  We use large-scale computer simulations to explore the non-equilibrium aging
dynamics in a microscopic model for colloidal gels. We find that gelation
resulting from a kinetically-arrested phase separation is accompanied by
`anomalous' particle dynamics revealed by superdiffusive particle motion and
compressed exponential relaxation of time correlation functions.
Spatio-temporal analysis of the dynamics reveals intermittent heterogeneities
producing spatial correlations over extremely large length scales. Our study is
the first microscopically-resolved model reproducing all features of the
spontaneous aging dynamics observed experimentally in soft materials.
