Soft question: I am a mathematician self-learning statistical mechanics. The (mathematical) literature is concentrated on lattice models like the Ising model and the lattice-gas model. I understand the value of these models. However, what are some the main obstacles to extending the results to more realistic models?

For a concrete example, why is it so hard to prove the existence of solid-liquid phase-transition for a collection of particles interacting with Lennard-Jones potential?