Complementary Densities of Lévy Walks: Typical and Rare Fluctuations

Rebenshtok, A.; Denisov, S. I.; Hänggi, Peter; Barkai, Eli

doi:10.1051/mmnp/201611306

Cited by 10 publications

(22 citation statements)

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“…. (38) This result was obtained in the limit of long times asymptotically that is the n−th term in the series is valid provided time is large. How large this time is depends on n: the higher n is, the larger times are needed for the validity of the asymptotic form.…”

Section: B Infinite Densitymentioning

confidence: 86%

“…The coefficients of the quadratic expansion near the maximum characterize thermodynamic fluctuations [36]. Thus in the case of finite dispersion there is one universal scaling with the limiting distribution (the special case of finite dispersion but power-law tail, α > 2, produces different limiting distribution [38]; we do not consider this case here). There are two different scalings that were found in the case of one-dimensional super-diffusive LWs [17,18].…”

Section: Introductionmentioning

confidence: 99%

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Limit theorems for Lévy walks inddimensions: rare and bulk fluctuations

Fouxon¹,

Denisov²,

Zaburdaev³

et al. 2017

J. Phys. A: Math. Theor.

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We consider super-diffusive Lévy walks in d 2 dimensions when the duration of a single step, i.e., a ballistic motion performed by a walker, is governed by a power-law tailed distribution of infinite variance and finite mean. We demonstrate that the probability density function (PDF) of the coordinate of the random walker has two different scaling limits at large times. One limit describes the bulk of the PDF. It is the d−dimensional generalization of the one-dimensional Lévy distribution and is the counterpart of central limit theorem (CLT) for random walks with finite dispersion. In contrast with the one-dimensional Lévy distribution and the CLT this distribution does not have universal shape. The PDF reflects anisotropy of the single-step statistics however large the time is. The other scaling limit, the so-called 'infinite density', describes the tail of the PDF which determines second (dispersion) and higher moments of the PDF. This limit repeats the angular structure of PDF of velocity in one step. Typical realization of the walk consists of anomalous diffusive motion (described by anisotropic d−dimensional Lévy distribution) intermitted by long ballistic flights (described by infinite density). The long flights are rare but due to them the coordinate increases so much that their contribution determines the dispersion. We illustrate the concept by considering two types of Lévy walks, with isotropic and anisotropic distributions of velocities. Furthermore, we show that for isotropic but otherwise arbitrary velocity distribution the d−dimensional process can be reduced to one-dimensional Lévy walk.

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Section: B Infinite Densitymentioning

confidence: 86%

Section: Introductionmentioning

confidence: 99%

Limit theorems for Lévy walks inddimensions: rare and bulk fluctuations

Fouxon¹,

Denisov²,

Zaburdaev³

et al. 2017

J. Phys. A: Math. Theor.

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show abstract

“…By taking the corresponding limits in Eqs. (28) and (30), respectively, we obtain the same asymptotic form as…”

Section: Complementarity Among Different Scaling Regimesmentioning

confidence: 74%

“…The mechanisms of the strong anomalous diffusion for Lévy walk are studied in detail in Refs. [28][29][30], where the probability density function (PDF) consists of two kinds of distributions-Lévy distribution in the central part and infinite density in the tail part. The infinite density is non-normalizable, the concept of which was thoroughly investigated as mathematical issues [31,32], and has been successfully applied to physics; for Lévy walk, it aims at characterizing the ballistic scaling (x ∼ t), which is complementary to the Lévy scaling in the central part of Lévy walk.…”

Section: Introductionmentioning

confidence: 99%

“…Since the case of pure Lévy walk has been fully studied in Refs. [28][29][30], the main objective of this paper is to try to discover new phenomena after introducing the Brownian phase in Lévy walk and to uncover the intrinsic mechanism by clear theoretical analysis. Intuitively, one can expect that at least three modes coexist in this two-state process.…”

Section: Introductionmentioning

confidence: 99%

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Strong anomalous diffusion in two-state process with Lévy walk and Brownian motion

Wang

Chen

Deng

2020

Phys. Rev. Research

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Strong anomalous diffusion phenomena are often observed in complex physical and biological systems, which are characterized by the nonlinear spectrum of exponents qν(q) by measuring the absolute q-th moment |x| q . This paper investigates the strong anomalous diffusion behavior of a two-state process with Lévy walk and Brownian motion, which usually serves as an intermittent search process. The sojourn times in Lévy walk and Brownian phases are taken as power law distributions with exponents α+ and α−, respectively. Detailed scaling analyses are performed for the coexistence of three kinds of scalings in this system. Different from the pure Lévy walk, the phenomenon of strong anomalous diffusion can be observed for this two-state process even when the distribution exponent of Lévy walk phase satisfies α+ < 1, provided that α− < α+. When α+ < 2, the probability density function (PDF) in the central part becomes a combination of stretched Lévy distribution and Gaussian distribution due to the long sojourn time in Brownian phase, while the PDF in the tail part (in the ballistic scaling) is still dominated by the infinite density of Lévy walk.

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