Enhanced diffusion in random velocity fields
G. Zumofen, J. Klafter, A. Blumen · Physical Review A · 1990
We study superlinear diffusion in a layered medium containing random velocity fields, where the mean-squared displacement grows as 〈${\mathit{x}}^{2}$(t)〉\ensuremath{\sim}${\mathit{t}}^{\mathrm{\ensuremath{\alpha}}}$ with \ensuremath{\alpha}>1 [S. Redner, Physica D 38, 287 (1989)]. For a two-dimensional system with preassigned random velocities in the longitudinal x direction and with diffusional motion in the transversal direction, we determine exactly the asymptotic behavior of the first three nontrivial moments ${\mathit{M}}_{2\mathit{m}}$=〈${\mathit{x}}^{2\mathit{m}}$(t)〉/〈${\mathit{x}}^{2}$(t)${\mathrm{〉}}^{\mathit{m}}$ of the displacement. Furthermore, we succeed in relating the diffusional problem to the one-dimensional trapping problem. We then are in a position to analyze the scaling form of the propagator P(x,t)\ensuremath{\sim}${\mathit{t}}^{\mathrm{\ensuremath{-}}3/4}$f(${\mathit{x}}^{3/4}$), where the function f(z) obeys a complicated stretched exponential behavior. We also generalize the problem to transverse motion on fractals and ultrametric spaces that leads to \ensuremath{\alpha} values that interpolate between 1 and 2. We support our theoretical analytical results by simulation calculations.