Weak chaos in the disordered nonlinear Schrödinger chain: Destruction of Anderson localization by Arnold diffusion

Abstract The subject of this study is the long-time equilibration dynamics of a strongly disordered one-dimensional chain of coupled weakly anharmonic classical oscillators. It is shown that chaos in this system has a very particular spatial structure: it can be viewed as a dilute gas of chaotic spots. Each chaotic spot corresponds to a stochastic pump which drives the Arnold diffusion of the oscillators surrounding it, thus leading to their relaxation and thermalization. The most important mechanism of equilibration at long distances is provided by random migration of the chaotic spots along the chain, which bears analogy with variable-range hopping of electrons in strongly disordered solids. The corresponding macroscopic transport equations are obtained.

[1]  J. Pöschel,et al.  Small divisors with spatial structure in infinite dimensional Hamiltonian systems , 1990 .

[2]  A. Iomin.  Subdiffusion in the nonlinear Schrödinger equation with disorder. , 2009, Physical review. E, Statistical, nonlinear, and soft matter physics.

[3]  L. Landau,et al.  statistical-physics-part-1 , 1958 .

[4]  A. Iomin.  Dynamics of wave packets for the nonlinear Schrödinger equation with a random potential. , 2009, Physical review. E, Statistical, nonlinear, and soft matter physics.

[5]  B. Altshuler,et al.  A finite-temperature phase transition for disordered weakly interacting bosons in one dimension , 2009, 0910.4534.

[6]  Anderson localization of elementary excitations in a one-dimensional Bose-Einstein condensate , 2006, cond-mat/0602622.

[7]  Observation of a localization transition in quasiperiodic photonic lattices. , 2008, Physical review letters.

[8]  A. Kolmogorov.  On conservation of conditionally periodic motions for a small change in Hamilton's function , 1954 .

[9]  H. Callen,et al.  Irreversibility and Generalized Noise , 1951 .

[10]  B. Chirikov.  A universal instability of many-dimensional oscillator systems , 1979 .

[11]  S. Flach.  Spreading of waves in nonlinear disordered media , 2010, 1001.2673.

[12]  C. Froeschlé,et al.  Stochasticity of dynamical systems with increasing number of degrees of freedom , 1975 .

[13]  S. Fishman,et al.  Perturbation theory for the nonlinear Schrödinger equation with a random potential , 2009, 0901.4951.

[14]  P. Marko,et al.  ABSENCE OF DIFFUSION IN CERTAIN RANDOM LATTICES , 2008 .

[15]  S. Fishman,et al.  On the Problem of Dynamical Localization in the Nonlinear Schrödinger Equation with a Random Potential , 2008 .

[16]  M. Modugno,et al.  Delocalization of a disordered bosonic system by repulsive interactions , 2009, 0910.5062.

[17]  D. Shepelyansky,et al.  Nonlinear delocalization on disordered Stark ladder , 2009, 0903.2103.

[18]  A. Dhar,et al.  Effect of phonon-phonon interactions on localization. , 2007, Physical review letters.

[19]  C. Varma,et al.  Hopping Conductivity in "One-Dimensional" Disordered Compounds , 1973 .

[20]  Zhifei Zhang,et al.  Long Time Anderson Localization for the Nonlinear Random Schrödinger Equation , 2008, 0805.3520.

[21]  M. Segev,et al.  Transport and Anderson localization in disordered two-dimensional photonic lattices , 2007, Nature.

[22]  A. Aspect,et al.  Direct observation of Anderson localization of matter waves in a controlled disorder , 2008, Nature.

[23]  George M. Zaslavsky.  Chaos in Dynamic Systems , 1985 .

[24]  T. V. Laptyeva,et al.  The crossover from strong to weak chaos for nonlinear waves in disordered systems , 2010, 1005.0485.

[25]  J. Fröhlich,et al.  Localization in disordered, nonlinear dynamical systems , 1986 .

[26]  G. Czycholl,et al.  Conductivity and localization of electron states in one dimensional disordered systems: Further numerical results , 1981 .

[27]  S. Flach,et al.  Spreading of wave packets in disordered systems with tunable nonlinearity. , 2010, Physical review. E, Statistical, nonlinear, and soft matter physics.

[28]  Rasmussen,et al.  Statistical mechanics of a discrete nonlinear system , 2000, Physical review letters.

[29]  S. Fishman,et al.  Localization length of stationary states in the nonlinear Schrödinger equation. , 2007, Physical review. E, Statistical, nonlinear, and soft matter physics.

[30]  G. Kopidakis,et al.  Absence of wave packet diffusion in disordered nonlinear systems. , 2007, Physical review letters.

[31]  G. G. Stokes.  "J." , 1890, The New Yale Book of Quotations.

[32]  A. Pikovsky,et al.  Spreading in disordered lattices with different nonlinearities , 2010, 1002.3470.

[33]  P. Wolynes,et al.  Quantization of the Stochastic Pump Model of Arnold Diffusion , 1997 .

[34]  K. Richter,et al.  Nonlinear transport of Bose-Einstein condensates through waveguides with disorder , 2005, cond-mat/0509446.

[35]  A. Pikovsky,et al.  Dynamical thermalization of disordered nonlinear lattices. , 2009, Physical review. E, Statistical, nonlinear, and soft matter physics.

[36]  A. Pikovsky,et al.  Destruction of Anderson localization by a weak nonlinearity. , 2007, Physical review letters.

[37]  Massimo Inguscio,et al.  Anderson localization of a non-interacting Bose–Einstein condensate , 2008, Nature.

[38]  C. Dutta,et al.  K-Edge Absorption Spectra of Ionic Potassium and ItsZ+1Analogy , 1980 .

[39]  S. Fishman,et al.  Spreading for the generalized nonlinear Schrödinger equation with disorder. , 2009, Physical review. E, Statistical, nonlinear, and soft matter physics.

[40]  A numerical and symbolical approximation of the Nonlinear Anderson Model , 2009, 0912.3906.

[41]  Relaxation to the invariant density for the kicked rotor , 1999, Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics.

[42]  D. Huse,et al.  Energy transport in disordered classical spin chains , 2009, 0905.4112.

[43]  Molina,et al.  Absence of localization in a nonlinear random binary alloy. , 1994, Physical review letters.

[44]  D. O. Krimer,et al.  Delocalization of wave packets in disordered nonlinear chains. , 2009, Physical review. E, Statistical, nonlinear, and soft matter physics.

[45]  P. Leboeuf,et al.  Dipole oscillations of a Bose-Einstein condensate in the presence of defects and disorder. , 2008, Physical Review Letters.

[46]  Chaotic destruction of Anderson localization in a nonlinear lattice , 2008 .

[47]  Franz Wegner,et al.  Anomaly in the band centre of the one-dimensional Anderson model , 1981 .

[48]  V. Vasil’ev,et al.  Waveguides with Random Inhomogeneities and Brownian Motion In the Lobachevsky Plane , 1959 .

[49]  B. M. Fulk.  MATH , 1992 .

[50]  D. Wiersma,et al.  Fifty years of Anderson localization , 2009 .

[51]  L. Galgani,et al.  Stochastic transition in a classical nonlinear dynamical system: A Lennard-Jones chain , 1976 .

[52]  A. Lichtenberg,et al.  Regular and Stochastic Motion , 1982 .

[53]  K. Mallick,et al.  Asymptotic localization of stationary states in the nonlinear Schrödinger equation. , 2008, Physical review. E, Statistical, nonlinear, and soft matter physics.

[54]  M. I. Molina,et al.  Transport of localized and extended excitations in a nonlinear Anderson model , 1998 .

[55]  G. Kopidakis,et al.  KAM tori in 1D random discrete nonlinear Schrödinger model? , 2010, 1007.1912.

[56]  D. O. Krimer,et al.  Delocalization and spreading in a nonlinear Stark ladder. , 2009, Physical review. E, Statistical, nonlinear, and soft matter physics.

[57]  Jean Bourgain,et al.  Quasi-periodic solutions of nonlinear random Schrödinger equations , 2008 .

[58]  Roberto Morandotti,et al.  Anderson localization and nonlinearity in one-dimensional disordered photonic lattices. , 2007, Physical review letters.

[59]  Nevill Mott,et al.  The theory of impurity conduction , 1961 .

[60]  V. V. Vecheslavov,et al.  Theory of fast arnold diffusion in many-frequency systems , 1993 .

[61]  D. O. Krimer,et al.  Universal spreading of wave packets in disordered nonlinear systems. , 2008, Physical review letters.

[62]  P. Lee.  Variable-Range Hopping in Finite One-Dimensional Wires , 1984 .

[63]  Numerical studies of variable-range hopping in one-dimensional systems , 2009, 0909.0320.

[64]  Nevill Mott,et al.  Conduction in non-crystalline materials , 1989 .

[65]  J. Bodyfelt,et al.  One-parameter scaling theory for stationary states of disordered nonlinear systems. , 2010, Physical review letters.

[66]  S. Gredeskul,et al.  Propagation and scattering of nonlinear waves in disordered systems , 1992 .

[67]  P. Schlagheck,et al.  Superfluidity versus Anderson localization in a dilute Bose gas. , 2007, Physical review letters.

[68]  D. Thouless.  A relation between the density of states and range of localization for one dimensional random systems , 1972 .

[69]  Shepelyansky.  Delocalization of quantum chaos by weak nonlinearity. , 1993, Physical review letters.

[70]  L. Galgani,et al.  Numerical computations on a stochastic parameter related to the Kolmogorov entropy , 1976 .

[71]  J. Kurkijärvi.  Hopping Conductivity in One Dimension , 1973 .

[72]  Miss A.O. Penney.  (b) , 1974, The New Yale Book of Quotations.

[73]  P. Schlagheck,et al.  Anderson localization of a weakly interacting one dimensional Bose gas , 2009, 0907.0098.

[74]  A. Pikovsky,et al.  Scaling of energy spreading in strongly nonlinear disordered lattices. , 2010, Physical review. E, Statistical, nonlinear, and soft matter physics.

[75]  A. Pikovsky,et al.  Scaling properties of weak chaos in nonlinear disordered lattices. , 2010, Physical review. E, Statistical, nonlinear, and soft matter physics.