Anomalous Diffusion of Inertial, Weakly Damped Particles

Abstract

The anomalous (i.e. non-Gaussian) dynamics of particles subject to a deterministic acceleration and a series of 'random kicks' is studied. Based on an extension of the concept of continuous time random walks to position-velocity space, a new fractional equation of the Kramers-Fokker-Planck type is derived. The associated collision operator necessarily involves a fractional substantial derivative, representing important nonlocal couplings in time and space. For the force-free case, a closed solution is found and discussed.

Submitted to Physical Review Letters

Cited by in corpus (45)

• Lévy walks
• Non-equilibrium steady states of stochastic processes with intermittent resetting
• From non-Brownian Functionals to a Fractional Schrödinger Equation
• Origin of hyperdiffusion in generalized Brownian motion
• Markovian embedding of non-Markovian superdiffusion
• On distributions of functionals of anomalous diffusion paths
• Tempered Fractional Feynman-Kac Equation
• On tempered fractional calculus with respect to functions and the associated fractional differential equations
• A fractional Feynman-Kac equation for weak ergodicity breaking
• Analytic approaches of the anomalous diffusion: a review
• High order algorithms for the fractional substantial diffusion equation with truncated Lévy flights
• Fluctuation relations for anomalous dynamics
• Numerical algorithms for the forward and backward fractional Feynman-Kac equations
• Disentangling sources of anomalous diffusion
• Anomalous processes with general waiting times: functionals and multi-point structure
• A fractional diffusion equation for two-point probability distributions of a continuous-time random walk
• Lévy-walk-like Langevin dynamics
• Single integro-differential wave equation for Lévy walk
• Weak Galilean invariance as a selection principle for coarse-grained diffusive models
• Feynman-Kac equation for anomalous processes with space- and time-dependent forces
• Lévy walk dynamics in an external harmonic potential
• A Fractional Diffusion Equation for an n-Dimensional Correlated Levy Walk
• Feynman-Kac equation revisited
• Continuous Time Random Walks with Internal Dynamics and Subdiffusive Reaction-Diffusion Equations
• Fractional compound Poisson processes with multiple internal states
• Fluctuation relations for anomalous dynamics generated by time-fractional Fokker-Planck equations
• Langevin formulation of a subdiffusive continuous time random walk in physical time
• Extended Poisson-Kac theory: A unifying framework for stochastic processes with finite propagation velocity
• Lagrangian statistics in forced two-dimensional turbulence
• Correction of BDFk for fractional Feynman-Kac equation with Lévy flight
• Nonergodicity of $d$-dimensional generalized Lévy walks and their relation to other space-time coupled models
• Langevin picture of anomalous diffusion processes in expanding medium
• Lévy-walk-like Langevin dynamics affected by a time-dependent force
• Continuous Markovian model for Levy random walks with superdiffusive and superballistic regimes
• Lévy walk dynamics in mixed potentials from the perspective of random walk theory
• Linear Boltzmann-like equation, describing non-classical particle transport, and related asymptotic solutions for small mean free paths
• Feynman-Kac Equations for Reaction and Diffusion Processes
• Random diffusivity processes in an external force field
• Anomalous diffusion for inertial particles under gravity in parallel flows
• Fractional Bhatnagar-Gross-Krook kinetic equation
• Subdiffusion in the Presence of Reactive Boundaries: A Generalized Feynman-Kac Approach
• Regional controllability of anomalous diffusion generated by the time fractional diffusion equations
• Dynamics of a randomly kicked particle
• Well-posedness and numerical algorithm for the tempered fractional ordinary differential equations
• Backward difference formula: The energy technique for subdiffusion equation