Abstract
We introduce non-local dynamics on directed networks through the construction of a fractional version of a non-symmetric Laplacian for weighted directed graphs. Furthermore, we provide an ...analytic treatment of fractional dynamics for both directed and undirected graphs, showing the possibility of exploring the network employing random walks with jumps of arbitrary length. We also provide some examples of the applicability of the proposed dynamics, including consensus over multi-agent systems described by directed networks.
In classical viscous fluids, turbulent eddies are known to be responsible for the rapid spreading of embedded particles. However, in inviscid quantum fluids where the turbulence is induced by a ...chaotic tangle of quantized vortices, dispersion of the particles can be achieved via a nonclassical mechanism, i.e., their binding to the evolving vortices. However, knowledge on how the vortices diffuse and spread in quantum-fluid turbulence is very limited, especially for the so-called ultraquantum turbulence (UQT) generated by a random tangle of vortices. Here we report a systematic numerical study of the apparent diffusion of vortices in UQT in superfluid helium-4 using the full Biot-Savart simulation. We reveal that the vortices in the superfluid exhibit a universal anomalous diffusion (superdiffusion) at small times, which transits to normal diffusion at large times. This behavior is found to be the result of a generic scaling property of the vortex velocity. Our simulation at finite temperatures also nicely reproduces recent experimental observations. Lastly, the knowledge obtained from this study may form the base for understanding turbulent transport and universal vortex dynamics in various quantum fluids.
Many man-made and natural processes involve the diffusion of microscopic particles subject to random or chaotic, random-like movements. Besides the normal diffusion characterized by a Gaussian ...probability density function, whose variance increases linearly in time, so-called anomalous-diffusion regimes can also take place. They are characterized by a variance growing slower (subdiffusive) or faster (superdiffusive) than normal. In fact, many different underlying processes can lead to anomalous diffusion, with qualitative differences between mechanisms producing subdiffusion and mechanisms resulting in superdiffusion. Thus, a general description, encompassing all three regimes and where the specific mechanisms of each system are not explicit, is desirable. Here, our goal is to present a simple method of data analysis that enables one to characterize a model-less diffusion process from data observation, by observing the temporal evolution of the particle spread. To generate diffusive processes in different regimes, we use a Monte-Carlo routine in which both the step-size and the time-delay of the diffusing particles follow Pareto (inverse-power law) distributions, with either finite or diverging statistical momenta. We discuss on the application of this method to real systems.
•We use the evolution of distributions to characterize normal or anomalous diffusion.•We considered space and time dynamics to test different regimes of diffusion.•The width of the distribution is related to the parameters of the random walk.•This analysis is independent of the mechanism responsible for the diffusion.•The technique is specially useful in superdiffusion, where the variance diverges.
The infiltration of fluids into porous media frequently presents anomalous features, in which the fluid front displacement varies in time with an exponent ν different from the expected Fickean value ...of 1/2. A variety of transport models in fractal media predict the anomalies in this process and in related diffusion problems, but the associated exponents are non universal, i.e they depend not only on the fractal dimensions Df but also on other geometric properties. Here we study the horizontal infiltration in layered porous media where the matrices of higher conductivity have fractal distributions (Df<1) of lower conductivity inclusions. When the conductivity contrast is high, this process exhibits universal superdiffusive infiltration with ν=1/1+Df. This result is first demonstrated for inclusion patterns modeled by Cantor sets, but we argue that it extends to any fractal distributions of the inclusions under the condition of no spatial anisotropy in the rescaling of the observation size. Randomized versions of the Cantor sets are presented as possible realizations of disordered fractal patterns and confirm the universal relation. By considering properties of typical granular media and various soils, numerical calculations indicate that this universal superdiffusive infiltration could be observed in physically realizable laboratory and field settings.
•A model of pressure driven infiltration into horizontal layered media is developed.•Distributions of low conductivity layers (inclusions) in the media are fractal.•Infiltration is superdiffusive with exponent depending only on the fractal dimension.•The same results are applicable to regular or random fractal organizations.•The superdiffusive behavior is realizable in field and laboratory settings.
We investigate the spatial discretization of a stochastic semilinear superdiffusion problem driven by fractionally integrated multiplicative space–time white noise. The white noise is characterized ...by its properties of being white in both space and time, and the time fractional derivative is considered in the Caputo sense with an order α∈ (1, 2). A spatial discretization scheme is introduced by approximating the space–time white noise with the Euler method in the spatial direction and approximating the second-order space derivative with the central difference scheme. By using the Green functions, we obtain both exact and approximate solutions for the proposed problem. The regularities of both the exact and approximate solutions are studied, and the optimal error estimates that depend on the smoothness of the initial values are established.
The nonlocality (superdiffusion) of turbulence is expressed in the empiric Richardson t3 scaling law for the mean square of the mutual separation of a pair of particles in a fluid or gaseous medium. ...The development of the theory of nonlocality of various processes in physics and other sciences based on the concept of Lévy flights resulted in Shlesinger and colleagues’ about the possibility of describing the nonlocality of turbulence using a linear integro-differential equation with a slowly falling kernel. The approach developed by us made it possible to establish the closeness of the superdiffusion parameter of plasma density fluctuations moving across a strong magnetic field in a tokamak to the Richardson law. In this paper, we show the possibility of a universal description of the characteristics of nonlocality of transfer in a stochastic medium (including turbulence of gases and fluids) using the Biberman–Holstein approach to examine the transfer of excitation of a medium by photons, generalized in order to take into account the finiteness of the velocity of excitation carriers. This approach enables us to propose a scaling that generalizes Richardson’s t3 scaling law to the combined regime of Lévy flights and Lévy walks in fluids and gases.
Fractional order diffusion equations are generalizations of classical diffusion equations, treating super-diffusive flow processes. In this paper, we examine a practical numerical method which is ...second-order accurate in time and in space to solve a class of initial-boundary value fractional diffusive equations with variable coefficients on a finite domain. An approach based on the classical Crank–Nicholson method combined with spatial extrapolation is used to obtain temporally and spatially second-order accurate numerical estimates. Stability, consistency, and (therefore) convergence of the method are examined. It is shown that the fractional Crank–Nicholson method based on the shifted Grünwald formula is unconditionally stable. A numerical example is presented and compared with the exact analytical solution for its order of convergence.
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