  We generalize the force-level, microscopic, Nonlinear Langevin Equation (NLE)
theory and its elastically collective generalization (ECNLE theory) of
activated dynamics in bulk spherical particle liquids to address the influence
of random particle pinning on structural relaxation. The simplest neutral
confinement model is analyzed for hard spheres where there is no change of the
equilibrium pair structure upon particle pinning. As the pinned fraction grows,
cage scale dynamical constraints are intensified in a manner that increases
with density. This results in the mobile particles becoming more transiently
localized, with increases of the jump distance, cage scale barrier and NLE
theory mean hopping time; subtle changes of the dynamic shear modulus are
predicted. The results are contrasted with recent simulations. Similarities in
relaxation behavior are identified in the dynamic precursor regime, including a
roughly exponential, or weakly supra-exponential, growth of the alpha time with
pinning fraction and a reduction of dynamic fragility. However, the increase of
the alpha time with pinning predicted by the local NLE theory is too small, and
severely so at very high volume fractions. The strong deviations are argued to
be due to the longer range collective elasticity aspect of the problem which is
expected to be modified by random pinning in a complex manner. A qualitative
physical scenario is offered for how the three distinct aspects that quantify
the elastic barrier may change with pinning. ECNLE theory calculations of the
alpha time are then presented based on the simplest effective-medium-like
treatment for how random pinning modifies the elastic barrier. The results
appear to be consistent with most, but not all, trends seen in recent
simulations. Key open problems are discussed with regards to both theory and
simulation.
