Fractional Laplacian in bounded domains

Zoia, Andrea; Rosso, Alberto; Kardar, Mehran

doi:10.1103/physreve.76.021116

Cited by 230 publications

(310 citation statements)

References 45 publications

(59 reference statements)

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“…First, the standard approach to deal with the finiteness of open disordered systems, via the extrapolation length, should be revisited to take into account non-exponential step length distributions. For power-law distributions, this problem is directly related to the treatment of boundaries in the presence of spatial nonlocality [35]. Second, our scaling analysis has been performed on a restricted thickness range while it would be interesting to investigate, both experimentally and theoretically, the limit towards infinite system size [36,37].…”

Section: Discussionmentioning

confidence: 99%

Walk dimension for light in complex disordered media

Savo

Burresi²,

Svensson³

et al. 2014

Phys. Rev. A

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Transport in complex systems is characterized by a fractal dimension -the walk dimension -that indicates the diffusive or anomalous nature of the underlying random walk process. Here we report on the experimental retrieval of this key quantity, using light waves propagating in disordered media. The approach is based on measurements of the time-resolved transmission, in particular on how the lifetime scales with sample size. We show that this allows one to retrieve the walk dimension and apply the concept to samples with varying degree of fractal heterogeneity. In addition, the method provides the first experimental demonstration of anomalous light dynamics in a random medium.

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Section: Discussionmentioning

confidence: 99%

Walk dimension for light in complex disordered media

Savo

Burresi²,

Svensson³

et al. 2014

Phys. Rev. A

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“…See e.g. [1,[9][10][11] for a discussion of some of those issues, specifically in connection with a correct spatial shape of the pertinent eigenfunctions.…”

Section: Cauchy Well Eigenvalue Problemmentioning

confidence: 99%

“…Currently, the status of undoubtful relevance have approximate statements (various estimates) pertaining to the asymptotic behavior of eigenfunctions at the well boundaries and estimates, of varied degree of accuracy, of the eigenvalues, c.f. [11] and [12]- [16].…”

Section: Remarkmentioning

confidence: 99%

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Solving fractional Schrödinger-type spectral problems: Cauchy oscillator and Cauchy well

Żaba

Garbaczewski

2014

Journal of Mathematical Physics

111

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This paper is a direct offspring of Ref. [1] where basic tenets of the nonlocally induced random and quantum dynamics were analyzed. A number of mentions was maid with respect to various inconsistencies and faulty statements omnipresent in the literature devoted to so-called fractional quantum mechanics spectral problems. Presently, we give a decisive computer-assisted proof, for an exemplary finite and ultimately infinite Cauchy well problem, that spectral solutions proposed so far were plainly wrong. As a constructive input, we provide an explicit spectral solution of the finite Cauchy well. The infinite well emerges as a limiting case in a sequence of deepening finite wells. The employed numerical methodology (algorithm based on the Strang splitting method) has been tested for an exemplary Cauchy oscillator problem, whose analytic solution is available. An impact of the inherent spatial nonlocality of motion generators upon computer-assisted outcomes (potentially defective, in view of various cutoffs), i.e. detailed eigenvalues and shapes of eigenfunctions, has been analyzed.2

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“…For α < 2, a test particle moving in the linear potential can change its position via extremely long, jump like excursions. This in turn requires the use of nonlocal boundary conditions in evaluation of the mean first passage time (MFPT) [23,29,30,31]. In this paragraph the above issue is taken care of when generating first passage times (FPTs) by Monte Carlo simulations.…”

Section: B Resonant Activationmentioning

confidence: 99%

Lévy stable noise-induced transitions: stochastic resonance, resonant activation and dynamic hysteresis

Dybiec¹,

Gudowska–Nowak²

2009

J. Stat. Mech.

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A standard approach to analysis of noise-induced effects in stochastic dynamics assumes a Gaussian character of the noise term describing interaction of the analyzed system with its complex surroundings. An additional assumption about the existence of timescale separation between the dynamics of the measured observable and the typical timescale of the noise allows external fluctuations to be modeled as temporally uncorrelated and therefore white. However, in many natural phenomena the assumptions concerning the abovementioned properties of "Gaussianity" and "whiteness" of the noise can be violated. In this context, in contrast to the spatiotemporal coupling characterizing general forms of non-Markovian or semi-Markovian Lévy walks, so called Lévy flights correspond to the class of Markov processes which still can be interpreted as white, but distributed according to a more general, infinitely divisible, stable and non-Gaussian law. Lévy noise-driven non-equilibrium systems are known to manifest interesting physical properties and have been addressed in various scenarios of physical transport exhibiting a superdiffusive behavior. Here we present a brief overview of our recent investigations aimed to understand features of stochastic dynamics under the influence of Lévy white noise perturbations. We find that the archetypal phenomena of noise-induced ordering are robust and can be detected also in systems driven by non-Gaussian, heavy-tailed fluctuations with infinite variance.

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Fractional Laplacian in bounded domains

Cited by 230 publications

References 45 publications

Walk dimension for light in complex disordered media

Walk dimension for light in complex disordered media

Solving fractional Schrödinger-type spectral problems: Cauchy oscillator and Cauchy well

Lévy stable noise-induced transitions: stochastic resonance, resonant activation and dynamic hysteresis

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