Lévy walks

Abstract

Random walk is a fundamental concept with applications ranging from quantum physics to econometrics. Remarkably, one specific model of random walks appears to be ubiquitous across many fields as a tool to analyze transport phenomena in which the dispersal process is faster than dictated by Brownian diffusion. The Lévy-walk model combines two key features, the ability to generate anomalously fast diffusion and a finite velocity of a random walker. Recent results in optics, Hamiltonian chaos, cold atom dynamics, biophysics, and behavioral science demonstrate that this particular type of random walk provides significant insight into complex transport phenomena. This review gives a self-consistent introduction to Lévy walks, surveys their existing applications, including latest advances, and outlines further perspectives.

Emergence of Lévy walks in systems of interacting individuals

“…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].…”

“…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.…”

Observing Power-Law Dynamics of Position-Velocity Correlation in Anomalous Diffusion

“…An intriguing system to look at in this context is that of anomalous dynamics for which the mean square displacement (MSD) scales as x 2 ∼ t 2α , with α = 1/2. This type of dynamics, found in a wide variety of systems in nature ranging from dynamics of "bubbles" in denaturing DNA molecules [1], through fluctuations in the stockmarket [2] to models describing brief awakenings in the course of a night's sleep [3], is generally non-universal and system-dependent [4][5][6].A uniquely interesting model system for the study of anomalous diffusion is that of cold atoms diffusing in a dissipative 1D lattice, closely related to Lévy walks and motion in logarithmic potentials, displaying such phenomena as the breakdown of ergodicity and of equipartition, memory effects and slow relaxation to equilibrium [6][7][8][9][10][11][12][13][14][15][16][17][18][19][20][21][22][23]. The major advantage of such a system is the high degree of control it enables over the physical parameters governing the dynamics.…”

“…4 (e), for two distinct scenarios, one where the velocities are initialized in some arbitrary initial size (orange triangles) and one where they are initialized at their steady-state value corresponding to each lattice depth, setting β = 0 (blue circles). The second simulation is a Lévy walk simulation [6], where particles are initialized in an uncorrelated Gaussian phase-space and proceed to perform walks of durations τ , drawn from a unity-scaled Lomax distribution ψ γ0 (τ ) = γ 0 /(1 + τ ) 1+γ0 . The width of the velocity distribution remains constant throughout the simulation (β = 0).…”

Solitons as candidates for energy carriers in Fermi-Pasta-Ulam lattices

“…The anomalous energy transport in it is attributed to the non-interaction of phonons. When nonlinear effects take place, the anomalous energy transport in 1D momentum-conserving lattices is mainly attributed to the Lévy walk of energy carriers on the microscopic level [31][32][33][34][35][36][37][38][39]. On the mesoscopic level, the theory of nonlinear fluctuating hydrodynamics (NFH) is a powerful tool to study the anomalous energy transport in 1D momentum-conserving lattices [40][41][42][43][44].…”