  We propose a theoretical framework for the dynamics of bulk isotropic
hard-sphere systems in the presence of randomly pinned particles and apply this
theory to supercooled water to validate it. Structural relaxation is mainly
governed by local and non-local activated process. As the pinned fraction
grows, a local caging constraint becomes stronger and the long range collective
aspect of relaxation is screened by immobile obstacles. Different responses of
the local and cooperative motions results in subtle predictions for how the
alpha relaxation time varies with pinning and density. Our theoretical analysis
for the relaxation time of water with pinned molecules quantitatively well
describe previous simulations. In addition, the thermal dependence of
relaxation for unpinned bulk water is also consistent with prior computational
and experimental data.
