Subdiffusion on a fractal comb

Abstract

])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...

The Role of Fractional Time-Derivative Operators on Anomalous Diffusion

“…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].…”

Diffusion and Subdiffusion of Interacting Particles on Comblike Structures

“…The particle can spend a long time exploring a tooth, which results in a subdiffusive motion along the backbone with x 2 (t) ∝ t α with α = 1/2. Since, numerous results have been obtained for this model [5][6][7][8][9][10][11][12], including the determination of the occupation time statistics [13], of mean first-passage times between two nodes of a finite comb [14] or the case of fractional Brownian walks on comb-like structures [15].In parallel, the comb model has been invoked to account for transport in real systems like spiny dendrites [10], diffusion of cold atoms [16] and mainly diffusion in crowded media like cells [17]. However, all existing studies have focused on single-particle diffusion, and interactions between particles have up to now been completely left aside.…”

“…The particle can spend a long time exploring a tooth, which results in a subdiffusive motion along the backbone with x 2 (t) ∝ t α with α = 1/2. Since, numerous results have been obtained for this model [5][6][7][8][9][10][11][12], including the determination of the occupation time statistics [13], of mean first-passage times between two nodes of a finite comb [14] or the case of fractional Brownian walks on comb-like structures [15].…”

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

“…Here, a relation between the transport exponent and the lattice potential depth is established in the framework of a fractional comb model [5]. We analyze an experimental observation of dependence of a diffusion exponent as a function of the lattice depth U trap , presented in Fig.…”

“…To this end, an analysis developed in [5] is applied. One carries out the Laplace transform in the time domainL[P (k, y, t)] =P (k, y, s).…”