Anomalous diffusion as a way the stochastic advection manifests itself

摘要

A theory is proposed to describe contaminant particle transport over highly disordered media due to advection in the stochastic fluid velocity field characterized by long-range correlations. The problem is treated in the most general case. The analysis is based upon the concepts of scale-invariance and Feynmann diagram technique. The particle may be considered diffusing, the diffusivity field being stochastic as well, but this irregularity doesn’t occur to result in affecting the power of dispersion-time dependence: it leads only to renormalization of effective diffusivity. On the contrary, velocity fluctuations may lead to anomalous (“superdiffusion”) behaviour. Namely, when in fluctuation velocity correlator < vi _(r→ ) vk _(r→′) >~| r→ − r→′| − 2he the index h is less than unity, the characteristic migration length R varies in time according to R ~ t1/(1+ h) , which is faster comparing with the classical law R ~ t1/ 2 . The heavy superdiffusion tails are shown to behave as t2 r(5 +2h) , i.e. are much weaker than in customary superdiffusion theory. The concentration equations with fractional derivatives are given, which describe the contaminant behaviour in the region adjacent to source and in the far distance region. No common differential equation exists for the whole space. For the case h >1 in the main space region the behaviour of the system corresponds to classical diffusion, and here stochastic advection results only in renormalization of the average diffusivity without any change in Fick’s law. But, except for the special values of h , at far distances power tails of concentration appear. In the borderline case of h = 1 the migration length behaves as R ~ t1/ 2 ln1/ 4 t , and the tail has the form t2 r7 .