Pre-asymptotic corrections to fractional diffusion equations

The motion of contaminant particles through complex environments such as fractured rocks or porous sediments is often characterized by anomalous diffusion: the spread of the transported quantity is found to grow sublinearly in time due to the presence of obstacles which hinder particle migration. The asymptotic behavior of these systems is usually well described by fractional diffusion, which provides an elegant and unified framework for modeling anomalous transport. We show that pre-asymptotic corrections to fractional diffusion might become relevant, depending on the microscopic dynamics of the particles. To incorporate these effects, we derive a modified transport equation and validate its effectiveness by a Monte Carlo simulation.     

Fractional nonlinear diffusion equation,solutions and anomalous diffusion

We devote this work to investigate the solutions of a generalized diffusion equation which contains spatial fractional derivatives and nonlinear terms. The presence of external forces and absorbent terms is also considered. The solutions found here can have a compact or long tail behavior and, in particular, for the last case in the asymptotic limit, we relate these solutions to the Lévy or Tsallis distributions. In addition, from the results presented here a rich class of diffusive processes, including normal and anomalous ones, can be obtained.     

A mixed SOC-turbulence model for nonlocal transport and Lévy-fractional Fokker–Planck equation

The phenomena of nonlocal transport in magnetically confined plasma are theoretically analyzed. A hybrid model is proposed, which brings together the notion of inverse energy cascade, typical of drift-wave- and two-dimensional fluid turbulence, and the ideas of avalanching behavior, associable with self-organized critical (SOC) behavior. Using statistical arguments, it is shown that an amplification mechanism is needed to introduce nonlocality into dynamics. We obtain a consistent derivation of nonlocal Fokker–Planck equation with space-fractional derivatives from a stochastic Markov process with the transition probabilities defined in reciprocal space. The hybrid model observes the Sparre Andersen universality and defines a new universality class of SOC.     

Lévy walks for turbulence: A numerical study

We present a numerical study of enhanced diffusion, for which the mean-squared displacement follows asymptotically r ²(t) t , > 1. We simulate continuous time random walks with waiting-time distributions which couple the spatial and temporal parameters; this gives rise to Lévy-walks. Our results confirm the theoretically predicted long-time behavior and demonstrate its temporal regime of validity. Furthermore, the simulations document the appearance of (parameter-dependent) transitions between regular and enhanced diffusion regimes.     

Effects of asymmetric Lévy noise in parameter-induced aperiodic stochastic resonance

We study properties of parameter-induced aperiodic stochastic resonance in the presence of asymmetric Lévy noise. The system performance is characterized by the bit error rate. Investigations are based on the numerical solution of the space-fractional Fokker-Planck equation and Monte Carlo simulations. After choosing the optimal detection threshold, it is shown that the skewness parameter β has very limited influence on the system performance. Under the same conditions, the system performance is slightly reduced with the increasing β. The stability index α has the dominant effects on the system performance. The lower value of α leads to the better system performance.     

Generalized Langevin equation driven by Lévy processes: A probabilistic, numerical and time series based approach

Lévy processes have been widely used to model a large variety of stochastic processes under anomalous diffusion. In this note we show that Lévy processes play an important role in the study of the Generalized Langevin Equation (GLE). The solution to the GLE is proposed using stochastic integration in the sense of convergence in probability. Properties of the solution processes are obtained and numerical methods for stochastic integration are developed and applied to examples. Time series methods are applied to obtain estimation formulas for parameters related to the solution process. A Monte Carlo simulation study shows the estimation of the memory function parameter. We also estimate the stability index parameter when the noise is a Lévy process.     

Temperature controlled Lévy flights of minority carriers in photoexcited bulk n-InP

We study by photoluminescence the spatial distribution of minority carriers (holes) arising from their anomalous photon-assisted diffusion upon photo-excitation at an edge of n-InP slab for temperatures ranging from 300 K to 78 K. Giant enhancement in the spread of holes — over distances exceeding 1 cm from the excitation edge — is seen at lower temperatures. We show that the experiment provides a realization of the “Lévy flight” random walk of holes, in which the Lévy distribution index γ   is controlled by the temperature. The variation γ(T)$\gamma (T)$ is close to that predicted earlier, γ=1−Δ/kT$\gamma =1-\Delta /kT$, where Δ(T)$\Delta (T)$ is the Urbach tailing parameter of the absorption spectra. This theoretical prediction is based on the assumption of a quasi-equilibrium intrinsic emission spectrum in the form due to van Roosbroeck and Shockley.     

Lévy flights in inhomogeneous environments

We study the long time asymptotics of probability density functions (pdfs) of Lévy flights in confining potentials that originate from inhomogeneities of the environment in which the flights take place. To this end we employ two model patterns of dynamical behavior: Langevin-driven and (Lévy-Schrödinger) semigroup-driven dynamics. It turns out that the semigroup modeling provides much stronger confining properties than the standard Langevin one. For computational and visualization purposes our observations are exemplified for the Cauchy driver and its response to external polynomial potentials (referring to Lévy oscillators), with respect to both dynamical mechanisms. We discuss the links of the Lévy semigroup motion scenario with that of random searches in spatially inhomogeneous media.     

Energy and number of collision fluctuations in inelastic gases

The two-dimensional Inelastic Maxwell Model (IMM) is studied by numerical simulations. It is shown how the inelasticity of collisions together with the fluctuations of the number of collisions undergone by a particle lead to energy fluctuations. These fluctuations are associated to a shrinking of the available phase space. We find the asymptotic scaling of these energy fluctuations and show how they affect the tail of the velocity distribution during long time intervals. We stress that these fluctuations relax like power laws on much slower time scales than the usual exponential relaxations taking place in kinetic theory.     

Scaling Properties of Field-Induced Superdiffusion in Continuous Time Random Walks

We consider a broad class of Continuous Time Random Walks (CTRW) with large fluctuations effects in space and time distributions: a random walk with trapping, describing subdiffusion in disordered and glassy materials, and a Lévy walk process, often used to model superdiffusive effects in inhomogeneous materials. We derive the scaling form of the probability distributions and the asymptotic properties of all its moments in the presence of a field by two powerful techniques, based on matching conditions and on the estimate of the contribution of rare events to power-law tails in a field.     

Time evolution of the probability distribution in stochastic and chaotic systems with enhanced diffusion

We perform a detailed study of the time evolution of the probability distribution for two processes displaying enhanced diffusion: a stochastic process named the Lévy walk and a deterministic chaotic process, the amplified climbing-sine map. The time evolution of the probability distribution differs in the two cases and carries information which is peculiar to the investigated process.     

Cascade-like and scaling behavior of wind velocity increments in the atmospheric surface layer

By using a large amount of data collected in the atmospheric surface layer, we analyze the probability density functions (PDFs), the probability of return and the moments of wind velocity increments. Results show that the PDFs change from the non-Gaussian long-tailed distributions to Gaussian with the increase of time scales. This is similar to what has been observed and interpreted as an indication of cascade in the fully developed homogeneous and isotropic turbulence. Besides, both the probability of return and the moments are found to be scaling with time scales. We then compare above results with the truncated Lévy flights and the log-normal PDF model. It is found that although both models show the cascade-like behavior in the PDFs and the scaling behavior in the probability of return and the moments under some conditions, they are not good enough for quantitatively describing the random process of wind velocity increments.     

Inertial effects, mass separation and rectification power in Lévy ratchets

We study the dynamics of particles in an external anisotropic periodic potential under the influence of additive white Lévy noise, in a general not overdamped situation. Different quantities characterizing directionality, coherence and dispersion are analyzed as functions of the mass and other systems parameters. We show that, while the current decreases monotonously with the stability index of the Lévy noise, there exists a particular intermediate value of such parameter (slightly dependent on the mass) that minimizes the time required to form a coherent particle package advancing in the preferred direction. Moreover, we show the possibility of observing mass separation. This means that particles of different masses may advance in opposite directions when influenced by the same ratchet potential and the same Lévy noise. Finally, we show that the ratio of the advanced distance to the total distance travelled constitutes a relevant measure for the rectification power, useful not only for Lévy ratchets but also for general ratchets systems. In particular, we find that it behaves quite similar to the rectification efficiency for standard models of rocking and flashing ratchets found in the literature.     

On the first passage time and leapover properties of Lévy motions

We investigate two coupled properties of Lévy stable random motions: the first passage times (FPTs) and the first passage leapovers (FPLs). While, in general, the FPT problem has been studied quite extensively, the FPL problem has hardly attracted any attention. Considering a particle that starts at the origin and performs random jumps with independent increments chosen from a Lévy stable probability law λ_α,β(x), the FPT measures how long it takes the particle to arrive at or cross a target. The FPL addresses a different question: given that the first passage jump crosses the target, then how far does it get beyond the target? These two properties are investigated for three subclasses of Lévy stable motions: (i) symmetric Lévy motions characterized by Lévy index α(0<α<2) and skewness parameter β=0, (ii) one-sided Lévy motions with 0<α<1, β=1, and (iii) two-sided skewed Lévy motions, the extreme case, 1<α<2, β=−1.     

热辐射输运问题的高效蒙特卡罗模拟方法

惯性约束聚变研究中,热辐射光子在介质中的输运以及热辐射光子与介质的相互作用是重要研究课题,蒙特卡罗方法是该类问题的重要研究手段之一.隐式蒙特卡罗方法虽然能正确地模拟热辐射在介质中的输运过程,但当模拟重介质(材料的吸收系数大)问题时,该方法花费的计算时间将变得很长,导致模拟效率很低.本文以离散扩散蒙特卡罗方法为基础,开发了"离散扩散蒙特卡罗方法辐射输运模拟程序",可以较好地解决重介质区的计算效率问题,但是离散扩散蒙卡罗方法在模拟轻介质区时精度不够高.辐射输运问题中通常既有轻介质也有重介质,为了能同时解决蒙特卡罗方法模拟的效率和精度问题,本文研究了离散扩散蒙特卡罗方法与隐式蒙特卡罗方法相结合的模拟方法,并提出了新的扩散区与输运区界面处理方法,研制了混合蒙特卡罗方法的辐射输运模拟程序.典型辐射输运问题模拟显示:在模拟重介质问题时,该程序能大幅缩短模拟时间,且能取得与隐式蒙特卡罗方法一致的结果;在模拟轻重介质均存在的问题时,与隐式蒙特卡罗方法相比,混合蒙特卡罗方法的模拟精度与其相当且计算效率同样能够得到显著提升.     

Power tails of index distributions in chinese stock market

The power α$\alpha$ of the Lévy tails of stock market fluctuations discovered in recent years are generally believed to be universal. We show that for the Chinese stock market this is not true, the powers depending strongly on anomalous daily index changes short before market closure, and weakly on the opening data.     

Human memory retrieval as Lévy foraging

When people attempt to recall as many words as possible from a specific category (e.g., animal names) their retrievals occur sporadically over an extended temporal period. Retrievals decline as recall progresses, but short retrieval bursts can occur even after tens of minutes of performing the task. To date, efforts to gain insight into the nature of retrieval from this fundamental phenomenon of semantic memory have focused primarily upon the exponential growth rate of cumulative recall. Here we focus upon the time intervals between retrievals. We expected and found that, for each participant in our experiment, these intervals conformed to a Lévy distribution suggesting that the Lévy flight dynamics that characterize foraging behavior may also characterize retrieval from semantic memory. The closer the exponent on the inverse square power-law distribution of retrieval intervals approximated the optimal foraging value of 2, the more efficient was the retrieval. At an abstract dynamical level, foraging for particular foods in one's niche and searching for particular words in one's memory must be similar processes if particular foods and particular words are randomly and sparsely located in their respective spaces at sites that are not known a priori. We discuss whether Lévy dynamics imply that memory processes, like foraging, are optimized in an ecological way.     

An exponential integrator for advection-dominated reactive transport in heterogeneous porous media

We present an exponential time integrator in conjunction with a finite volume discretisation in space for simulating transport by advection and diffusion including chemical reactions in highly heterogeneous porous media representative of geological reservoirs. These numerical integrators are based on the variation of constants solution and solving the linear system exactly. This is at the expense of computing the exponential of the stiff matrix comprising the finite volume discretisation. Using real Léja points or a Krylov subspace technique compared to standard finite difference-based time integrators. We observe for a variety of example applications that numerical solutions with exponential methods are generally more accurate and require less computational cost. They hence comprise an efficient and accurate method for simulating non-linear advection-dominated transport in geological formations.     

Exact solutions for nonlinear fractional anomalous diffusion equations

This work is devoted to investigating exact solutions of generalized nonlinear fractional diffusion equations with external force and absorption. We first investigate the nonlinear anomalous diffusion equations with one-fractional derivative and then multi-fractional ones. In both situations, we obtain the corresponding exact solution, its diffusive behavior, and the sufficient and necessary conditions for solutions satisfying the boundary condition W(±∞,t)=0 and the sharp initial condition W(x,0)=δ(x).     

Pricing bounds for discrete arithmetic Asian options under Lévy models

Analytical bounds for Asian options are almost exclusively available in the Black-Scholes framework. In this paper we derive bounds for the price of a discretely monitored arithmetic Asian option when the underlying asset follows an arbitrary Lévy process. Explicit formulas are given for Kou’s model, Merton’s model, the normal inverse Gaussian model, the CGMY model and the variance gamma model. The results are compared with the comonotonic upper bound, existing numerical results, Monte carlo simulations and in the case of the variance gamma model with an existing lower bound. The method outlined here provides lower and upper bounds that are quick to evaluate, and more accurate than existing bounds.