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 between finite-sized particles in a compressible fluid on the basis of the linearized Navier-Stokes equation. The theoretical results show that at high frequencies, the compressibility of the fluid has a significant impact on the frequency-dependent pair interactions. The predictions of hydrodynamic theory are compared to molecular dynamics simulations of two nanocolloids 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.

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