Trajectory end point distribution of a test particle in a turbulent liquid

In a recent paper the mean square displacement (MSD), <R^2(T)>, of a particle carried by a turbulent liquid over time T has been shown to be proportional to T^6/5, meaning that the motion of the particle is slightly super-diffusive. In some cases, though, it might be important to have more information than that provided by that law. An example would be the distribution of pollutants as a function of time by turbulent flow. Here small amounts of material reaching relatively large distances are of importance. This motivates our interest in the full distribution of the location of particles swept by the fluid as a function of time. The distribution depends on the distance through the dimensionless quantity X^2=R^2/<R^2(T)> . We find that for small X, the distribution P(R,T) is proportional to exp(-aX^2) but at its tail when X is large it behaves as exp(-bX^2/3) . pacs numbers-02.50.Ey, 05.40.Fb