Continuum mechanics from molecular dynamics via adiabatic time and length scale separation

We show how to construct, by exploiting adiabatic time and length scale separation between atomistic and continuum mechanics, a multiscale scheme for continuum dynamics free from macroscopic constitutive modeling. To do so, we introduce a new set of degrees of freedom that simultaneously represent the macroscopic and the microscopic dynamics, based on a space tessellation. In this new formulation, the dynamics of the macroscopic fields steers the microscopic particle dynamics by producing the conditions under which they evolve and, concurrently, the particle dynamics drives the evolution of the macroscopic fields by providing them with atomistically based constitutive information. Under conditions of adiabatic separation both in time and length scales, it is possible to decimate the tessellating cells and let the macroscopic equations of motion be driven by the time average of the terms coupling them with the microscopic degrees of freedom.