Anomalous diffusion and Hall effect on comb lattices

Burioni, Raffaella; Cassi, Davide; Giusiano, G.; Regina, S.

doi:10.1103/physreve.67.016116

Cited by 16 publications

(16 citation statements)

References 11 publications

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“…The fractional operator is also responsible for introducing a nonlinear time dependence in the mean square displacement of the system [17]. Thus, a large class of complex phenomena can be effectively described by extending the standard differential operator to a non-integer order [25][26][27][28][29][30][31][32][33][34]; indeed, as pointed out by West [35], the fractional calculus provides a suitable framework to deal with complex systems. Recently, researchers have made and promoted remarkable progress toward improving experimental techniques for investigating diffusive processes, mainly illustrated by the developments in the single-particle tracking technique [36][37][38][39].…”

Section: Introductionmentioning

confidence: 99%

The Role of Fractional Time-Derivative Operators on Anomalous Diffusion

2017

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The generalized diffusion equations with fractional order derivatives have shown be quite efficient to describe the diffusion in complex systems, with the advantage of producing exact expressions for the underlying diffusive properties. Recently, researchers have proposed different fractional-time operators (namely: the Caputo-Fabrizio and Atangana-Baleanu) which, differently from the well-known Riemann-Liouville operator, are defined by non-singular memory kernels. Here we proposed to use these new operators to generalize the usual diffusion equation. By analyzing the corresponding fractional diffusion equations within the continuous time random walk framework, we obtained waiting time distributions characterized by exponential, stretched exponential, and power-law functions, as well as a crossover between two behaviors. For the mean square displacement, we found crossovers between usual and confined diffusion, and between usual and sub-diffusion. We obtained the exact expressions for the probability distributions, where non-Gaussian and stationary distributions emerged. This former feature is remarkable because the fractional diffusion equation is solved without external forces and subjected to the free diffusion boundary conditions. We have further shown that these new fractional diffusion equations are related to diffusive processes with stochastic resetting, and to fractional diffusion equations with derivatives of distributed order. Thus, our results suggest that these new operators may be a simple and efficient way for incorporating different structural aspects into the system, opening new possibilities for modeling and investigating anomalous diffusive processes.

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Section: Introductionmentioning

confidence: 99%

The Role of Fractional Time-Derivative Operators on Anomalous Diffusion

2017

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“…Following [18] and [19] we find the analytical values for the spectral dimension for fractals A-C [20]:…”

Section: We Consider the Anderson Hamiltonianmentioning

confidence: 99%

“…This gives three scaling relations for ω. Following [19], recurrence relations are determined by the transformation matrix…”

Section: We Consider the Anderson Hamiltonianmentioning

confidence: 99%

Critical conductance distribution in various dimensions

Travěnec

Markoš

2002

Phys. Rev. B

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We study numerically the metal -insulator transition in the Anderson model on various lattices with dimension 2 < d ≤ 4 (bifractals and Euclidian lattices). The critical exponent ν and the critical conductance distribution are calculated. We confirm that ν depends only on the spectral dimension. The other parameters -critical disorder, critical conductance distribution and conductance cummulants -depend also on lattice topology. Thus only qualitative comparison with theoretical formulae for dimension dependence of the cummulants is possible.PACS numbers: 71.30.+h, 71.23.An, 72.15.Rn It is commonly accepted, though not proved, that metal -insulator transitions (MIT) can be described by oneparameter scaling theory [1]. The critical exponent ν which describes the divergence of correlation length at MIT depends only on the system dimension for a chosen universality class. Microscopic details of models do not affect it. This was confirmed by numerical analysis of quasi-one-dimensional (Q1D) systems [2]. Theoretical dependence of ν on dimension d = 2 + ε was found in [3,4] for small ε. Numerically, ν(ε) was studied on bifractals [5].The conductance g was originally chosen as the order parameter in the scaling theory [1]. Soon it became clear, that the absence of self-averaging of g in the critical region must be taken into account [6,7]. The shape of the critical conductance distribution P (g) in 3D models was numerically analysed in detail [8,9,10,11,12]. Contrary to the critical exponent, P (g) is not universal. Its shape depends not merely on the dimension [10] and physical symmetry [11] but also on boundary conditions [12] and even anisotropy [13]. Nevertheless, for a given physical model the mean conductance and resistance follow one parameter scaling [14].Analytical theory of MIT is restricted to systems with dimension close to the lower critical dimension: 2 + ε with ε ≪ 1 [15]. In spite of predicted non-universality of higher order conductance cummulants δg n ,(1) the distribution P (g) should be universal in the infinite system size limit [16]. For small ε, the bulk of the distribution is approximately Gaussian near the mean value g . The parameters of Gaussian peak,agree with the estimation of the first cummulants (1). For large g, theory predicts long power-law tailNumerical results for 3D systems show completely different P (g), indicating that theory is not applicable to large ε [8].In this Letter we present the critical exponent and the critical conductance distribution for Anderson model obtained numerically on three bifractal lattices with dimension 2 < d < 3. They all possess the same fractal dimension d f = log 3/ log 2 + 1. Two of these lattices have the same spectral dimension d s . Their critical exponents are identical within error bars. This confirms the universality of MIT. The shape of the P (g), and the value of the critical disorder, depend not only on d s but also on the lattice topology. This novel non-universality of P (g) is found also by numerical analysis of two different 3D...

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“…On the contrary, in the presence of a non-zero drift [8], the emergence of a dynamical crossover is connected to the breaking of the FDR. Indeed, what we found in the infinite L comb model is…”

Section: Comb: Diffusion and Response Functionmentioning

confidence: 99%

Einstein Relation in Systems with Anomalous Diffusion

et al. 2013

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We discuss the role of non-equilibrium conditions in the context of anomalous dynamics. We study in detail the response properties in different models, featuring subdiffusion and superdiffusion: in such models, the presence of currents induces a violation of the Einstein relation. We show how in some of them it is possible to find the correlation function proportional to the linear response, in other words, we have a generalized fluctuation-response relation.

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Anomalous diffusion and Hall effect on comb lattices

Cited by 16 publications

References 11 publications

The Role of Fractional Time-Derivative Operators on Anomalous Diffusion

The Role of Fractional Time-Derivative Operators on Anomalous Diffusion

Critical conductance distribution in various dimensions

Einstein Relation in Systems with Anomalous Diffusion

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