Anomalous diffusion and random search in xyz-comb: exact results

Mathematical Modelling of Natural Phenomena, 2016

We consider a generalised diffusion equation in two dimensions for modeling diffusion on a comblike structures. We analyse the probability distribution functions and we derive the mean squared displacement in x and y directions. Different forms of the memory kernels (Dirac delta, power-law, and distributed order) are considered. It is shown that anomalous diffusion may occur along both x and y directions. Ultraslow diffusion and some more general diffusive processes are observed as well. We give the corresponding continuous time random walk model for the considered two dimensional diffusion-like equation on a comb, and we derive the probability distribution functions which subordinate the process governed by this equation to the Wiener process.

2021

Trifce Sandev, 2 Alexander Iomin, Holger Kantz, Ralf Metzler, 5 and Aleksei Chechkin 6, ∗ Max Planck Institute for the Physics of Complex Systems, Nöthnitzer Strasse 38, 01187 Dresden, Germany Radiation Safety Directorate, Partizanski odredi 143, P.O. Box 22, 1020 Skopje, Macedonia Department of Physics, Technion, Haifa 32000, Israel Institute for Physics and Astronomy, University of Potsdam, D-14776 Potsdam-Golm, Germany Department of Physics, Tampere University of Technology, FI-33101 Tampere, Finland Akhiezer Institute for Theoretical Physics, Kharkov 61108, Ukraine (Dated: July 4, 2021)

Chaos, Solitons & Fractals, 2016

We present an exact analytical result on ultra-slow diffusion by solving a Fokker-Planck equation, which describes anomalous transport in a three dimensional (3D) comb. This 3D cylindrical comb consists of a cylinder of discs of ether infinite or finite radius, threaded on a backbone. It is shown that the ultraslow particle spreading along the backbone is described by the mean squared displacement (MSD) of the order of ln(t). This phenomenon takes place only for normal two dimensional diffusion inside the infinite secondary branches (discs). When the secondary branches have finite boundaries, the ultra-slow motion is a transient process and the asymptotic behavior is normal diffusion. In another example, when anomalous diffusion takes place in the secondary branches, a destruction of ultra-slow (logarithmic) diffusion takes place as well. As the result, one observes "enhanced" subdiffusion with the MSD ∼ t 1−α ln t, where 0 < α < 1.

Physical Review E

A two-dimensional (2D) comb model is proposed to characterize reaction-ultraslow diffusion of tracers both in backbones (x direction) and side branches (y direction) of the comblike structure with two memory kernels. The memory kernels include Dirac delta, power-law, and logarithmic and inverse Mittag-Leffler (ML) functions, which can also be considered as the structural functions in the time structural derivative. Based on the comb model, ultraslow diffusion on a fractal comb structure is also investigated by considering spatial fractal geometry of the backbone volume. The mean squared displacement (MSD) and the corresponding concentration of the tracers, i.e., the solution of the comb model, are derived for reactive and conservative tracers. For a fractal structure of backbones, the derived MSDs and corresponding solutions depend on the backbone's fractal dimension. The proposed 2D comb model with different kernel functions is feasible to describe ultraslow diffusion in both the backbone and side branches of the comblike structure.

Physica A: Statistical Mechanics and its Applications, 2002

From the generalized scheme of random walks on the comb-like structure, one can obtain the fractional Fokker-Planck equation in the domain of fractal time evolution with critical exponent ÿ (0 ¡ ÿ 6 1). The operator method for the moments associated with the density p(x; t) is used to solve the obtained equation. It is shown that due to ÿngers di usion has an anomalous character, the ÿrst two moments increase sub-linearly in time.

Physical Review E, 2015

Combs are a simple caricature of various types of natural branched structures, which belong to the category of loopless graphs and consist of a backbone and branches. We study continuous time random walks on combs and present a generic method to obtain their transport properties. The random walk along the branches may be biased, and we account for the effect of the branches by renormalizing the waiting time probability distribution function for the motion along the backbone. We analyze the overall diffusion properties along the backbone and find normal diffusion, anomalous diffusion, and stochastic localization (diffusion failure), respectively, depending on the characteristics of the continuous time random walk along the branches.

The problem of finding the distribution of functional of a trajectory of a particle executing a random walk in a disordered medium containing both traps and obstacles is considered. As a model of a disordered medium, the Schirmacher model, which is the combination of the random barrier model and the multiple-trapping model, is used. Forward and backward Feynman-Kac equations with the boundary conditions at discontinuity points are formulated. As an example, the distribution of the residence time in a half-space is obtained. It is shown that the anomalous subdiffusion due to traps and that due to obstacles give very different distributions.

Physical Review E, 2003

In this paper we study the effects of a magnetic field on the discrete time random walk of a classical charged particle moving on a comb lattice. We develop an analytical technique to study the Lorentz force effects on the asymptotic diffusion laws. This approach also allows the description of the combined action of an electric and a magnetic field ͑Hall effect͒. The generalization to other comblike branched structures is discussed.

Journal of Physics A: Mathematical and Theoretical

We study the diffusive motion of a test particle in a two-dimensional comb structure consisting of a main backbone channel with continuously distributed side branches, in the presence of stochastic Markovian resetting to the initial position of the particle. We assume that the motion along the infinitely long branches is biased by a confining potential. The crossover to the steady state is quantified in terms of a large deviation function, which is derived for the first time for comb structures in present paper. We show that the relaxation region is demarcated by a nonlinear "light-cone" beyond which the system is evolving in time. We also investigate the first-passage times along the backbone and calculate the mean first-passage time and optimal resetting rate. Backbone diffusion and first-passage dynamics in a comb structure with confining branches.. . 2 [12, 13, 14, 15]. Sometimes for α > 2 the term hyperdiffusion is used [16, 17, 18]. We note that the case α = 1 in heterogeneous media does not necessarily imply that the process has a Gaussian probability density function (PDF), instead, for instance, exponential or stretched Gaussian forms may be observed [19, 20]. We also note that the MSD may also grow exponentially, for a multiplicative noise such as geometric Brownian motion or heterogeneous diffusion processes [21, 22, 23, 24], or logarithmically in strongly disordered environments [25, 26]. A by-now classical model for heterogeneous systems, popularised by Mandelbrot, are fractals, such as the Sierpiński gasket [27, 28, 29]. However, such ideal mathematical fractals are often insufficient to adequately describe real fractals such as networks of rivers, blood vessels, or nerve fibres, for which random fractals such as percolation clusters are more appropriate [27, 28, 29]. In many cases such structures have a characteristic backbone from which various branches emerge [28, 29]. A highly effective model addressing transport on such random loopless structures is a comb, in which infinite branches branch off the central backbone. The comb model was introduced to understand anomalous transport in percolation clusters [30, 31, 32, 33, 34]. Now, comb-like models are widely employed to describe various experimental applications. Comb-like structures are particularly important from a biophysical point of view as they provide a way to address transport along spiny dendrites [35, 36, 37], in which the transport properties crucially depend on the underlying geometry [38]. Similar approaches are being used in the modelling of river basins with their often very ramified geometry [39, 40]. In fact, long time retention data of tracers in water catchments reveal scaling exponents consistent with comb dynamics [41, 42]. Depending on the specific setting the geometry of comb structures effects both subdiffusion [43, 44, 45], including ultraslow diffusion [46], and superdiffusion [17, 47, 48]. The nontrivial nature of transport along a comb is discernible from the fact that motion along the branches results in a long-range memory for motion along the backbone which is generically responsible for the anomalous behaviour of transport [49]. In fact, the comb model can be regarded as the discrete version of a continuous time random walk, in which the return time distribution from a side branch to the main backbone effects power-law waiting times with diverging mean [33], and thus weak ergodicity breaking and ageing effects [50, 51]. Given the wide relevance and interesting properties of comb-like, loopless structures, these represent very powerful mathematical constructs to address motion in heterogeneous media. We here combine the analysis of diffusion on a comb with the idea of stochastic resetting. The concept of stochastic resetting (SR) has attracted considerable attention in non-equilibrium statistical physics [52]. In SR a moving particle is reset, i.e., returned to its initial location at regular or stochastic intervals. This results in a non-equilibrium steady-state even in cases in which the system under consideration does not relax to a steady-state in absence of any resets, see, e.g., free Brownian motion in d dimensions [53, 54]. The effect of resetting is particularly relevant for the first-passage properties of the motion of interest [55, 56, 57]. Indeed, even in generic cases in finite domains

The comb model is a simplified description for anomalous diffusion under geometric constraints. It represents particles spreading out in a two-dimensional space where the motions in the x-direction are allowed only when the y coordinate of the particle is zero. Here, we propose an extension for the comb model via Langevin-like equations driven by fractional Gaussian noises (longrange correlated). By carrying out computer simulations, we show that the correlations in the y-direction affect the diffusive behavior in the x-direction in a non-trivial fashion, resulting in a quite rich diffusive scenario characterized by usual, superdiffusive or subdiffusive scaling of second moment in the x-direction. We further show that the long-range correlations affect the probability distribution of the particle positions in the x-direction, making their tails longer when noise in the y-direction is persistent and shorter for anti-persistent noise. Our model thus combines and allows the study/analysis of the interplay between Content from this work may be used under the terms of the Creative Commons Attribution 3.0 licence. Any further distribution of this work must maintain attribution to the author(s) and the title of the work, journal citation and DOI.