Topological approximation of the nonlinear Anderson model

Milovanov, Alexander V.; Iomin, Alexander

doi:10.1103/physreve.89.062921

Cited by 21 publications

(78 citation statements)

References 92 publications

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“…We reiterate that non-diagonal elements V k,m1,m2,m3 characterize couplings between each four eigenstates with wave numbers k, m 1 , m 2 , and m 3 . The comprehension of Hamiltonian character of the dynamics paves the way for a consistency analysis of the various transport scenarios behind the Anderson localization problem (with the topology of resonance overlap taken into account) [10,11]. To this end, the transport problem for the wave function becomes essentially a topological problem in phase space.…”

Section: The Three-step Topological Approachmentioning

confidence: 99%

“…Then the percolation transition threshold can be translated into a critical value of the nonlinearity control parameter, such that above this value the field spreads to infinity, and is dynamically localized in spite of these nonlinearities otherwise. This critical value when account is taken for hierarchical geometry of the Cayley tree is found to be β c = 1/ ln 2 ≈ 1.4427 [10,11], a fancy number representing the topology of nonlinear interaction posed by the quadratic power term. It was argued based on a random walk approach that in vicinity of the criticality the spreading of the wave field is subdiffusive in the limit t → +∞, and that the second moments grow with time as a power law…”

Section: Introductionmentioning

confidence: 99%

“…This localization-delocalization transition bears signatures, enabling to associate it with a percolation transition on the infinite Cayley tree (Bethe lattice) [10,11]. The main idea here is that delocalization occurs through infinite clusters of chaotic states on a Bethe lattice, with occupancy probabilities decided by the strength of nonlinear interaction.…”

Section: Introductionmentioning

confidence: 99%

“…In a recent investigation of NLSE with disorder, we have shown [11] that the critical strength destroying localization is only preserved through dynamics, if s = 1. For s > 1 (and similarly for 0 < s < 1, a regime not considered here), the critical strength is dynamic in that it involves a dependence on the number of already excited modes (the latter are the exponentially localized modes of the linear disordered lattice).…”

Section: Introductionmentioning

confidence: 99%

“…The phenomena of critical spreading find their significance in some connection with the general problem [12] of transport along separatrices of dynamical systems with many degrees of freedom and are mathematically related to a description [13][14][15][16] in terms of Hamiltonian pseudochaos (random non-chaotic dynamics with zero Lyapunov exponents) [17,18] and time-fractional diffusion equations. For s > 1, the phenomena of field spreading are limited to finite clusters at the onset of delocalization [11]. Mathematically, this regime of field spreading is complicated by the fact that finiteness of clusters on which the transport processes concentrate conflicts with the assumptions of threshold percolation, breaking the universal scaling laws [19][20][21][22] which pertain to the infinite clusters.…”

Section: Introductionmentioning

confidence: 99%

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

Milovanov¹,

Iomin²

2015

DNC

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This study is concerned with destruction of Anderson localization by a nonlinearity of the powerlaw type. We suggest using a nonlinear Schrödinger 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 nonlinear Anderson model and, if the exponent of the power nonlinearity is either integer or halfinteger, 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.

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Section: The Three-step Topological Approachmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

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

Milovanov¹,

Iomin²

2015

DNC

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25 Years of Self-Organized Criticality: Solar and Astrophysics

et al. 2014

Self Cite

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Shortly after the seminal paper "Self-Organized Criticality: An explanation of 1/f noise" by Bak et al. (1987), the idea has been applied to solar physics, in "Avalanches and the Distribution of Solar Flares" by Lu and Hamilton (1991). In the following years, an inspiring cross-fertilization from complexity theory to solar and astrophysics took place, where the SOC concept was initially applied to solar flares, stellar flares, and magnetospheric substorms, and later extended to the radiation belt, the heliosphere, lunar craters, the asteroid belt, the Saturn ring, pulsar glitches, soft X-ray repeaters, blazars, black-hole objects, cosmic rays, and boson clouds. The application of SOC concepts has been performed by numerical cellular automaton simulations, by analytical calculations of statistical (powerlaw-like) distributions based on physical scaling laws, and by observational tests of theoretically predicted size distributions and waiting time distributions. Attempts have been undertaken to import physical models into the numerical SOC toy models, such as the discretization of magneto-hydrodynamics (MHD) processes. The novel applications stimulated also vigorous debates about the discrimination between SOC models, SOC-like, and non-SOC processes, such as phase transitions, turbulence, random-walk diffusion, percolation, branching processes, network theory, chaos theory, fractality, multi-scale, and other complexity phenomena. We review SOC studies from the last 25 years and highlight new trends, open questions, and future challenges, as discussed during two recent ISSI workshops on this theme

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25 Years of Self-organized Criticality: Concepts and Controversies

et al. 2015

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Introduced by the late Per Bak and his colleagues, self-organized criticality (SOC) has been one of the most stimulating concepts to come out of statistical mechanics and condensed matter theory in the last few decades, and has played a significant role in the development of complexity science. SOC, and more generally fractals and power laws, have attracted much comment, ranging from the very positive to the polemical. The other papers (Aschwanden et al. in Space Sci. Rev., 2014, this issue; McAteer et al. in Space Sci. Rev., 2015, this issue; Sharma et al. in Space Sci. Rev. 2015, in preparation) in this special issue showcase the considerable body of observations in solar, magnetospheric and fusion plasma inspired by the SOC idea, and expose the fertile role the new paradigm has played in approaches to modeling and understanding multiscale plasma instabilities. This very broad impact, and the necessary process of adapting a scientific hypothesis to the conditions of a given physical system, has meant that SOC as studied in these fields has sometimes differed significantly from the definition originally given by its creators. In Bak's own field of theoretical physics there are significant observational and theoretical open questions, even 25 years on (Pruessner 2012). One aim of the present review is to address the dichotomy between the great reception SOC has received in some areas, and its shortcomings, as they became manifest in the controversies it triggered. Our article tries to clear up what we think are misunderstandings of SOC in fields more remote from its origins in statistical mechanics, condensed matter and dynamical systems by revisiting Bak, Tang and Wiesenfeld's original papers

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Topological approximation of the nonlinear Anderson model

Cited by 21 publications

References 92 publications

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

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

25 Years of Self-Organized Criticality: Solar and Astrophysics

25 Years of Self-organized Criticality: Concepts and Controversies

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