Probability density, long time trajectories

The problem of vacancy-mediated tracer diffusion in two dimensions has been investigated for a long time [40-44] and several different methods (simulation, analytical models, enumeration of trajectories, etc.) can be used to address it. The mathematics of this type of diffusion was solved first for the simplest case [41], when the diffusion of the vacancy is unbiased (all diffusion barriers are equal the tracer atom is identical to the other atoms), the lattice is two-dimensional and infinite. There is a single vacancy present that makes a nearest-neighbor move in a random direction at regular time intervals and has an infinite lifetime, as there are no traps. The solution is constructed by separating the motion of the tracer and that of the vacancy. The correlation between the moves of the tracer atom is calculated from the probability that the vacancy returns to the tracer from a direction, which is equal, perpendicular or opposite to its previous departure. The probability density distribution of the tracer atom spreads with... 

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