Fractional diffusion equation for an<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>n</mml:mi></mml:math>-dimensional correlated Lévy walk

Taylor-King, Jake P.; Klages, Rainer; Fedotov, Sergei; Gorder, Robert A. Van

doi:10.1103/physreve.94.012104

Cited by 21 publications

(29 citation statements)

References 53 publications

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“…Different to the product and XY -models, this is a radially symmetric function which naturally follows from the microscopic isotropy of the walk. Mathematically, the expression above is a generalization of the Lévy distribution to two dimensions [27,30]. However, from the physics point of view, it provides the generalization of the Einstein relation and relates the generalized diffusion constant K γ to the physical parameters of the 2d process, v 0 , τ 0 and γ.…”

Section: Governing Equationsmentioning

confidence: 99%

Superdiffusive Dispersals Impart the Geometry of Underlying Random Walks

et al. 2016

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It is recognised now that a variety of real-life phenomena ranging from diffuson of cold atoms to motion of humans exhibit dispersal faster than normal diffusion. Lévy walks is a model that excelled in describing such superdiffusive behaviors albeit in one dimension. Here we show that, in contrast to standard random walks, the microscopic geometry of planar superdiffusive Lévy walks is imprinted in the asymptotic distribution of the walkers. The geometry of the underlying walk can be inferred from trajectories of the walkers by calculating the analogue of the Pearson coefficient. Introduction. The Lévy walk (LW) model [1-3] was developed to describe spreading phenomena that were not fitting the paradigm of Brownian diffusion [4]. Still looking as a random walk, see Fig. 1, but with a very broad distribution of excursions' lengths, the corresponding processes exhibit dispersal faster than in the case of normal diffusion. Conventionally, this difference is quantified with the mean squared displacement (MSD), r 2 (t) ∝ t α , and the regime with α > 1 is called superdiffusion. Examples of such systems range from cold atoms moving in dissipative optical lattices [5] to T cells migrating in the brain tissue [6]. Most of the existing theoretical results, however, were derived for one dimensional LW processes [3]. In contrast, real life phenomena -biological motility (from bacteria [7] to humans [8] and autonomous robots [9,10]), animal foraging [11,12] and search [13] -happen in two dimensions. Somewhat surprisingly, generalizations of the Lévy walks to two dimensions are still virtually unexplored.

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Section: Governing Equationsmentioning

confidence: 99%

Superdiffusive Dispersals Impart the Geometry of Underlying Random Walks

et al. 2016

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“…Lévy walk [1][2][3] is an important concept which describes a wide spectrum of physical and biological processes involving stochastic transport [4][5][6][7][8][9][10]. Cold atoms moving in dissipative optical lattices [11], endosomal active transport in living cells [12], and T-cells migrating in the brain tissue [13] are just several examples where Lévy walks were reported.…”

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confidence: 99%

Emergence of Lévy walks in systems of interacting individuals

Fedotov¹,

Korabel²

2017

Phys. Rev. E

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We propose a model of superdiffusive Lévy walk as an emergent nonlinear phenomenon in systems of interacting individuals. The aim is to provide a qualitative explanation of recent experiments [G. Ariel et al., Nat. Commun. 6, 8396 (2015)] revealing an intriguing behavior: swarming bacteria fundamentally change their collective motion from simple diffusion into a superdiffusive Lévy walk dynamics. We introduce microscopic mean-field kinetic equations in which we combine two key ingredients: (1) alignment interactions between individuals and (2) non-Markovian effects. Our interacting run-and-tumble model leads to the superdiffusive growth of the mean-squared displacement and the power-law distribution of run length with infinite variance. The main result is that the superdiffusive behavior emerges as a cooperative effect without using the standard assumption of the power-law distribution of run distances from the inception. At the same time, we find that the collision and repulsion interactions lead to the density-dependent exponential tempering of power-law distributions. This qualitatively explains the experimentally observed transition from superdiffusion to the diffusion of mussels as their density increases [M. [15][16][17][18]. Recently an intriguing behavior of swarming bacteria was found: they fundamentally change their collective motion from simple diffusion into a superdiffusive Lévy walk dynamics [19]. The extraordinary feature of this movement is that the emergence of a superdiffusive motility is a result of the interactions between bacteria rather than the standard mechanism of controlling the individual frequency of tumbling. However, it is still an open question how Lévy walks emerge in systems of interacting self-propelled particles. The current theory of Lévy walk [8] assumes noninteracting particles and power-law distribution of traveled distances from the inception.The collective behavior of large groups of interacting individuals such as bird flocks, fish schools, and the collective migration of cells or bacteria is another rapidly growing area of active matter research [20][21][22][23][24][25][26][27][28][29][30][31][32][33][34][35][36][37][38]. There exist two main types of models used for a collective behavior: (1) Lagrangian models describing the movements of self-propelled particles in terms of nonlinear equations for the positions and velocities of all particles [20][21][22][23], and (2) kinetic models involving partial differential equations for the population densities [24][25][26][27][28][29][30]. In this Rapid Communication we are only concerned with the nonlinear kinetic and macroscopic equations. They have been used to describe interactions between individuals and investigate the formation of a large variety of spatiotemporal self-organized aggregations. Most of these models converge to the density-dependent diffusive transport and do not lead to non-Markovian Lévy walks.In this Rapid Communication we propose a kinetic nonlinear Lévy walk model for interacting individuals in which...

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“…In recent years, Lévy walks [8] attracted attention in modelling movement patterns of living things [9], from sub-cellular [10][11][12] to organism [13][14][15] scales. Until now, Lévy walks have been mostly described by coupled continuous time random walks (CTRW) [8], various fractional PDEs [16][17][18][19][20][21][22] and integro-differential equations [23,24]. These approaches require power-law distributed running times with divergent first and second moments as an ab initio assumption.…”

Section: Introductionmentioning

confidence: 99%

Self-reinforcing directionality generates truncated Lévy walks without the power-law assumption

et al. 2021

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We introduce a persistent random walk model with finite velocity and self-reinforcing directionality, which explains how exponentially distributed runs self-organize into truncated Lévy walks observed in active intracellular transport by Chen et. al. [Nat. mat., 2015]. We derive the nonhomogeneous in space and time, hyperbolic PDE for the probability density function (PDF) of particle position. This PDF exhibits a bimodal density (aggregation phenomena) in the superdiffusive regime, which is not observed in classical linear hyperbolic and Lévy walk models. We find the exact solutions for the first and second moments and criteria for the transition to superdiffusion.

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Fractional diffusion equation for ann-dimensional correlated Lévy walk

Cited by 21 publications

References 53 publications

Superdiffusive Dispersals Impart the Geometry of Underlying Random Walks

Superdiffusive Dispersals Impart the Geometry of Underlying Random Walks

Emergence of Lévy walks in systems of interacting individuals

Self-reinforcing directionality generates truncated Lévy walks without the power-law assumption

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