What is a solid?

By definition, a solid must have a rigid microscopic reference state around which a microscopic Cauchy–Born lattice dynamics expansion of the mechanical energy is possible and consistent. This means that atomic solids are described by a unique permutation of the atoms. The atoms can be labeled by their respective equilibrium positions. The correlation functions which are important in the description of solids are between labeled particles and the Gibbs ensemble which appears naturally is the “specific phase”. This is in contrast to the description of liquids which are naturally described by Gibbs’ “generic phases” and for which the correlation functions describe averages over permutations. The reason for this is that permutation symmetry is an active symmetry in fluids because the averaging over permutations is ergodic. We argue that the symmetry broken at solidification is permutation symmetry which becomes non-ergodic in the solid. Thus, the classical permutation entropy associated with the N! permutations of N particles can be regarded as a quenched missing entropy closely analogous to other missing entropies. We use a generalized Lindemann criterion to analyze the self-consistency of the expansion procedure.