Operator Levy motion and multiscaling anomalous diffusion
The long-term limit motions of individual heavy-tailed (power-law) particle jumps that characterize anomalous diffusion may have different scaling rates in different directions. Operator stable motions {l_brace}Y(t):t{>=}0{r_brace} are limits of d-dimensional random jumps that are scale-invariant according to c{sup H}Y(t)=Y(ct), where H is a dxd matrix. The eigenvalues of the matrix have real parts 1/{alpha}{sub j}, with each positive {alpha}{sub j}{<=}2. In each of the j principle directions, the random motion has a different Fickian or super-Fickian diffusion (dispersion) rate proportional to t{sup 1/{alpha}{sub j}}. These motions have a governing equation with a spatial dispersion operator that is a mixture of fractional derivatives of different order in different directions. Subsets of the generalized fractional operator include (i) a fractional Laplacian with a single order {alpha} and a general directional mixing measure m({theta}); and (ii) a fractional Laplacian with uniform mixing measure (the Riesz potential). The motivation for the generalized dispersion is the observation that tracers in natural aquifers scale at different (super-Fickian) rates in the directions parallel and perpendicular to mean flow.
- Sponsoring Organization:
- (US)
- OSTI ID:
- 40205361
- Journal Information:
- Physical Review E, Vol. 63, Issue 2; Other Information: DOI: 10.1103/PhysRevE.63.021112; Othernumber: PLEEE8000063000002021112000001; 127102PRE; PBD: Feb 2001; ISSN 1063-651X
- Publisher:
- The American Physical Society
- Country of Publication:
- United States
- Language:
- English
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