Superdiffusive transport due to random velocity fields

The motion of a random walk 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, 〈x2(t)〉, grows faster than linearly with time. For a two-dimensional medium with layers of random velocities in the x-direction, 〈x2(t)〉 is found to increase as t2v with 2v=32. The probability distribution of displacements appears to fit the form P(x,t)∼t−34exp[−(x/t34)δ], with δ≲1.7. However,a Lifshitz argument for the tail of the distribution suggests that δ≤43. This discrepancy is yet to be fully resolved. We also discuss some intriguing properties in a model with isotropic velocity fields. In two dimensions, we find v=43, while δ=3. These values obey the general scaling relation δ=(1−v)−1.