  Particle motion of a Lennard-Jones supercooled liquid near the glass
transition is studied by molecular dynamics simulations. We analyze the wave
vector dependence of relaxation times in the incoherent self scattering
function and show that at least three different regimes can be identified and
its scaling properties determined. The transition from one regime to another
happens at characteristic length scales. The lengthscale associated with the
onset of Fickian diffusion corresponds to the maximum size of heterogeneities
in the system, and the characterisitic timescale is several times larger than
the alpha relaxation time. A second crossover lengthscale is observed, which
corresponds to the typical time and length of heterogeneities, in agreement
with results from four point functions. The different regimes can be traced
back in the behavior of the van Hove distribution of displacements, which shows
a characteristic exponential regime in the heterogeneous region before the
crossover to gaussian diffusion and should be observable in experiments.
  Our results show that it is possible to obtain characteristic length scales
of heterogeneities through the computation of two point functions at different
times.
