The problem is to minimize the energy density, total energy divided by N, in the limit that N goes to infinity. This is similar to the sphere packing problem which asks for the best arrangement of hard spheres which maximizes the packing density. The difficulty of these problems stems from the fact that there are local arrangements that do better than one can do in a global arrangement.
Statistical mechanics provides a method for dealing with such frustrated systems.
We give some simple examples in which one can prove that the energy density is minimized by a periodic structure and speculate on whether this method can be applied to the sphere packing problem.