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Register a new userEmergence of L{e}vy Walks in Systems of Interacting Individuals
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Recent experiments (G. Ariel, et al., Nature Comm. 6, 8396 (2015)) revealed an intriguing behavior of swarming bacteria: they fundamentally change their collective motion from simple diffusion into a superdiffusive L{e}vy walk dynamics. We introduce a nonlinear non-Markovian persistent random walk model that explains the emergence of superdiffusive L{e}vy walks. We show that the alignment interaction between individuals can lead 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 nonlinear collective phenomenon, rather than due to the standard assumption of the power law distribution of run distances from the inception. At the same time, we find that the repulsion/collision effects lead to the density dependent exponential tempering of power law distributions. This qualitatively explains experimentally observed transition from superdiffusion to the diffusion of mussels as their density increases (M. de Jager et al., Proc. R. Soc. B 281, 20132605 (2014)).
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Intermittent stochastic processes appear in a wide field, such as chemistry, biology, ecology, and computer science. This paper builds up the theory of intermittent continuous time random walk (CTRW) and L{e}vy walk, in which the particles are stochastically reset to a given position with a resetting rate $r$. The mean squared displacements of the CTRW and L{e}vy walks with stochastic resetting are calculated, uncovering that the stochastic resetting always makes the CTRW process localized and L{e}vy walk diffuse slower. The asymptotic behaviors of the probability density function of Levy walk with stochastic resetting are carefully analyzed under different scales of $x$, and a striking influence of stochastic resetting is observed.
Levy walk process is one of the most effective models to describe superdiffusion, which underlies some important movement patterns and has been widely observed in the micro and macro dynamics. From the perspective of random walk theory, here we investigate the dynamics of Levy walks under the influences of the constant force field and the one combined with harmonic potential. Utilizing Hermite polynomial approximation to deal with the spatiotemporally coupled analysis challenges, some striking features are detected, including non Gaussian stationary distribution, faster diffusion, and still strongly anomalous diffusion, etc.
The transport of self-propelled particle confined in corrugated channel with L{e}vy noise is investigated. The parameters of L{e}vy noise(i.e., the stability index, the asymmetry parameter, the scale parameter, the location parameter) and the parameters of confined corrugated channel(i.e., the compartment length, the channel width and the bottleneck size) have joint effects on the system. There exits flow reverse phenomena with increasing mean parameter. Left distribution noise will induce $-x$ directional transport and right distribution noise will induce $+x$ directional transport. The distribution skewness will effect the moving direction of the particle. The average velocity shows complex behavior with increasing stability index. The angle velocity and the angle Gaussian noise have little effects on the particle transport.
L{e}vy walk is a popular and more `physical model to describe the phenomena of superdiffusion, because of its finite velocity. The movements of particles are under the influences of external potentials almost at anytime and anywhere. In this paper, we establish a Langevin system coupled with a subordinator to describe the L{e}vy walk in the time-dependent periodic force field. The effects of external force are detected and carefully analyzed, including nonzero first moment (even though the force is periodic), adding an additional dispersion on the particle position, the consistent influence on the ensemble- and time-averaged mean-squared displacement, etc. Besides, the generalized Klein-Kramers equation is obtained, not only for the time-dependent force but also for space-dependent one.
Motivated by the emph{L{e}vy foraging hypothesis} -- the premise that various animal species have adapted to follow emph{L{e}vy walks} to optimize their search efficiency -- we study the parallel hitting time of L{e}vy walks on the infinite two-dimensional grid.We consider $k$ independent discrete-time L{e}vy walks, with the same exponent $alpha in(1,infty)$, that start from the same node, and analyze the number of steps until the first walk visits a given target at distance $ell$.We show that for any choice of $k$ and $ell$ from a large range, there is a unique optimal exponent $alpha_{k,ell} in (2,3)$, for which the hitting time is $tilde O(ell^2/k)$ w.h.p., while modifying the exponent by an $epsilon$ term increases the hitting time by a polynomial factor, or the walks fail to hit the target almost surely.Based on that, we propose a surprisingly simple and effective parallel search strategy, for the setting where $k$ and $ell$ are unknown:The exponent of each L{e}vy walk is just chosen independently and uniformly at random from the interval $(2,3)$.This strategy achieves optimal search time (modulo polylogarithmic factors) among all possible algorithms (even centralized ones that know $k$).Our results should be contrasted with a line of previous work showing that the exponent $alpha = 2$ is optimal for various search problems.In our setting of $k$ parallel walks, we show that the optimal exponent depends on $k$ and $ell$, and that randomizing the choice of the exponents works simultaneously for all $k$ and $ell$.
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