Hybrid model for combined particle and continuum dynamics

Differential equations for continuum fields describe many macroscopic phenomena. Hydrodynamics, for example, is described by the Navier-Stokes equations, and their solutions depend on boundary conditions. However, boundary conditions are set by the interactions at the atomistic or molecular scale. We introduce a "hybrid model" that permits a continuum description in one region to be coupled to an atomistic description in another region. The coupling is symmetric in the sense that the fluxes of the conserved quantities are continuous across the particle-field interface. As an example, we couple a Lennard-Jones liquid and the compressible Navier-Stokes equations and show that the hybrid model is consistent with hydrodynamic predictions.