Superdiffusion in random velocity fields

We discuss the superdiffusive motion of a random walk in a medium containing random velocity fields. For a two-dimensional layered medium with y-dependent random velocities in the x direction u(y), 〈${\mathit{x}}^2$(t)〉∼${\mathit{t}}^{2\nu }$, with 2ν=3/2, 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)〉∼${\mathit{t}}^{\mathrm{-}3/4}$f(x/${\mathit{t}}^{3/4}$), where v(u)∼exp(-${\mathit{u}}^{\mathrm{\delta }}$) for u≫1, with δ=4/3. For an isotropic two-dimensional medium with ${\mathit{u}}_{\mathit{x}}$(y)=f(y) and ${\mathit{u}}_{\mathit{y}}$(x)=f(x), we find ν=2/3 and δ=1-ν${)}^{\mathrm{-}1}$=3.