Lévy walk dynamics in an external harmonic potential

Xu, Pengbo; Zhou, Tian; Metzler, Ralf; Deng, Weihua

doi:10.1103/physreve.101.062127

Cited by 20 publications

(26 citation statements)

References 80 publications

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“…Moreover, the first moment in comoving coordinate x(t) = 0 since R 1 (s) = 0, which indicates the process in comoving coordinate is symmetric. It should be noted that the normalization of the PDF W (x, t) can also be verified through Hermite polynomials expansion and the result of R 0 = 1/ √ π, the detailed calculations of which can be found in [40,41].…”

Section: Hermite Polynomial Approximation To L éVy Walk In Non-static...mentioning

confidence: 87%

“…In the following, to theoretically obtain the behaviors of moments in comoving and physical coordinates, we utilize Hermite orthogonal polynomials to approach PDF [40][41][42].…”

Section: L éVy Walk In a Non-static Mediamentioning

confidence: 99%

“…Another quantity to characterize the property of PDF for stochastic process is kurtosis K, which measures the tails of the PDF. For a symmetric process with zero spatial average, the kurtosis can be defined as [41,42]…”

Section: Exponential Scale Factormentioning

confidence: 99%

“…Then using Hermite polynomials expansion established in [40] solves the equation to calculate the statistical observables. The method of Hermite orthogonal polynomials expansion is good at calculating MSD even for some cases that are unsolvable by ordinary method of integral transform, such as Lévy walks moving under external potential [41,42]. In addition, for the Langevin equation describing the diffusion processes under the action of external potential [43][44][45][46][47], Hermite polynomials expansion is also a doable method [45,[48][49][50].…”

Section: Introductionmentioning

confidence: 99%

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Lévy walk dynamics in non-static media

Zhou,

Xu,

Deng

2021

Preprint

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Almost all the media the particles move in are non-static. Depending on the expected resolution of the studied dynamics and the amplitude of the displacement of the media, sometimes the nonstatic behaviours of the media can not be ignored. In this paper, we build the model describing Lévy walks in non-static media, where the physical and comoving coordinates are connected by scale factor. We derive the equation governing the probability density function of the position of the particles in comoving coordinate. Using the Hermite orthogonal polynomial expansions, some statistical properties are obtained, such as mean squared displacements (MSDs) in both coordinates and kurtosis. For some representative non-static media and Lévy walks, the asymptotic behaviors of MSDs in both coordinates are analyzed in detail. The stationary distributions and mean first passage time for some cases are also discussed through numerical simulations.

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Section: Hermite Polynomial Approximation To L éVy Walk In Non-static...mentioning

confidence: 87%

“…In the following, to theoretically obtain the behaviors of moments in comoving and physical coordinates, we utilize Hermite orthogonal polynomials to approach PDF [40][41][42].…”

Section: L éVy Walk In a Non-static Mediamentioning

confidence: 99%

Section: Exponential Scale Factormentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Lévy walk dynamics in non-static media

Zhou,

Xu,

Deng

2021

Preprint

Self Cite

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“…Although the generalized Kramers–Fokker–Planck equation of Lévy walk in arbitrary external potentials is given in [ 25 , 26 ], it seems hard to uncover more concrete characters due to the spatiotemporal coupled property. Hermite orthogonal polynomials approximation is a complementary method to integral transform introduced in [ 69 ] to solve the spatiotemporal coupled problem of Lévy walk and Lévy walk process under the action of external potential is detailedly discussed in [ 70 , 74 ] by this brand new method. Moreover, Langevin equations [ 16 ] provide a feasible framework to solve the problem with external potentials.…”

Section: Introductionmentioning

confidence: 99%

Lévy Walk Dynamics in an External Constant Force Field in Non-Static Media

Zhou

Deng

2022

J Stat Phys

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Based on the recognition of the huge change of the transport properties for diffusion particles in non-static media, we consider a Lévy walk model subjected to an external constant force in non-static media. Since the physical and comoving coordinates of non-static media are related by scale factor, we equivalently transfer the process from physical coordinate into comoving coordinate and derive the master equation governing the probability density function of the position of the particles in comoving coordinate. Utilizing the Hermite orthogonal polynomial expansions, some statistical properties are obtained, including the asymptotic behaviors of the first two moments in both coordinates and kurtosis. For some representative types of non-static media and Lévy walks, the striking and interesting phenomena originating from the interplay between non-static media, external force, and intrinsic stochastic motion are observed. The stationary distribution are also analyzed for some cases through numerical simulations.

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Active impedance control based adaptive locomotion for a bionic hexapod robot

Zhu,

Liu,

Yuan

et al. 2024

Journal of Field Robotics

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In recent years, with the continuous development of human exploration of the natural world, there has been a growing demand across various fields for robots capable of free movement in diverse environments. In this study, we address the issue of compliant control for a hexapod robot in diverse environments and propose a novel control method based on an adaptive impedance model for position control. Our approach enables the hexapod robot to stabilize foot force on complex terrains while preserving balance and body height. Specifically, we analyze the algorithm's parameters and stability by establishing the hexapod robot's structural and impedance control models. To tackle this challenge, we introduce an adaptive impedance control algorithm that estimates environmental parameters using Lyapunov's asymptotic stability theorem and achieves tracking of actual foot‐end forces to desired foot forces. Furthermore, to ensure body stability and height, we incorporate attitude feedback and body feedback. Experimental results from foot force control experiments conducted on a multilegged robot demonstrate that our proposed algorithm enhances the adaptability and robustness of the robot. This research holds significant implications for the stable control of hexapod robots in complex environments and has promising practical applications.

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Lévy walk dynamics in an external harmonic potential

Cited by 20 publications

References 80 publications

Lévy walk dynamics in non-static media

Lévy walk dynamics in non-static media

Lévy Walk Dynamics in an External Constant Force Field in Non-Static Media

Active impedance control based adaptive locomotion for a bionic hexapod robot

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