Exotic Ground States of Directional Pair Potentials via Collective-Density Variables
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Exotic Ground States of Directional Pair Potentials via Collective-Density Variables Stephen Martis · Étienne Marcotte · Frank H. Stillinger · Salvatore Torquato
Received: 16 July 2012 / Accepted: 10 October 2012 / Published online: 26 October 2012 © Springer Science+Business Media New York 2012
Abstract Collective-density variables have proved to be a useful tool in the prediction and manipulation of how spatial patterns form in the classical many-body problem. Previous work has employed properties of collective-density variables along with a robust numerical optimization technique to find the classical ground states of many-particle systems subject to radial pair potentials in one, two and three dimensions. That work led to the identification of ordered and disordered classical ground states. In this paper, we extend these collectivecoordinate studies by investigating the ground states of directional pair potentials in two dimensions. Our study focuses on directional potentials whose Fourier representations are non-zero on compact sets that are symmetric with respect to the origin and zero everywhere else. We choose to focus on one representative set that has exotic ground-state properties: two circles whose centers are separated by some fixed distance. We obtain ground states for this “two-circle” potential that display large void regions in the disordered regime. As more degrees of freedom are constrained the ground states exhibit a collapse of dimensionality characterized by the emergence of filamentary structures and linear chains. This collapse of dimensionality has not been observed before in related studies. Keywords Ground states · Collective coordinates · Directional potentials S. Martis · É. Marcotte · S. Torquato () Department of Physics, Princeton University, Princeton, NJ 08544, USA e-mail: [email protected] F.H. Stillinger · S. Torquato Department of Chemistry, Princeton University, Princeton, NJ 08544, USA S. Torquato Princeton Center for Theoretical Science, Princeton University, Princeton, NJ 08544, USA S. Torquato Princeton Institute for the Science and Technology of Materials, Princeton University, Princeton, NJ 08544, USA S. Torquato Program in Applied and Computational Mathematics, Princeton University, Princeton, NJ 08544, USA
Exotic Ground States of Directional Pair Potentials
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1 Introduction The Fourier components of a set of position or density variables are commonly referred to as collective-density variables and the resultant space is called the collective-coordinate space. The sort of analysis involving collective-density variables naturally arises in the context of the N -body problem, where a collective-coordinate transformation will often simplify seemingly intractable problems. The reader is referred to Feynman’s discussion of the superfluid properties of 4 He or a series of papers of Percus and Yevick for illustrative examples of their use [1–3]. Recently, there has been renewed interest in collective-density variables and the associated ground states for a c
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