  Hydrodynamic interactions play an important role in many areas of soft matter
science. In simulations with implicit solvent, various techniques such as
Brownian or Stokesian dynamics explicitly include hydrodynamic interactions a
posteriori by using hydrodynamic diffusion tensors derived from the Stokes
equation. However, this equation assumes the interaction to be instantaneous
which is an idealized approximation and only valid on long time scales. In the
present paper, we go one step further and analyze the time-dependence of
hydrodynamic interactions in a compressible fluid on the basis of the
linearized Navier-Stokes equation. The theoretical results show that the
compressibility of the fluid has a significant impact on frequency-dependent
pair interactions.
  The predictions of the hydrodynamic theory are compared to molecular dynamics
simulations of two solid spheres in a Lennard-Jones fluid. For this system we
reconstruct memory functions by extending the inverse Volterra technique. The
simulation data agree very well with the theory, therefore, the theory can be
used to implement dynamically consistent hydrodynamic interactions in the
increasingly popular field of non-Markovian modeling.
