Physical Review Letters · 1999 · 776 citations · 29 references
Natural SciencesPhysicsStochastic ProcessesTheoretical PhysicsThermodynamicsBoltzmann Transport EquationBrownian MotionDiffusion ProcessesDissipative SystemsStatistical MechanicsStochastic Differential EquationsAnomalous DiffusionAnomalous Diffusion EffectsBoltzmann DistributionBoltzmann EquationsFokker - Planck EquationFokker-planck EquationsFokker-planck TheoryNonlinear Systems
The study introduces a fractional Fokker‑Planck equation that models a particle’s stochastic evolution under a nonlinear external force and a thermal bath. The equation is illustrated by a harmonic‑potential example, showing how the model captures dynamics in a concrete setting. In the force‑free case the model predicts subdiffusion, satisfies generalized Einstein relations, yields a Boltzmann stationary distribution, and shows single‑mode relaxation following a Mittag‑Leffler decay.
We introduce a fractional Fokker-Planck equation describing the stochastic evolution of a particle under the combined influence of an external, nonlinear force and a thermal heat bath. For the force-free case, a subdiffusive behavior is recovered. The equation is shown to obey generalized Einstein relations, and its stationary solution is the Boltzmann distribution. The relaxation of single modes is shown to follow a Mittag-Leffler decay. We discuss the example of a particle in a harmonic potential.
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Fractional diffusion and wave equations
W. R. Schneider, Walter Wyss · Journal of Mathematical Physics · 1989 · 1.1K citations
Stochastic pathway to anomalous diffusion
J. Klafter, A. Blumen, Michael F. Shlesinger · Physical review. A, General physics · 1987 · 724 citations
Engineering, Mathematics, Physics +15