Authors: Stephen Martis Étienne Marcotte Frank H Stillinger Salvatore Torquato
Publish Date: 2012/10/26
Volume: 150, Issue: 3, Pages: 414-431
Abstract
Collectivedensity variables have proved to be a useful tool in the prediction and manipulation of how spatial patterns form in the classical manybody problem Previous work has employed properties of collectivedensity variables along with a robust numerical optimization technique to find the classical ground states of manyparticle 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 nonzero 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 groundstate properties two circles whose centers are separated by some fixed distance We obtain ground states for this “twocircle” 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 studiesWe are pleased to offer this contribution to honor M Fisher J Percus and B Widom whose own remarkable contributions have vividly demonstrated the originality intrinsic to statistical mechanics This work was supported by the Office of Basic Energy Sciences US Department of Energy under Grant No DEFG0204ER46108We derive a simple relation transform describing how the Fourier transform of a function in ddimensional Euclidean space ℝ d behaves under invertible affine transformations We also present selected results for the ellipse potential described in Sect 3 The ellipse potential is described by the dimensionless aspect ratio b/a where b is the length of the minor axis and a is the length of the major axis of the potential in kspace We go on to show that the ellipse potential is related to the previously studied circle potential by a simple affine transformation which allows for a simple analysis of the ground state patternsIn the case of the ellipse potential the region on which Ck is constrained can be expressed as a circular exclusion region to which we have applied invertible affine transformations Since the groundstate properties of the circular exclusion region are well known in light of previous studies we can use this information combined with this affine relation to draw conclusions about the groundstate properties of the ellipse potential We emphasize that the affine transformation is not applied to the particles themselves and therefore does not allow for the particles to rotateGround state configurations were generated over a range of aspect ratios for the ellipse potential at a fixed χ value Comparing the pair correlation function of the ground states of a potential of given aspect ratio to that of the isotropic circle potential it is apparent that the approximate affine relation derived in the previous section holds In addition if χ is increased to values near but still below 05 the particles develop what resemble hard cores and we see the emergence of perfect nematic order which is by construction
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