The random walk's guide to anomalous diffusion: a fractional dynamics approach
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Cites background or methods from "The random walk's guide to anomalou..."
...The MFPT diverges both in the absence of a bias and under a constant drift, pertaining to both finite as well as semi-infinite domains (Metzler and Klafter 2000e, Rangarajan and Ding 2000a, 2000b, 2003, Scher et al 2002a, Barkai 2001)22....
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...Conversely, in the overdamped case, the FKKE (14) corresponds in position space to the fractional Fokker– Planck equation (FFPE) (Metzler et al 1999a, 1999b, Metzler and Klafter 2000b, 2000c) ∂P ∂t = 0D1−αt ( − ∂ ∂x F (x) mηα + Kα ∂2 ∂x2 ) P(x, t), (16) where ηα ≡ ητα/τ ∗ and Kα ≡ kBT /(mηα)....
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...In this limit, one can obtain simple reductions of the H-function solution, in the following three cases (Metzler and Nonnenmacher 2002): (i) Cauchy propagator α = µ = 1, P(x, t) = 1 2πK11 t 1 1 + x2 /( K11 t ) (52) with the long-tailed asymptotics P(x, t) ∼ (2π)−1x−2....
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...…the LF in the presence of the drift V defined by the equation ∂ ∂t P (x, t) = ( ∂ ∂x V + Kµ ∂µ ∂|x|µ ) P(x, t) (35) is the solution PV =0(x, t) of equation (34) taken at position x − V t , i.e., PV (x, t) = PV =0(x−V t, t) (Jespersen et al 1999, Metzler and Compte 2000, Metzler and Klafter 2000a)....
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...We note that the solution of certain classes of fractional equations is intimately related to the Fox H-function and related special functions (Mathai and Saxena 1978, Srivastava et al 1982, Saxena and Saigo 2001)....
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"The random walk's guide to anomalou..." refers background in this paper
...In fact, the LeH vy #ight trajectory can be assigned a fractal dimension d f "k [38,42,89] and is commonly supposed to be an e$cient search strategy of living organisms [141,142,146]....
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