported in part by the Israel Science Foundation (ISF) and by the ISSI project “Self-Organized Criticality and Turbulence” (Bern, Switzerland).

References

  1. [1]  Anderson, P.W. (1958), Absence of diffusion in certain random lattices, Physical Review, 109, 1492-1506.
  2. [2]  Akkermans, E. and Montambaux, G. (2006), Mesoscopic Physics of Electrons and Photons, Cambridge University Press, Cambridge.
  3. [3]  Weaver, R.L. (1990), Anderson localization of ultrasound, Wave Motion, 12, 129-142.
  4. [4]  Störzer,M., Gross, P, Aegerter, C.M., and Maret, G. (2006), Observation of the critical regime near Anderson localization of light, Physical Review Letters, 96, 063904.
  5. [5]  Schwartz, T., Bartal, G., Fishman, S., and Segev, M. (2007), Transport and Anderson localization in disordered twodimensional photonic lattices, Nature (London), 446, 52-55.
  6. [6]  Billy, J., Josse, V., Zuo, Z., Bernard, A., Hambrecht, B., Lugan, P., Clément, D., Sanchez-Palencia, L., Bouyer, P., and Aspect, A. (2008), Direct observation of Anderson localization of matter waves in a controlled disorder, Nature (London), 453, 891-894.
  7. [7]  Abou-Chacra, R., Anderson, P.W., and Thouless, D.J. (1973),A selfconsistent theory of localization, Journal of Physics C (Solid State Physics), 6, 1734-1752.
  8. [8]  Shepelyansky, D.L. (1993), Delocalization of quantum chaos by weak nonlinearity, Physical Review Letters, 70, 1787- 1791.
  9. [9]  Pikovsky, A.S. and Shepelyansky, D.L. (2008), Destruction of Anderson localization by a weak nonlinearity, Physical Review Letters, 100, 094101.
  10. [10]  Milovanov, A.V. and Iomin, A. (2012), Localization-delocalization transition on a separatrix system of nonlinear Schrödinger equation with disorder, Europhysics Letters, 100, 10006.
  11. [11]  Milovanov, A.V. and Iomin, A. (2014), Topological approximation of the nonlinear Anderson model, Physical Review E, 89, 062921.
  12. [12]  Chirikov, B.V. and Vecheslavov, V.V. (1997) Arnold diffusion in large systems, Journal of Experimental and Theoretical Physics, 112, 616-624.
  13. [13]  Milovanov, A.V. (2009), Pseudochaos and low-frequency percolation scaling for turbulent diffusion in magnetized plasma, Physical Review E, 79, 046403.
  14. [14]  Milovanov, A.V. (2010), Self-organized criticality with a fishbone-like instability cycle, Europhysics Letters, 89, 60004.
  15. [15]  Milovanov, A. V. (2011), Dynamic polarization random walk model and fishbone-like instability for self-organized critical systems, New Journal of Physics, 13, 043034.
  16. [16]  Milovanov, A.V. (2013), Percolation Models of Self-Organized Critical Phenomena, Chapter 4, Self-Organized Criticality Systems, ed. Aschwanden, M. J., Open Academic Press, Berlin, pp. 103-182.
  17. [17]  Lyubomudrov, O., Edelman, M., and Zaslavsky, G.M. (2003), Pseudochaotic systems and their fractional kinetics, International Journal of Modern Physics B, 17, 4149-4167.
  18. [18]  Zaslavsky, G.M. (2002), Chaos, fractional kinetics, and anomalous transport, Physics Reports, 371, 461-580.
  19. [19]  Havlin, S. and ben-Avraham, D. (2002), Diffusion in disordered media, Advances in Physics, 51, 187-292.
  20. [20]  Havlin, S. and ben-Avraham, D. (2002), Diffusion and Reactions in Fractals and Disordered Systems, Cambridge University Press, Cambridge.
  21. [21]  Nakayama, T., Yakubo, K., and Orbach, R.L. (1994), Dynamical properties of fractal networks: Scaling, numerical simulations, and physical realizations, Reviews of Modern Physics, 66, 381-443.
  22. [22]  Zelenyi, L.M. and Milovanov, A.V. (2004), Fractal topology and strange kinetics: from percolation theory to problems in cosmic electrodynamics, Physics-Uspekhi, 47, 749-788.
  23. [23]  Hirsch, M.W. (1997), Differential Topology, Springer, New York.
  24. [24]  Cox, D., Little, J., and O'Shea, D. (1998), Using Algebraic Geometry, Springer-Verlag, New York.
  25. [25]  Gefen, Y., Aharony, A., and Alexander, S. (1983), Anomalous diffusion on percolating clusters, Physical Review Letters, 50, 77-80.
  26. [26]  Milovanov, A.V. (1997), Topological proof for the Alexander-Orbach conjecture, Physical Review E, 56, 2437-2446.
  27. [27]  Schroeder, M.R. (1991), Fractals, Chaos, Power Laws: Minutes from an Infinite Paradise, Freeman, New York.
  28. [28]  Coniglio, A. (1982), Cluster structure near the percolation threshold, Journal of Physics A, 15, 3829-3844.
  29. [29]  O'Shaughnessy, B. and Procaccia, I. (1985), Analytical solutions for diffusion on fractal objects, Physical Review Letters, 54, 455-458.
  30. [30]  Iomin, A. (2010), Subdiffusion in the nonlinear Schrödinger equation with disorder, Physical Review E, 81, 017601.
  31. [31]  Iomin, A. (2013), Dynamics in nonlinear Schrödinger equation with dc bias: From subdiffusion to Painlevé transcendent, Mathematical Modelling of Natural Phenomena, 8, 88-99.
  32. [32]  Flach, S., Krimer, D.O., and Skokos, Ch. (2009), Universal spreading of wave packets in disordered nonlinear systems, Physical Review Letters, 102, 024101.
  33. [33]  Skokos, Ch. and Flach, S. (2010), Spreading of wave packets in disordered systems with tunable nonlinearity, Physical Review E, 82, 016208.
  34. [34]  Nash, C. and Sen, S. (1987), Topology and Geometry for Physicists, Academic Press, London.
  35. [35]  Fomenko, A.T. and Fuks, D.B. (1989), A Course of Homotopic Topology, Nauka, Moscow.
  36. [36]  Podlubny, I. (1999), Fractional Differential Equations, Academic Press, San Diego.
  37. [37]  Iomin, A. (2009), Dynamics of wave packets for the nonlinear Schrödinger equation with a random potential, Physical Review E, 80, 022103.
  38. [38]  Iomin, A. (2011), Fractional-time Schrödinger equation: Fractional dynamics on a comb, Chaos, Solitons & Fractals, 44, 348-352.
  39. [39]  Metzler, R. and Klafter, J. (2000), The random walk's guide to anomalous diffusion: a fractional dynamics approach, Physics Reports, 339, 1-77.
  40. [40]  Shlesinger, M.F., Zaslavsky, G.M., and Klafter, J. (1993), Strange kinetics, Nature (London), 363, 31-37.
  41. [41]  Sokolov I.M., Klafter, J., and Blumen, A. (2002), Fractional kinetics, Physics Today, 55, 48-54.
  42. [42]  Metzler, R. and Klafter, J. (2004), The restaurant at the end of the random walk: recent developments in the description of anomalous transport by fractional dynamics, Journal of Physics A: Mathematical and General, 37, R161-R208.
  43. [43]  Metzler, R., Jeon, J.-H., Cherstvy, A.G., and Barkai, E. (2014), Anomalous diffusion models and their properties: non-stationarity, non-ergodicity, and ageing at the centenary of single particle tracking, Physical Chemistry Chemical Physics, 16, 24128-24164.