Friction dominated dynamics of interacting particles locally close to a crystallographic lattice

We study a system of particles, in general d-dimensional space, that interact by means of pair potential and adjust their positions according to the gradient flow dynamics induced by the total energy of the system. We consider the case when the range of the interaction is of the same order as the mean interparticle distance. It is also assumed that particles, locally, are located close to some crystallographic lattice. An appropriate system of equations that describes the evolution of macroscopic deformation of the crystallographic lattice, as well as the system that describe the evolution of the main crystallographic directions is derived. Well posedness of the derived system is studied as well as the stability of the particle system. Same examples of potentials that yield stable and unstable systems are given.