2000/02/01 by David A. Benson, Stephen W. Wheatcraft, Mark M. Meerschaert · 4 citations
Engineering · Mathematics · #Anomalous diffusion #Brownian motion #Continuous-time random walk #Differential equation #Dimension (graph theory) #Dispersion (optics) #Fluid Dynamics and Turbulent Flows #Fokker–Planck equation #Fractional Brownian motion #Fractional Differential Equations Solutions #Fractional calculus #Geometry #Lévy flight #Mathematical analysis #Mathematics #Order (exchange) #Particle Dynamics in Fluid Flows #Physics #Pure mathematics #Quantum mechanics #Random walk #Scaling #Skewness #Space (punctuation) #Statistics
paper · pdf · doi:10.1029/2000wr900032
openalex publication_date 2000/02/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/23
A governing equation of stable random walks is developed in one dimension. This Fokker‐Planck equation is similar to, and contains as a subset, the second‐order advection dispersion equation (ADE) except that the order (α) of the highest derivative is fractional (e.g., the 1.65th derivative). Fundamental solutions are Lévy's α‐stable densities that resemble the Gaussian except that they spread proportional to time 1/α , have heavier tails, and incorporate any degree of skewness. The measured variance of a plume undergoing Lévy motion would grow faster than Fickian plume, at a rate of time 2/α , where 0 < α ≤ 2. The equation is parsimonious since the parameters are not functions of time or distance. The scaling behavior of plumes that undergo Lévy motion is accounted for by the fractional derivatives, which are appropriate measures of fractal functions. In real space the fractional derivatives are integrodifferential operators, so the fractional ADE describes a spatially nonlocal process that is ergodic and has analytic solutions for all time and space.