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Negative superdiffusion due to inhomogeneous convection
Phys. Rev. E 71, 061101 – Published 7 June, 2005
DOI: https://doi.org/10.1103/PhysRevE.71.061101
Abstract
Fractional transport of particles on a comb structure in the presence of an inhomogeneous convection flow is studied [Baskin and Iomin, Phys. Rev. Lett. 93, 120603 (2004)]. The large scale asymptotics is considered. It is shown that a contaminant spreads superdiffusively in the direction opposite to the convection flow. Conditions for the realization of this effect are discussed in detail.
Article Text
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This asymptotics is valid for , only. When , the first term of the order of is the most important. In this case the large scale asymptotics corresponds to subdiffusion [11].
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This approximation was used independently by Liouville and Green. In quantum mechanics this approximation is known as the Wentzel-Kramers-Brillouin (WKB) approximation. However, we quote [19]: “the contribution of these authors was not the construction of the approximation (which was already known), but the determination of connection formulas for linking exponential and oscillatory LG approximations across a turning point on the real axis.”
- In this connection, it should be underlined that the observed solution reflects the theorem on the unique relation between the Laplace transform and the probability distribution function (PDF), which reads that distinct PDFs have distinct Laplace transforms; see W. Feller, An Introduction to Probability Theory and Its Applications (John Wiley and Sons, New York, 1971), Vol. 2, Chap. 13.