  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.
