Infinite densities for Lévy walks

Abstract

Motion of particles in many systems exhibits a mixture between periods of random diffusive like events and ballistic like motion. In many cases, such systems exhibit strong anomalous diffusion, where low order moments $<|x(t){|}^q>$ with $q$ below a critical value $q_c$ exhibit diffusive scaling while for $q>q_c$ a ballistic scaling emerges. The mixed dynamics constitutes a theoretical challenge since it does not fall into a unique category of motion, e.g., the known diffusion equations and central limit theorems fail to describe both aspects. In this paper we resolve this problem by resorting to the concept of infinite density. Using the widely applicable Lévy walk model, we find a general expression for the corresponding non-normalized density which is fully determined by the particles velocity distribution, the anomalous diffusion exponent $\alpha$ and the diffusion coefficient $K_{\alpha }$. We explain how infinite densities play a central role in the description of dynamics of a large class of physical processes and discuss how they can be evaluated from experimental or numerical data.

Phys. Rev. E, in press

References in corpus (12)

• The large deviation approach to statistical mechanics
• Non-ergodic Intensity Correlation Functions for Blinking Nano Crystals
• Distribution of Time-Averaged Observables for Weak Ergodicity Breaking
• Probing rare physical trajectories with Lyapunov weighted dynamics
• Numerical test of hydrodynamic fluctuation theory in the Fermi-Pasta-Ulam chain
• Pesin-Type Identity for Weak Chaos
• Weakly non-ergodic Statistical Physics
• Photon Counting Statistics For Blinking CdSe-ZnS Quantum Dots: A Levy Walk Process
• Markovian embedding of fractional superdiffusion
• Space-time velocity correlation function for random walks
• Anomalous transmission and drifts in one-dimensional Levy structures
• Transport properties of Lévy walks: an analysis in terms of multistate processes

Cited by in corpus (6)

• Lévy walks
• Optimization and universality of Brownian search in quenched heterogeneous media
• Anomalous dispersion in correlated porous media: A coupled continuous time random walk approach
• Solvable continuous time random walk model of the motion of tracer particles through porous media
• Fluctuations around equilibrium laws in ergodic continuous-time random walks
• Non-stationary Markovian Replication Process causing Diverse Diffusions