Lévy flights with power-law absorption

Cattivelli, Luca; Agliari, Elena; Sartori, Fabio; Cassi, Davide

doi:10.1103/physreve.92.042156

Cited by 3 publications

(6 citation statements)

References 42 publications

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“…the target is discovered with a power-law decaying probabilityin Ref. [63]. In that case, if the stable index α < 1, the search still can be absolutely reliable (P = 1), if the scaling exponent characterising an absorption probability β ≤ α, i.e., the target localisation or absorption probability is delocalised stronger than the LF process.…”

Section: Discussionmentioning

confidence: 99%

Search reliability and search efficiency of combined Lévy–Brownian motion: long relocations mingled with thorough local exploration

Palyulin

Chechkin

Klages

et al. 2016

J. Phys. A: Math. Theor.

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Abstract. A combined dynamics consisting of Brownian motion and Lévy flights is exhibited by a variety of biological systems performing search processes. Assessing the search reliability of ever locating the target and the search efficiency of doing so economically of such dynamics thus poses an important problem. Here we model this dynamics by a one-dimensional fractional Fokker-Planck equation combining unbiased Brownian motion and Lévy flights. By solving this equation both analytically and numerically we show that the superposition of recurrent Brownian motion and Lévy flights with stable exponent α < 1, by itself implying zero probability of hitting a point on a line, lead to transient motion with finite probability of hitting any point on the line. We present results for the exact dependence of the values of both the search reliability and the search efficiency on the distance between the starting and target positions as well as the choice of the scaling exponent α of the Lévy flight component.Search reliability and search efficiency of combined Lévy-Brownian motion 2 Figure 1. Sketch of intermittent search in one dimension: a searcher (blue) proceeds by a combination of Lévy jumps and Brownian steps until it finds the target (red). Physically, Lévy jumps decorrelate the motion, leading the searcher to sites not previously visited. Brownian motion, instead, provides a thorough local search at the price of oversampling, see text.

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Section: Discussionmentioning

confidence: 99%

Search reliability and search efficiency of combined Lévy–Brownian motion: long relocations mingled with thorough local exploration

Palyulin

Chechkin

Klages

et al. 2016

J. Phys. A: Math. Theor.

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“…Moreover, the probability of encounter along a tooth remains a(Y ) ∼ 1/Y , consequently also in these cases the two-particle transience is ensured (see Sec. II and the condition of the paper [32]) as proven in [15,23].…”

Section: Two Random Walkers On Bundled Structuresmentioning

confidence: 89%

“…This model was studied in detail in [32], where it is shown that the Lévy flight characterized by a jump distribution h(ξ) ∼ ξ −µ−1 in the presence of traps with distribution scaling as a(ξ) ∼ ξ −α has a finite probability of never being absorbed when the displacement exponent µ (in our case µ = 1/2) is lower than the absorption exponent α (in our case α = 1). Since here this condition is fulfilled, we recover the two-particle transience on combs, that is, the two particles have a finite probability of never meeting, regardless of their starting position.…”

Section: Two Random Walkers On Combsmentioning

confidence: 93%

“…The probability a(∆ y ) to visit the encounter line before leaving the plane ∆ x = 0 scales as ∼ 1/∆ y [21]. Therefore, recalling again that for a walker with jump distribution h(ξ) ∼ ξ −µ−1 in the presence of traps with distribution scaling as a(ξ) ∼ ξ −α , the overall absorption is not certain as long as µ < α [32], we get that the walker on M has a finite probability not to meet the encounter line. This basically equals to state the two particle transience of the comb.…”

Section: Mapping the Two-particle Problem Into A One-particle Problemmentioning

confidence: 99%

“…Indeed, the jump sizes and the waiting times are coupled and the process is better described in terms of a Coupled Continuous Time Random Walk (CCTRW). However, as shown in[32], the absorption probability depends only on a(Y ) and h(Y ) and not on the distribution ψ(t). Therefore, the description of the process in terms of Lévy flights is perfectly workable.…”

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confidence: 90%

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Two-particle problem in comblike structures

Agliari

Cassi²,

Cattivelli

et al. 2016

Phys. Rev. E

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Encounters between walkers performing a random motion on an appropriate structure can describe a wide variety of natural phenomena ranging from pharmacokinetics to foraging. On homogeneous structures the asymptotic encounter probability between two walkers is (qualitatively) independent of whether both walkers are moving or one is kept fixed. On infinite comb-like structures this is no longer the case and here we deepen the mechanisms underlying the emergence of a finite probability that two random walkers will never meet, while one single random walker is certain to visit any site. In particular, we introduce an analytical approach to address this problem and even more general problems such as the case of two walkers with different diffusivity, particles walking on a finite comb and on arbitrary bundled structures, possibly in the presence of loops. Our investigations are both analytical and numerical and highlight that, in general, the outcome of a reaction involving two reactants on a comb-like architecture can be strongly different according to whether both reactants are moving (no matter their relative diffusivities) or only one, and according to the density of short-cuts among the branches.

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Control of dynamics via identical time-lagged stochastic inputs

Bolhasani

Azizi

Abdollahpour

et al. 2020

Chaos: An Interdisciplinary Journal of Nonlinear Science

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We investigate the impact of a stochastic forcing, comprised of a sum of time-lagged copies of a single source of noise, on the system dynamics. This type of stochastic forcing could be made artificially, or it could be the result of shared upstream inputs to a system through different channel lengths. By means of a rigorous mathematical framework, we show that such a system is, in fact, equivalent to the classical case of a stochastically-driven dynamical system with time-delayed intrinsic dynamics but without a time lag in the input noise. We also observe a resonancelike effect between the intrinsic period of the oscillation and the time lag of the stochastic forcing, which may be used to determine the intrinsic period of oscillations or the inherent time delay in dynamical systems with oscillatory behavior or delays. As another useful application of imposing time-lagged stochastic forcing, we show that the dynamics of a system can be controlled by changing the time lag of this stochastic forcing, in a fashion similar to the classical case of Pyragas control via delayed feedback. To confirm these results experimentally, we set up a laser diode system with such stochastic inputs, which effectively behaves as a Langevin system. As in the theory, a peak emerged in the autocorrelation function of the output signal that could be tuned by the lag of the stochastic input. Our findings, thus, indicate a new approach for controlling useful instabilities in dynamical systems.

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Lévy flights with power-law absorption

Cited by 3 publications

References 42 publications

Search reliability and search efficiency of combined Lévy–Brownian motion: long relocations mingled with thorough local exploration

Search reliability and search efficiency of combined Lévy–Brownian motion: long relocations mingled with thorough local exploration

Two-particle problem in comblike structures

Control of dynamics via identical time-lagged stochastic inputs

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