We use large-scale computer simulations to explore the nonequilibrium 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.Understanding the complex mechanisms underlying the formation and stability of colloidal gels remains a challenge, despite the diversity of existing applications exploiting their mechanical properties. Gels are lowdensity structures forming percolating networks, with bonds that are either permanent (chemical gels) or transient (physical gels) [1][2][3][4][5][6]. A prevalent method to form physical gels follows a nonequilibrium route by quenching a homogeneous fluid into a phase coexistence region [1,[7][8][9][10][11][12], which generates bicontinuous structures. When the dense phase forms an amorphous solid, the phase separation is kinetically hindered [13][14][15][16] and a gel forms. The microscopic dynamical and structural properties of these nonequilibrium gels evolve slowly with time.This aging dynamics has been the subject of many experimental studies, which established that aging in colloidal gels is 'anomalous' [17,18], i.e. it differs qualitatively from the aging observed in conventional glassy materials, such as polymer and colloidal glasses [19,20]. Scattering experiments [21][22][23][24][25] report that time correlation functions are described by compressed exponential relaxations. Such behaviour differs from the (exponential) diffusive dynamics in simple liquids or the (stretched exponential) decay observed in glassy fluids [26]. In addition, compressed exponentials are seen to emerge only for large enough displacements, with the relaxation timescale τ (q) crossing over from τ ∼ q −2 , characteristics of diffusion, to τ ∼ q −1 , characteristic of ballistic dynamics, with decreasing the scattering wavevector q. Finally, spatiallyresolved dynamic measurements revealed the existence of long-ranged correlations extending up to the system size [27,28], again contrasting with the much smaller-ranged dynamic heterogeneity observed in glassy materials [29]. Such peculiar behaviour is hypothesized to follow from the infrequent release of 'internal stresses' that relax the fractal network [18], but this interpretation remains to be confirmed by direct observation. This overall phenomenology has been reported for laponite, carbon black, micellar polycrystals, multilamellar vesicles, implying it is generic to a large class of soft materials [18].A microscopic perspective via theoretical modeling is also missing. To study this problem, one needs a model with a...