Failure of crystallization for generalized Lennard–Jones potentials and coarse graining to a rotating stars problem in one dimension. (June 2023)

• Record Type: Journal Article
• Authors: Crismale, V.; Kubin, A.; Ninno, A.; Ponsiglione, M.
• Is Part Of: Nonlinear analysis. Volume 231(2023)
• Publication Date: 2023-06
• Subjects: 49J45 -- 70C20 -- 82D25; Variational methods -- Crystallization; Mathematical analysis -- Periodicals; Functional analysis -- Periodicals; Nonlinear theories -- Periodicals; Analyse mathématique -- Périodiques; Analyse fonctionnelle -- Périodiques; Théories non linéaires -- Périodiques; Functional analysis; Mathematical analysis; Nonlinear theories; Periodicals; Electronic journals; 515.7248
• DOI: 10.1016/j.na.2022.113046
• Languages: English
• ISSNs: 0362-546X

Abstract

Abstract: This paper deals with ground states for systems governed by generalized Lennard–Jones potentials L J p, 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.