Tempered stable Lévy motion and transient super-diffusion

Baeumer, Boris; Meerschaert, Mark M.

doi:10.1016/j.cam.2009.10.027

Cited by 243 publications

(258 citation statements)

References 21 publications

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“…The next step is to generate the tempered stable random variable using the exponential rejection method of Baeumer and Meerschaert [47]. For example, to generate the α-order tempered stable random variable dξ i in (5), we first draw a random variable Z = D 1/α Y , and an exponentially distributed random variable W with mean λ −1 .…”

Section: Discussionmentioning

confidence: 99%

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Particle-tracking simulation of fractional diffusion-reaction processes

Zhang

Papelis

2011

Phys. Rev. E

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Abstract. Computer simulation of reactive transport in heterogeneous systems remains a challenge, due to the multi-scale nature of reactive dynamics and the non-Fickian behavior of transport. This study develops a fully Lagrangian approach via particle tracking to describe the reactive transport controlled by the tempered super-or sub-diffusion. In the particle-tracking algorithm, the local-scale reaction is affected by the interaction radius between adjacent reactants, whose motion can be simulated by the Langevin equations corresponding to the tempered stable models. Lagrangian simulation results show that the transient superdiffusion enhances reaction by enhancing the degree of mixing of reactants. The proposed particle-tracking scheme can also be extended conveniently to multi-scaling super-diffusion. For the case of transient sub-diffusion, the trapping of solutes in the immobile phase can either decrease or accelerate the reaction rate, depending on the initial condition of reactant particles. Further practical applications show that the new solver efficiently captures bimolecular reactions observed in laboratories.

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Section: Discussionmentioning

confidence: 99%

“…The Riemann-Liouville fractional derivative is used here, so that the shifted Grünwald approximation can be applied [47]. The model (26) can be discretized using an implicit finite difference scheme…”

Section: Discussionmentioning

confidence: 99%

Particle-tracking simulation of fractional diffusion-reaction processes

Zhang

Papelis

2011

Phys. Rev. E

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“…The exponentially damped Lévy density, therefore, needs to be renormalized and recentred. More details on these issues can be found in [34]. Figure 1 shows three sample LF trajectories whose displacement lengths have been drawn from a truncated Lévy distribution.…”

Section: Modelling the Anomalous Dispersal Of Honeybeesmentioning

confidence: 99%

A Lévy-flight diffusion model to predict transgenic pollen dispersal

Vallaeys

Tyson

Lane³

et al. 2017

J. R. Soc. Interface.

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The containment of genetically modified (GM) pollen is an issue of significant concern for many countries. For crops that are bee-pollinated, model predictions of outcrossing rates depend on the movement hypothesis used for the pollinators. Previous work studying pollen spread by honeybees, the most important pollinator worldwide, was based on the assumption that honeybee movement can be well approximated by Brownian motion. A number of recent studies, however, suggest that pollinating insects such as bees perform Lévy flights in their search for food. Such flight patterns yield much larger rates of spread, and so the Brownian motion assumption might significantly underestimate the risk associated with GM pollen outcrossing in conventional crops. In this work, we propose a mechanistic model for pollen dispersal in which the bees perform truncated Lévy flights. This assumption leads to a fractional-order diffusion model for pollen that can be tuned to model motion ranging from pure Brownian to pure Lévy. We parametrize our new model by taking the same pollen dispersal dataset used in Brownian motion modelling studies. By numerically solving the model equations, we show that the isolation distances required to keep outcrossing levels below a certain threshold are substantially increased by comparison with the original predictions, suggesting that isolation distances may need to be much larger than originally thought.

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“…where {U(t), t ≥ 0} is a totally skewed α−stable Lévy motion with the stability index α, [23,40]. By using tail approximation of stable density, [41], we obtain the following:…”

Section: Tempered Stable Subordinatormentioning

confidence: 99%

Arithmetic Brownian motion subordinated by tempered stable and inverse tempered stable processes

Wyłomańska

2012

Physica A: Statistical Mechanics and its Applications

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In the last decade the subordinated processes have become popular and found many practical applications. Therefore in this paper we examine two processes related to time-changed (subordinated) classical Brownian motion with drift (called arithmetic Brownian motion). The first one, so called normal tempered stable, is related to the tempered stable subordinator, while the second one -to the inverse tempered stable process. We compare the main properties (such as probability density functions, Laplace transforms, ensemble averaged mean squared displacements) of such two subordinated processes and propose the parameters' estimation procedures. Moreover we calibrate the analyzed systems to real data related to indoor air quality.

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Tempered stable Lévy motion and transient super-diffusion

Cited by 243 publications

References 21 publications

Particle-tracking simulation of fractional diffusion-reaction processes

Particle-tracking simulation of fractional diffusion-reaction processes

A Lévy-flight diffusion model to predict transgenic pollen dispersal

Arithmetic Brownian motion subordinated by tempered stable and inverse tempered stable processes

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