Identities for correlation functions in classical statistical mechanics and the problem of crystal states (Q2194195)

In this dedication note celebrating the 90th birthday of Joel Lebowitz, the author revives some ideas developed in the 80ties of the last century. Specifically an issue of crystal states, while addressed within the mathematical formalism of classical equilibrium statistical mechanics. The major issue becomes as yet unsolved problem of the crystal state existence status at nonzero temperature. The crystalline state of matter as idealized by a classical system of point particles in \(\mathbb{R}^{\nu }\). Translationally invariant interactions between particles are assumed, but the condition of rotational invariance is not in use. In dimension \(\nu \geq 3\) one expects the existence of crystal Gibbs states where the translation invariance is broken, albeit no example is known where this has been rigorously proved. (In dimension \(\nu \leq 2\) the Mermin-Wagner theorem excludes long-range order for short range interactions.) For suitable particle interactions there are no obvious reasons for singularities to occur in the dependence of the crystal equilibrium state on the activity \(z\). It is thus argued that there is in fact a real analytic dependence of the pressure and there exists a translationally invariant crystal state on \(z\), provided we assume a certain cluster property (finite correlation length). This does not provide a rigorous proof, since it depends on unproved technical assumptions. Nonetheless this non-rigorous discussion of how the cluster properties contributes to the yet uncompleted existence proof for crystal states.