Calculus of Variations and Geometric Measure Theory

V. Crismale - A. Kubin - A. Ninno - M. Ponsiglione

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

created by ponsiglio on 15 Dec 2021
modified by crismale on 18 Apr 2023


Published Paper

Inserted: 15 dec 2021
Last Updated: 18 apr 2023

Journal: Nonlinear Analysis
Volume: 231
Pages: Paper no. 113046
Year: 2023


This paper deals with ground states for systems governed by generalized Lennard-Jones potentials $LJ^{p,q}(r):= r^{-p} - r^{-q}$, for $0<q<1<p$. The energy per particle diverges to $-\infty$ as the number $N$ of particles diverges. As a consequence, the average distance between particles vanishes as $N\to +\infty$. After suitable scaling, we prove that such a model converges, as $N\to +\infty$ and in the sense of $\Gamma$-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.