Evolution of particle separation in slowly decorrelating velocity fields

We consider the evolution of the separation distance between two particles advected by a random velocity field with slowly decaying temporal and spatial correlations in the weak coupling regime. It has been shown in [5] that the motion of a single particle converges to a fractional Brownian motion on a time scale δ−γ with some γ < 2, which is shorter than the classical diffusive time scale δ−2 (see [9]). In the present paper we prove that unlike the single particle position, the two-particle separation behaves diffusively, and evolves on the classical time scale δ−2, even when the random flow is slowly decorrelating in time and space. The results of this paper illustrate that the flows under consideration display both diffusive and superdiffusive transport on different time scales for various physical quantities.