Superdiffusion in random velocity fields

Stochastic transport in a medium containing random, but spatially correlated velocity fields is discussed. This type of disorder generally leads to superdiffusive behavior in which the mean-square displacement of a random walk, 〈x2(t)〉, grows faster than linearly with time. For a two-dimentional layered medium with y-dependent random velocities in the x-direction ux(y), 〈x2(x)〉∼t2ν with v=34, and with strong sample-to-sample fluctuations. The probability distribution of displacements, averaged over environments, takes a non-Gaussian scaling form at large time, 〈P(x, t)〉∼-34ƒ(x/t34), where ƒ(u)∼exp(-uδ) for u⪢1, with δ=43. For an isotropic two-dimensional medium with ux(y) having the same statistical properties, we find v=23 and δ=(1−ν)-1=3. For the layered medium, the moments of the time for a random walker to first reach a distance x in the longitudinal direction increases as 〈tn〉1n∼x43, possibly modified by logarithmic corrections, however. The probability that the walk has not reached a distance x in a time t decreases asymptotically as e-tτ, with τ∼x2, indicating that more than a single time scale is needed to account for first passage properties.