Variational evolution of one-dimensional Lennard-Jones systems

Scope of this paper is twofold: on one hand we continue the study of LennardJones systems from the standpoint of variational principles, on the other hand these allow to provide a non-trivial example within the theory of minimizing movements. In a one-dimensional static setting, Lennard-Jones systems have been shown to be equivalent to energies of Fracture Mechanics using the notion of equivalence by Γ-convergence [8, 9]. Here we prove that this equivalence also holds as gradient-flow type dynamics are concerned. Within the theory of minimizing movements, the scaled Lennard-Jones energies we consider are an example of a sequence of non-convex functionals for which Γ-convergence and gradient-flow dynamics commute. We start by briefly recalling the minimizing-movement scheme. Typically, we are given an ‘energy functional’ F , defined on a space X, whose (local) minimizers provide the stable configurations of the system. As an answer to the problem of modeling the evolution from a given initial state u, in [11] (see also [1, 5]) a general scheme is proposed, based on an iterative-minimization process. More precisely, in the particular case in which X is a Hilbert space, we fix a ‘time step’ τ > 0 and consider the sequence (uτ )k recursively defined by letting uτ = u 0 and uτ (k ≥ 1) be a minimizer of the penalized functional