We use a long, all-atom molecular dynamics (MD) simulation combined with theoretical modeling to investigate the dynamics of selected lipid atoms and lipid molecules in a hydrated diyristoyl-phosphatidylcholine (DMPC) lipid bilayer. From the analysis of a 0.1 $mu$s MD trajectory we find that the time evolution of the mean square displacement, [delta{r}(t)]^2, of lipid atoms and molecules exhibits three well separated dynamical regions: (i) ballistic, with [delta{r}(t)]^2 ~ t^2 for t < 10 fs; (ii) subdiffusive, with [delta{r}(t)]^2 ~ t^{beta} with beta<1, for 10 ps < t < 10 ns; and (iii) Fickian diffusion, with [delta{r}(t)]^2 ~ t for t > 30 ns. We propose a memory function approach for calculating [delta{r}(t)]^2 over the entire time range extending from the ballistic to the Fickian diffusion regimes. The results are in very good agreement with the ones from the MD simulations. We also examine the implications of the presence of the subdiffusive dynamics of lipids on the self-intermediate scattering function and the incoherent dynamics structure factor measured in neutron scattering experiments.
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