The fractional Laplacian as a limiting case of a self-similar spring model and applications to n-dimensional anomalous diffusion

Michelitsch, Thomas; Maugin, Gérard A.; Nowakowski, Andrzej; Nicolleau, F.; Mujibur, Rahman

doi:10.2478/s13540-013-0052-5

Cited by 20 publications

(38 citation statements)

References 32 publications

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“…Let us consider transition matrix (23) for large ( n s=1 (p s − q s ) 2 ) 1 2 = | p − q| >> 1 on large lattices and finite times. We then can evaluate (22) by introducing the vector k with the components k j = ϕ j | p − q| where the integration limits can be put ±∞ to obtain the leading asymptotic contribution…”

Section: Asymptotic Behavior Of Fractional Transition Matrix: Emergenmentioning

confidence: 99%

Fractional random walk lattice dynamics

Michelitsch¹,

Collet²,

Riascos³

et al. 2017

J. Phys. A: Math. Theor.

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We analyze time-discrete and continuous 'fractional' random walks on undirected regular networks with special focus on cubic periodic lattices in n = 1, 2, 3, .. dimensions. The fractional random walk dynamics is governed by a master equation involving fractional powers of Laplacian matrices L α 2 where α = 2 recovers the normal walk. First we demonstrate that the interval 0 < α ≤ 2 is admissible for the fractional random walk. We derive analytical expressions for fractional transition matrix and closely related the average return probabilities. We further obtain the fundamental matrix Z (α) , and the mean relaxation time (Kemeny constant) for the fractional random walk. The representation for the fundamental matrix Z (α) relates fractional random walks with normal random walks. We show that the fractional transition matrix elements exihibit for large cubic n-dimensional lattices a power law decay of an n-dimensional infinite space Riesz fractional derivative type indicating emergence of Lévy flights. As a further footprint of Lévy flights in the n-dimensional space, the fractional transition matrix and fractional return probabilities are dominated for large times t by slowly relaxing long-wave modes leading to a characteristic t − n α -decay. It can be concluded that, due to long range moves of fractional random walk, a small world property is emerging increasing the efficiency to explore the lattice when instead of a normal random walk a fractional random walk is chosen.

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Section: Asymptotic Behavior Of Fractional Transition Matrix: Emergenmentioning

confidence: 99%

Fractional random walk lattice dynamics

Michelitsch¹,

Collet²,

Riascos³

et al. 2017

J. Phys. A: Math. Theor.

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“…Introducing the new vector valued integration variable ξ = κp (ξ j = pκ j , ∀j = 1, .., n) we can write for the infinite lattice integral of (28) by utilizing spherical polar coordinates p = p e p ( e p · e p = 1, p 2 = n j p 2 j ) , e.g. [12,13], and holds only for 0 < α < 2. Relation (40) does not hold for α = 2 where the fractional Laplacian takes asymptotically for α → 2 − 0 the localized singular distributional representation (−∆)δ n (p) of the standard Laplacian.…”

Section: Asymptotic Behaviormentioning

confidence: 99%

“…We can identify the asymptotic representation (39), (40) with the kernel of Riesz fractional derivative (fractional Laplacian) of the nD infinite space. For a more detailed discussion of properties we refer to [12,13].…”

Section: Asymptotic Behaviormentioning

confidence: 99%

A fractional generalization of the classical lattice dynamics approach

Michelitsch

Collet

Riascos

et al. 2016

Chaos, Solitons & Fractals

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We develop physically admissible lattice models in the harmonic approximation which define by Hamilton's variational principle fractional Laplacian matrices of the forms of power law matrix functions on the n-dimensional periodic and infinite lattice in n = 1, 2, 3, .. dimensions. The present model which is based on Hamilton's variational principle is confined to conservative non-dissipative isolated systems. The present approach yields the discrete analogue of the continuous space fractional Laplacian kernel. As continuous fractional calculus generalizes differential operators such as the Laplacian to non-integer powers of Laplacian operators, the fractional lattice approach developed in this paper generalized difference operators such as second difference operators to their fractional (non-integer) powers. Whereas differential operators and difference operators constitute local operations, their fractional generalizations introduce nonlocal long-range features. This is true for discrete and continuous fractional operators. The nonlocality property of the lattice fractional Laplacian matrix allows to describe numerous anomalous transport phenomena such as anomalous fractional diffusion and random walks on lattices. We deduce explicit results for the fractional Laplacian matrix in 1D for finite periodic and infinite linear chains and their Riesz fractional derivative continuum limit kernels.The fractional lattice Laplacian matrix contains for α = 2 the classical local lattice approach with well known continuum limit of classic local standard elasticity, and for other integer powers to gradient elasticity. We also present a generalization of the fractional Laplacian matrix to n-dimensional cubic periodic (nD tori) and infinite lattices. We show that in the continuum limit the fractional Laplacian matrix yields the well-known kernel of the Riesz fractional Laplacian derivative being the kernel of the fractional power of Laplacian operator. In this way we demonstrate the interlink of the fractional lattice approach with existing continuous fractional calculus. The developed approach appears to be useful to analyze fractional random walks on lattices as well as fractional wave propagation phenomena in lattices.

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“…Let us also note that for the description of a complex inner microstructure shown in Fig. 1d, a more complex mathematical analysis may be required such as [81] for self-similar surfaces or as [87,88] for fractal-like solids.…”

Section: Introductionmentioning

confidence: 99%

On effective properties of materials at the nano- and microscales considering surface effects

Eremeyev

2015

Acta Mech

166

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In the last years, the rapid increase in the technical capability to control and design materials at the nanoscale has pushed toward an intensive exploitation of new possibilities concerning optical, chemical, thermoelectrical and electronic applications. As a result, new materials have been developed to obtain specific physical properties and performances. In this general picture, it was natural that the attention toward mechanical characterization of the new structures was left, in a sense, behind. Anyway, once the theoretically designed objects proceed toward concrete manufacturing and applications, an accurate and general description of their mechanical properties becomes more and more scientifically relevant. The aim of the paper is therefore to discuss new methods and techniques for modeling the behavior of nanostructured materials considering surface/interface properties, which are responsible for the main differences between nano-and macroscale, and to determine their actual material properties at the macroscale. Our approach is intended to study the mechanical properties of materials taking into account surface properties including possible complex inner microstructure of surface coatings. We use the Gurtin-Murdoch model of surface elasticity. We consider the inner regular and irregular surface thin coatings (i.e., ordered or disordered nanofibers arrays) and present few examples of averaged 2D properties of them. Since the actual 2D properties depend not only on the mechanical properties of fibers or other elements of a coating, but also on the interaction forces between them, the analysis also includes information on the geometry of the microstructure of the coating, on mechanical properties of elements and on interaction forces. Further we use the obtained 2D properties to derive the effective properties of solids and structures at the macroscale, such as the bending stiffness or Young's modulus.

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The fractional Laplacian as a limiting case of a self-similar spring model and applications to n-dimensional anomalous diffusion

Cited by 20 publications

References 32 publications

Fractional random walk lattice dynamics

Fractional random walk lattice dynamics

A fractional generalization of the classical lattice dynamics approach

On effective properties of materials at the nano- and microscales considering surface effects

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