Rectified Brownian transport in corrugated channels: Fractional Brownian motion and Lévy flights

2012 ◽  
Vol 137 (17) ◽  
pp. 174101 ◽  
Author(s):  
Bao-quan Ai ◽  
Zhi-gang Shao ◽  
Wei-rong Zhong
Keyword(s):  
Brownian Motion ◽  
Fractional Brownian Motion ◽  
Lévy Flights ◽  
Levy Flights

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.

scholarly journals Target Search for Transcription Factors on DNA Chains

2019 ◽  
Vol 20 (2) ◽  
pp. 295
Author(s):  
Luiz Otávio Rodrigues Alves Sereno ◽  
José Luiz Acebal
Keyword(s):  
Brownian Motion ◽  
Transcription Factors ◽  
Lattice Model ◽  
Model Parameters ◽  
Search Efficiency ◽  
Lévy Flights ◽  
Target Search ◽  
Grid Model ◽  
Levy Flights ◽  
Dna Chain

Among many structures in the cells of living beings, there are proteins called transcription factors (TF) that are responsible to inhibit or promote the transcription of the DNA. To accomplish their function, the transcription factors perform aleatory searches around the cytoplasm (for prokaryotic cells) and along the DNA chain as well for specific targets located in the DNA. Its movement fits into the class of anomalous Brownian. The efficiency in TFs search has implications in the cellular copy and in protection against viruses, hence the knowledge of the mechanism is of great interest. In the present work, we study the searching process of the TFs by simulating the anomalous Brownian motion through the cytoplasm and DNA chain by means of Levy flights through a lattice model and through a  free grid model. The final distribution of positions of the TF are obtained. The search efficiency is investigated in terms of the model parameters.

scholarly journals A Lévy-flight diffusion model to predict transgenic pollen dispersal

2017 ◽  
Vol 14 (126) ◽  
pp. 20160889 ◽  
Author(s):  
Valentin Vallaeys ◽  
Rebecca C. Tyson ◽  
W. David Lane ◽  
Eric Deleersnijder ◽  
Emmanuel Hanert
Keyword(s):  
Brownian Motion ◽  
Diffusion Model ◽  
Mechanistic Model ◽  
Pollen Dispersal ◽  
Lévy Flights ◽  
Levy Flights ◽  
Modelling Studies ◽  
Isolation Distances ◽  
Model Equations ◽  
Transgenic Pollen

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.

scholarly journals Cutoff Thermalization for Ornstein–Uhlenbeck Systems with Small Lévy Noise in the Wasserstein Distance

2021 ◽  
Vol 184 (3) ◽  
Author(s):  
G. Barrera ◽  
M. A. Högele ◽  
J. C. Pardo
Keyword(s):  
Brownian Motion ◽  
Linear Chain ◽  
Wasserstein Distance ◽  
Growth Patterns ◽  
Lévy Noise ◽  
Lévy Flights ◽  
Cutoff Phenomenon ◽  
Levy Flights ◽  
Generalized Eigenvectors ◽  
Levy Noise

AbstractThis article establishes cutoff thermalization (also known as the cutoff phenomenon) for a class of generalized Ornstein–Uhlenbeck systems $$(X^\varepsilon _t(x))_{t\geqslant 0}$$ ( X t ε ( x ) ) t ⩾ 0 with $$\varepsilon $$ ε -small additive Lévy noise and initial value x. The driving noise processes include Brownian motion, $$\alpha $$ α -stable Lévy flights, finite intensity compound Poisson processes, and red noises, and may be highly degenerate. Window cutoff thermalization is shown under mild generic assumptions; that is, we see an asymptotically sharp $$\infty /0$$ ∞ / 0 -collapse of the renormalized Wasserstein distance from the current state to the equilibrium measure $$\mu ^\varepsilon $$ μ ε along a time window centered on a precise $$\varepsilon $$ ε -dependent time scale $$\mathfrak {t}_\varepsilon $$ t ε . In many interesting situations such as reversible (Lévy) diffusions it is possible to prove the existence of an explicit, universal, deterministic cutoff thermalization profile. That is, for generic initial data x we obtain the stronger result $$\mathcal {W}_p(X^\varepsilon _{t_\varepsilon + r}(x), \mu ^\varepsilon ) \cdot \varepsilon ^{-1} \rightarrow K\cdot e^{-q r}$$ W p ( X t ε + r ε ( x ) , μ ε ) · ε - 1 → K · e - q r for any $$r\in \mathbb {R}$$ r ∈ R as $$\varepsilon \rightarrow 0$$ ε → 0 for some spectral constants $$K, q>0$$ K , q > 0 and any $$p\geqslant 1$$ p ⩾ 1 whenever the distance is finite. The existence of this limit is characterized by the absence of non-normal growth patterns in terms of an orthogonality condition on a computable family of generalized eigenvectors of $$\mathcal {Q}$$ Q . Precise error bounds are given. Using these results, this article provides a complete discussion of the cutoff phenomenon for the classical linear oscillator with friction subject to $$\varepsilon $$ ε -small Brownian motion or $$\alpha $$ α -stable Lévy flights. Furthermore, we cover the highly degenerate case of a linear chain of oscillators in a generalized heat bath at low temperature.

From Brownian motion to self-avoiding walks and Lévy flights

2013 ◽  
Vol 216 (1) ◽  
pp. 1-2
Author(s):  
Christian von Ferber ◽  
Yurij Holovatch ◽  
Ihor Mryglod ◽  
Gleb Oshanin
Keyword(s):  
Brownian Motion ◽  
Lévy Flights ◽  
Levy Flights ◽  
Self Avoiding Walks

scholarly journals A Wavelet-Based Almost-Sure Uniform Approximation of Fractional Brownian Motion with a Parallel Algorithm

2014 ◽  
Vol 51 (1) ◽  
pp. 1-18 ◽  
Author(s):  
Dawei Hong ◽  
Shushuang Man ◽  
Jean-Camille Birget ◽  
Desmond S. Lun
Keyword(s):  
Brownian Motion ◽  
Convergence Rate ◽  
Fractional Brownian Motion ◽  
Parallel Algorithm ◽  
Uniform Approximation ◽  
Haar Wavelets ◽  
Hurst Index ◽  
Sample Paths ◽  
Vanishing Moment ◽  
Uniform Expansion

We construct a wavelet-based almost-sure uniform approximation of fractional Brownian motion (FBM) (Bt(H))_t∈[0,1] of Hurst index H ∈ (0, 1). Our results show that, by Haar wavelets which merely have one vanishing moment, an almost-sure uniform expansion of FBM for H ∈ (0, 1) can be established. The convergence rate of our approximation is derived. We also describe a parallel algorithm that generates sample paths of an FBM efficiently.

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