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.
Recently, Kovács et al. considered a Mittag‐Leffler Euler integrator for a stochastic semilinear Volterra integral‐differential equation with additive noise and proved the strong convergence error ...estimates see SIAM J. Numer. Anal. 58(1) 2020, pp. 66‐85. In this article, we shall consider the Mittag‐Leffler integrators for more general models: stochastic semilinear subdiffusion and superdiffusion driven by fractionally integrated additive noise. The mild solutions of our models involve four different Mittag‐Leffler functions. We first consider the existence, uniqueness and the regularities of the solutions. We then introduce the full discretization schemes for solving the problems. The temporal discretization is based on the Mittag‐Leffler integrators and the spatial discretization is based on the spectral method. The optimal strong convergence error estimates are proved under the reasonable assumptions for the semilinear term and for the regularity of the noise. Numerical examples are given to show that the numerical results are consistent with the theoretical results.
Purpose
To develop an anomalous (non‐Gaussian) diffusion model for characterizing skeletal muscle perfusion using multi‐b‐value DWI.
Theory and methods
Fick’s first law was extended for describing ...tissue perfusion as anomalous superdiffusion, which is non‐Gaussian diffusion exhibiting greater particle spread than that of the Gaussian case. This was accomplished using a space‐fractional derivative that gives rise to a power‐law relationship between mean squared displacement and time, and produces a stretched exponential signal decay as a function of b‐value. Numerical simulations were used to estimate parameter errors under in vivo conditions, and examine the effect of limited SNR and residual fat signal. Stretched exponential DWI parameters, α and D, were measured in thigh muscles of 4 healthy volunteers at rest and following in‐magnet exercise. These parameters were related to a stable distribution of jump‐length probabilities and used to estimate microvascular volume fractions.
Results
Numerical simulations showed low dispersion in parameter estimates within 1.5% and 1%, and bias errors within 3% and 10%, for α and D, respectively. Superdiffusion was observed in resting muscle, and to a greater degree following exercise. Resting microvascular volume fraction was between 0.0067 and 0.0139 and increased between 2.2‐fold and 4.7‐fold following exercise.
Conclusions
This model captures superdiffusive molecular motions consistent with perfusion, using a parsimonious representation of the DWI signal, providing approximations of microvascular volume fraction comparable with histological estimates. This signal model demonstrates low parameter‐estimation errors, and therefore holds potential for a wide range of applications in skeletal muscle and elsewhere in the body.
Femtosecond laser‐induced photoexcitation of ferromagnet (FM)/heavy metal (HM) heterostructures has attracted attention by emitting broadband terahertz frequencies. The phenomenon relies on the ...formation of an ultrafast spin current, which is primarily attributed to the direct photoexcitation of the FM layer. However, during the process, the FM layer also experiences a secondary excitation led by the hot electrons from the HM layer that travel across the FM/HM interface and transfer additional energy in the FM. Thus, the generated secondary spins enhance the total spin current formation and lead to amplified spintronic terahertz emission. These results emphasize the significance of the secondary spin current, which even exceeds the primary spin currents when FM/HM heterostructures with thicker HM are used. An analytical model is developed to provide deeper insights into the microscopic processes within the individual layers, underlining the generalized ultrafast superdiffusive spin‐transport mechanism.
The presence of a secondary spin current (js') that significantly contributes to the amplification of terahertz emission in ferromagnet (FM)/heavy metal (HM) heterostructures is explored through precise control of the thickness of the HM layer.
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