Numerical simulations of anomalous diffusion

Accelerating and Retarding Anomalous Diffusion

In this paper Gaussian models of retarded and accelerated anomalous diffusion are considered. Stochastic differential equations of fractional order driven by single or multiple fractional Gaussian noise terms are introduced to describe retarding and accelerating subdiffusion and superdiffusion. Short and long time asymptotic limits of the mean squared displacement of the stochastic processes associated with the solutions of these equations are studied. Specific cases of these equations are shown to provide possible descriptions of retarding or accelerating anomalous diffusion.

Anomalous diffusion models: different types of subordinator distribution

Subordinated processes play an important role in modeling anomalous diffusion-type behavior. In such models the observed constant time periods are described by the subordinator distribution. Therefore, on the basis of the observed time series, it is possible to conclude on the main properties of the subordinator. In this paper we analyze the anomalous diffusion models with three types of subordinator distribution: α-stable, tempered stable and gamma. We present similarities and differences between the analyzed processes and point at their main properties (like the behavior of moments or the me…

具有双尺度反常扩散的若干积分-微分方程的平稳解存在性

Existence of stationary solutions for some integro-differential equations with the double scale anomalous diffusion

本文针对R³中含不同分数阶负拉普拉斯算子之和的双尺度反常扩散积分-微分方程，借助不动点技术及无界域无Fredholm性质椭圆算子的可解性条件，证明了其平稳解的存在性。

通过迭代定量均匀化实现反常扩散：核心思想综述

Anomalous diffusion via iterative quantitative homogenization: an overview of the main ideas

本综述介绍了研究者构建的具备完全湍流速度场特性的不可压缩矢量场，其对应标量平流-扩散方程呈反常扩散，并提出通过重整化涡粘度反向级联研究反常扩散的分析框架。

Infinite invariant densities for anomalous diffusion in optical lattices and other logarithmic potentials

We solve the Fokker-Planck equation for Brownian motion in a logarithmic potential. When the diffusion constant is below a critical value the solution approaches a non-normalizable scaling state, reminiscent of an infinite invariant density. With this non-normalizable density we obtain the phase diagram of anomalous diffusion for this important process. We briefly discuss the consequence for a range of physical systems including atoms in optical lattices and charges in vicinity of long polyelectrolytes. Our work explains in what sense the infinite invariant density and not Boltzmann's equilibr…

Models for characterizing the transition among anomalous diffusions with different diffusion exponents

Based on the theory of continuous time random walks (CTRW), we build the models of characterizing the transitions among anomalous diffusions with different diffusion exponents, often observed in natural world. In the CTRW framework, we take the waiting time probability density function (PDF) as an infinite series in three parameter Mittag-Leffler functions. According to the models, the mean squared displacement of the process is analytically obtained and numerically verified, in particular, the trend of its transition is shown; furthermore the stochastic representation of the process is presen…

独立于瑞利数的一般反常扩散的稳定效应

Stabilising effect of generic anomalous diffusion independent of the Rayleigh number

本文通过线性与非线性稳定性分析，研究多孔介质质量对流中超扩散反常扩散模型的稳定作用，揭示不同记忆函数可使系统越过经典 Rayleigh-Bénard 阈值保持稳定。

关于Kraichnan模型及时间相关变体中的反常扩散

On anomalous diffusion in the Kraichnan model and correlated-in-time variants

本文基于PDE方法简明证明了Kraichnan模型中被动标量的反常扩散现象，给出了时间相关变体反常扩散失效的实例，并证明了标量反常扩散蕴含流场ODE轨迹的非唯一性。

A Fractional Lie Group Method For Anomalous Diffusion Equations

Lie group method provides an efficient tool to solve a differential equation. This paper suggests a fractional partner for fractional partial differential equations using a fractional characteristic method. A space-time fractional diffusion equation is used as an example to illustrate the effectiveness of the Lie group method.

Ultrametric Cantor sets and Origin of Anomalous Diffusion

The anomalous mean square fluctuations are shown to arise naturally from the ordinary diffusion equation interpreted scale invariantly in a formalism endowing real numbers with a nonarchimedean multiplicative structure. A variable $t$ approaching 0 linearly in the ordinary analysis is shown to enjoy instead a sublinear $t\log t^{-1}$ flow in the presence of this scale invariant structure. Diffusion on an ultrametric Cantor set is also generically subdiffusive with the above sublinear mean square deviation. The present study seems to offer a new interpretation of a possible emergence of complex…

Analytical Study of Anomalous Diffusion by Randomly Modulated Dipole

This paper derives the Fokker-Planck (FP) equation for a particle moving in potential by a randomly modulated dipole. The FP equation describes the anomalous diffusion observed in the companion paper [1] and breaks the conservation of the total probability at the singularity by the dipole. It also shows anisotropic diffusion, which is typical in fluid turbulence. We need to modify the probability density by introducing a mechanism to recover the particle. After the modification, the latent fractal dimension matches with the results of the companion paper. We hope that our model gives a new exa…