Failure of crystallization for generalized Lennard–Jones potentials and coarse graining to a rotating stars problem in one dimension

This paper deals with ground states for systems governed by generalized Lennard– Jones potentials LJp,q(r) := r−p −r−q, for 0 < q < 1 < p. The energy per particle diverges to −∞ as the number N of particles diverges. As a consequence, the average distance between particles vanishes as N → +∞. After suitable scaling, we prove that such a model converges, as N → +∞ and in the sense of Γ- convergence, to a rotating stars model; the effective energy is given by the sum of a repulsive pressure term and an attractive nonlocal interaction functional. The ground states of such a limit energy have non constant density. As a consequence, for the generalized Lennard–Jones potentials considered here, crystallization does not occur in any reasonable sense.