scholarly journals Slices of the Anomalous Phase Cube Depict Regions of Sub- and Super-Diffusion in the Fractional Diffusion Equation

Mathematics ◽  
2021 ◽  
Vol 9 (13) ◽  
pp. 1481
Author(s):  
Richard L. Magin ◽  
Ervin K. Lenzi
Keyword(s):  
Brownian Motion ◽  
Diffusion Equation ◽  
Fractional Derivative ◽  
Phase Diagrams ◽  
Anomalous Diffusion ◽  
Lévy Process ◽  
Three Dimensional ◽  
Levy Process ◽  
Diffusion Phase ◽  
Non Gaussian

Fractional-order time and space derivatives are one way to augment the classical diffusion equation so that it accounts for the non-Gaussian processes often observed in heterogeneous materials. Two-dimensional phase diagrams—plots whose axes represent the fractional derivative order—typically display: (i) points corresponding to distinct diffusion propagators (Gaussian, Cauchy), (ii) lines along which specific stochastic models apply (Lévy process, subordinated Brownian motion), and (iii) regions of super- and sub-diffusion where the mean squared displacement grows faster or slower than a linear function of diffusion time (i.e., anomalous diffusion). Three-dimensional phase cubes are a convenient way to classify models of anomalous diffusion (continuous time random walk, fractional motion, fractal derivative). Specifically, each type of fractional derivative when combined with an assumed power law behavior in the diffusion coefficient renders a characteristic picture of the underlying particle motion. The corresponding phase diagrams, like pages in a sketch book, provide a portfolio of representations of anomalous diffusion. The anomalous diffusion phase cube employs lines of super-diffusion (Lévy process), sub-diffusion (subordinated Brownian motion), and quasi-Gaussian behavior to stitch together equivalent regions.

Fractional generalizations of filtering problems and their associated fractional Zakai equations

2014 ◽  
Vol 17 (3) ◽  
Author(s):  
Sabir Umarov ◽  
Frederick Daum ◽  
Kenric Nelson
Keyword(s):  
Brownian Motion ◽  
Fractional Derivative ◽  
Lévy Process ◽  
Nonlinear Filtering ◽  
Levy Process ◽  
Zakai Equation ◽  
Filtering Problem

AbstractIn this paper we discuss fractional generalizations of the filtering problem. The ”fractional” nature comes from time-changed state or observation processes, basic ingredients of the filtering problem. The mathematical feature of the fractional filtering problem emerges as the Riemann-Liouville or Caputo-Djrbashian fractional derivative in the associated Zakai equation. We discuss fractional generalizations of the nonlinear filtering problem whose state and observation processes are driven by time-changed Brownian motion or/and Lévy process.

scholarly journals Caputo Fractional Derivative and Quantum-Like Coherence

Entropy ◽  
2021 ◽  
Vol 23 (2) ◽  
pp. 211
Author(s):  
Garland Culbreth ◽  
Mauro Bologna ◽  
Bruce J. West ◽  
Paolo Grigolini
Keyword(s):  
Diffusion Equation ◽  
Fractional Derivative ◽  
Anomalous Diffusion ◽  
Caputo Fractional Derivative ◽  
Quantum Coherence ◽  
Time Derivative ◽  
The Other ◽  
Diffusion Equations ◽  
Self Organization ◽  
Ordinary Time

We study two forms of anomalous diffusion, one equivalent to replacing the ordinary time derivative of the standard diffusion equation with the Caputo fractional derivative, and the other equivalent to replacing the time independent diffusion coefficient of the standard diffusion equation with a monotonic time dependence. We discuss the joint use of these prescriptions, with a phenomenological method and a theoretical projection method, leading to two apparently different diffusion equations. We prove that the two diffusion equations are equivalent and design a time series that corresponds to the anomalous diffusion equation proposed. We discuss these results in the framework of the growing interest in fractional derivatives and the emergence of cognition in nature. We conclude that the Caputo fractional derivative is a signature of the connection between cognition and self-organization, a form of cognition emergence different from the other source of anomalous diffusion, which is closely related to quantum coherence. We propose a criterion to detect the action of self-organization even in the presence of significant quantum coherence. We argue that statistical analysis of data using diffusion entropy should help the analysis of physiological processes hosting both forms of deviation from ordinary scaling.

Anomalous diffusion equation using a new general fractional derivative within the Miller–Ross kernel

2020 ◽  
Vol 34 (27) ◽  
pp. 2050289
Author(s):  
Yiying Feng ◽  
Jiangen Liu
Keyword(s):  
Analytical Solution ◽  
Diffusion Equation ◽  
Laplace Transform ◽  
Fractional Derivative ◽  
Anomalous Diffusion ◽  
Fractional Integral ◽  
Liouville Type

In view of the generalization of Miller–Ross kernel in the sense of Riemann–Liouville type, we propose the new definitions of the general fractional integral (GFI) and general fractional derivative (GFD) to discuss the anomalous diffusion equation, which is distinct from those classic calculus operators. The obtained analytical solution of the application described in the graph is effective and accurate making the use of Laplace transform.

scholarly journals The Class of Distributions Associated with the Generalized Pollaczek-Khinchine Formula

2012 ◽  
Vol 49 (3) ◽  
pp. 883-887 ◽  
Author(s):  
Offer Kella
Keyword(s):  
Brownian Motion ◽  
Stationary Distribution ◽  
Lévy Process ◽  
Random Variables ◽  
Levy Process ◽  
Independent Random Variables ◽  
Reflected Brownian Motion ◽  
Distribution Of The Maximum ◽  
Workload Distribution ◽  
Stationary Workload

The goal is to identify the class of distributions to which the distribution of the maximum of a Lévy process with no negative jumps and negative mean (equivalently, the stationary distribution of the reflected process) belongs. An explicit new distributional identity is obtained for the case where the Lévy process is an independent sum of a Brownian motion and a general subordinator (nondecreasing Lévy process) in terms of a geometrically distributed sum of independent random variables. This generalizes both the distributional form of the standard Pollaczek-Khinchine formula for the stationary workload distribution in the M/G/1 queue and the exponential stationary distribution of a reflected Brownian motion.

FRACTIONAL DIFFUSION EQUATION, QUANTUM SUBDYNAMICS AND EINSTEIN'S THEORY OF BROWNIAN MOTION

2007 ◽  
Vol 21 (23n24) ◽  
pp. 3993-3999
Author(s):  
SUMIYOSHI ABE
Keyword(s):  
Brownian Motion ◽  
Quantum Theory ◽  
Fractional Calculus ◽  
Diffusion Equation ◽  
Master Equation ◽  
Anomalous Diffusion ◽  
Fractional Diffusion ◽  
Fractional Diffusion Equation ◽  
Quantum Master Equation ◽  
Objective System

The fractional diffusion equation for describing the anomalous diffusion phenomenon is derived in the spirit of Einstein's 1905 theory of Brownian motion. It is shown how naturally fractional calculus appears in the theory. Then, Einstein's theory is examined in view of quantum theory. An isolated quantum system composed of the objective system and the environment is considered, and then subdynamics of the objective system is formulated. The resulting quantum master equation is found to be of the Lindblad type.

scholarly journals Brownian motion with variable drift: 0-1 laws, hitting probabilities and Hausdorff dimension

2012 ◽  
Vol 153 (2) ◽  
pp. 215-234 ◽  
Author(s):  
YUVAL PERES ◽  
PERLA SOUSI
Keyword(s):  
Brownian Motion ◽  
Hausdorff Dimension ◽  
Lévy Process ◽  
Dirichlet Space ◽  
Levy Process ◽  
Standard Brownian Motion ◽  
Double Points ◽  
Hitting Probabilities

AbstractBy the Cameron–Martin theorem, if a function f is in the Dirichlet space D, then B + f has the same a.s. properties as standard Brownian motion, B. In this paper we examine properties of B + f when fD. We start by establishing a general 0-1 law, which in particular implies that for any fixed f, the Hausdorff dimension of the image and the graph of B + f are constants a.s. (This 0-1 law applies to any Lévy process.) Then we show that if the function f is Hölder(1/2), then B + f is intersection equivalent to B. Moreover, B + f has double points a.s. in dimensions d ≤ 3, while in d ≥ 4 it does not. We also give examples of functions which are Hölder with exponent less than 1/2, that yield double points in dimensions greater than 4. Finally, we show that for d ≥ 2, the Hausdorff dimension of the image of B + f is a.s. at least the maximum of 2 and the dimension of the image of f.

scholarly journals Three-dimensional Hausdorff derivative diffusion model for isotropic/anisotropic fractal porous media

2018 ◽  
Vol 22 (Suppl. 1) ◽  
pp. 1-6 ◽  
Author(s):  
Wei Cai ◽  
Wen Chen ◽  
Fajie Wang
Keyword(s):  
Porous Media ◽  
Fundamental Solution ◽  
Diffusion Equation ◽  
Diffusion Model ◽  
Anomalous Diffusion ◽  
Three Dimensional ◽  
Fractal Structures ◽  
Anisotropic Porous Media ◽  
Convection Diffusion Equation ◽  
Time Space

The anomalous diffusion in fractal isotropic/anisotropic porous media is characterized by the Hausdorff derivative diffusion model with the varying fractal orders representing the fractal structures in different directions. This paper presents a comprehensive understanding of the Hausdorff derivative diffusion model on the basis of the physical interpretation, the Hausdorff fractal distance and the fundamental solution. The concept of the Hausdorff fractal distance is introduced, which converges to the classical Euclidean distance with the varying orders tending to 1. The fundamental solution of the 3-D Hausdorff fractal derivative diffusion equation is proposed on the basis of the Hausdorff fractal distance. With the help of the properties of the Hausdorff derivative, the Huasdorff diffusion model is also found to be a kind of time-space dependent convection-diffusion equation underlying the anomalous diffusion behavior.

scholarly journals LÉVY FLIGHT SUPERDIFFUSION: AN INTRODUCTION

2008 ◽  
Vol 18 (09) ◽  
pp. 2649-2672 ◽  
Author(s):  
A. A. DUBKOV ◽  
B. SPAGNOLO ◽  
V. V. UCHAIKIN
Keyword(s):  
Brownian Motion ◽  
Characteristic Function ◽  
Lévy Process ◽  
Levy Process ◽  
Lévy Flight ◽  
Lévy Flights ◽  
Levy Flight ◽  
Barrier Crossing ◽  
Lévy Motion ◽  
Levy Flights

After a short excursion from the discovery of Brownian motion to the Richardson "law of four thirds" in turbulent diffusion, the article introduces the Lévy flight superdiffusion as a self-similar Lévy process. The condition of self-similarity converts the infinitely divisible characteristic function of the Lévy process into a stable characteristic function of the Lévy motion. The Lévy motion generalizes the Brownian motion on the base of the α-stable distributions theory and fractional order derivatives. Further development on this idea lies on the generalization of the Langevin equation with a non-Gaussian white noise source and the use of functional approach. This leads to the Kolmogorov's equation for arbitrary Markovian processes. As a particular case we obtain the fractional Fokker–Planck equation for Lévy flights. Some results concerning stationary probability distributions of Lévy motion in symmetric smooth monostable potentials, and a general expression to calculate the nonlinear relaxation time in barrier crossing problems are derived. Finally, we discuss the results on the same characteristics and barrier crossing problems with Lévy flights, recently obtained by different approaches.

MULTISCALING TRANSPORT EQUATION ON FRACTALS

Fractals ◽  
1999 ◽  
Vol 07 (02) ◽  
pp. 105-111
Author(s):  
QIUHUA ZENG ◽  
HOUQIANG LI
Keyword(s):  
Brownian Motion ◽  
Diffusion Equation ◽  
Transport Equation ◽  
Fractional Brownian Motion ◽  
Three Dimensional ◽  
Dimensional Case ◽  
Conservation Of Mass ◽  
Fractal Media ◽  
General Transport ◽  
Fractional Transport

The general transport equation of multiscaling disordered fractal media in three-dimensional case is derived from conservation of mass. Multiscaling fractional transport equation is obtained on the basis of discussing Brownian motion, fractional Brownian motion and standard diffusion equation of fractals, which is consistent with the result of literature.

ORTHOGONAL DECOMPOSITIONS FOR LÉVY PROCESSES WITH AN APPLICATION TO THE GAMMA, PASCAL, AND MEIXNER PROCESSES

2003 ◽  
Vol 06 (01) ◽  
pp. 73-102 ◽  
Author(s):  
EUGENE LYTVYNOV
Keyword(s):  
Brownian Motion ◽  
Poisson Process ◽  
Lévy Process ◽  
Lévy Processes ◽  
Random Variable ◽  
Levy Processes ◽  
Levy Process ◽  
Nuclear Space ◽  
Multiple Stochastic Integrals ◽  
Orthogonal Decompositions

It is well known that between all processes with independent increments, essentially only the Brownian motion and the Poisson process possess the chaotic representation property (CRP). Thus, a natural question appears: What is an appropriate analog of the CRP in the case of a general Lévy process. At least three approaches are possible here. The first one, due to Itô, uses the CRP of the Brownian motion and the Poisson process, as well as the representation of a Lévy process through those processes. The second approach, due to Nualart and Schoutens, consists of representing any square-integrable random variable as a sum of multiple stochastic integrals constructed with respect to a family of orthogonalized centered power jumps processes. The third approach, never applied before to the Lévy processes, uses the idea of orthogonalization of polynomials with respect to a probability measure defined on the dual of a nuclear space. The main aims of this paper are to develop the three approaches in the case of a general (ℝ-valued) Lévy process on a Riemannian manifold and (what is more important) to understand a relationship between these approaches. We apply the obtained results to the gamma, Pascal, and Meixner processes, in which case the analysis related to the orthogonalized polynomials becomes essentially simpler and richer than in the general case.

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