Superdiffusion on a Comb Structure

Baskin, E.; Iomin, Alexander

doi:10.1103/physrevlett.93.120603

Cited by 99 publications

(84 citation statements)

References 21 publications

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“…It is a particular example of a non-Markovian phenomenon, which was explained in the framework of continuous time random walks [2,[4][5][6][7]. In the last decade the comb model has been extensively studied for understanding of different realizations of nonMarkovian random walks both continuous [8][9][10] and discrete [11].…”

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confidence: 99%

Subdiffusion on a fractal comb

Iomin

2011

Phys. Rev. E

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])Subdiffusion on a fractal comb is considered. A mechanism of subdiffusion with a transport exponent different from 1/2 is suggested. It is shown that the transport exponent is determined by the fractal geometry of the comb.PACS numbers: 05.40.Fb, 05.45.Df A comb model was introduced for understanding of anomalous transport in percolating clusters [1,2] and it was considered as a toy model for a porous medium used for exploration of low dimensional percolation clusters [1,3], as well. It is a particular example of a non-Markovian phenomenon, which was explained in the framework of continuous time random walks [2,[4][5][6][7]. In the last decade the comb model has been extensively studied for understanding of different realizations of nonMarkovian random walks both continuous [8][9][10] and discrete [11].Usually, anomalous diffusion on the comb is described by the 2D distribution function P = P (x, y, t), and a special behavior is that the displacement in the x-direction is possible only along the structure axis (x-axis at y = 0). Therefore, diffusion in the x-direction is highly inhomogeneous. Namely, the diffusion coefficient is D xx =Dδ(y), while the diffusion coefficient in the y-direction (along fingers) is a constant D yy = D. Therefore, this inhomogeneous diffusion is described by the Fokker-Planck equation in the dimensionless time and coordinateŝ( 1) It is obtained by the rescaling with relevant combinations of the comb parameters D andD, such that the dimensionless time and coordinates areThe fractional transport along the structure x axis is described by the transporting contaminant distribution p(x, t) = ∞ −∞ P (x, y, t)dy. It was shown [12] that Eq. (1) is equivalent to the fractional Fokker-Planck equationfrom where subdiffusion can be immediately obtained:t is a fractional time derivative, which is a formal notation of an integral with a power law memory kernel. For 0 < α < 1 it readsSubdiffusive mechanism with an arbitrary transport exponent was also suggested by changing either the boundary conditions for diffusion in the fingers [13][14][15][16], or introducing a dependence of the diffusion coefficient on time and space [17]. In this paper we consider a fractal comb, when diffusion is highly inhomogeneous along the y fingers, as well. Namely, it takes palace for those coordinates of the x axis, which belong to a fractal set S ν (x), and is defined by a characteristic function χ(x), such that D yy = Dχ(x), where χ(x) = 1, if x ∈ S ν (x) and χ(x) = 0, if x / ∈ S ν (x). The fractal set S ν (x) is a random fractal with a fractal dimension 0 < ν < 1 embedded in the 1D Euclidian space (of the x axis). Such generalization of the comb model to a discrete (fractal) comb model for consideration of fractional transport in discrete systems is more realistic situation for theoretical studies of transport properties in discrete systems with complicated topology including fractal ones like porous discrete media [18], electronic transport in semiconductors with a discrete distribution of traps, cancer deve...

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confidence: 99%

Subdiffusion on a fractal comb

Iomin

2011

Phys. Rev. E

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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].…”

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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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“…1), where we use that d = Θ(U trap ) is the Heaviside function. One easily checks [7] the full width at half the maximum (FWHM) of the atom distribution. For U trap = 0 one has FWHM = x 2 = t that corresponds to ballistic motion, while for d = 1 one finds normal diffusion with FWHM = t Here we use the definition…”

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confidence: 99%

Superdiffusive comb: Application to experimental observation of anomalous diffusion in one dimension

Iomin

2012

Phys. Rev. E

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A possible mechanism of superdiffusion of ultra-cold atoms in a one-dimensional polarization optical lattice, observed experimentally in [Phys. Rev. Lett. 108, 093002 (2012)], is suggested. The analysis is based on a consideration of anomalous diffusion in a fractal comb [Phys. Rev. E 83, 052106 (2011)]. It is shown that the transport exponent is determined by the fractal geometry of the comb due to recoil distributions resulting in Lévy flights of atoms.

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Superdiffusion on a Comb Structure

Cited by 99 publications

References 21 publications

Subdiffusion on a fractal comb

Subdiffusion on a fractal comb

The Role of Fractional Time-Derivative Operators on Anomalous Diffusion

Superdiffusive comb: Application to experimental observation of anomalous diffusion in one dimension

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