Collective Coordinate Control of Density Distributions

Abstract

Real collective density variables $C(\mathbf{k})$ [c.f. Eq.\ref{Equation3})] in many-particle systems arise from non-linear transformations of particle positions, and determine the structure factor $S(\mathbf{k})$, where $\mathbf{k}$ denotes the wave vector. Our objective is to prescribe $C(\mathbf{k})$ and then to find many-particle configurations that correspond to such a target $C(\mathbf{k})$ using a numerical optimization technique. Numerical results reported here extend earlier one- and two-dimensional studies to include three dimensions. In addition, they demonstrate the capacity to control $S(\mathbf{k})$ in the neighborhood of $|\mathbf{k}|=$ 0. The optimization method employed generates multi-particle configurations for which $S(\mathbf{k})\propto |\mathbf{k}{|}^{\alpha }$, $|\mathbf{k}|\le K$, and $\alpha =$ 1, 2, 4, 6, 8, and 10. The case $\alpha =$ 1 is relevant for the Harrison-Zeldovich model of the early universe, for superfluid ${}^{4}He$, and for jammed amorphous sphere packings. The analysis also provides specific examples of interaction potentials whose classical ground state are configurationally degenerate and disordered.

26 pages, 8 figures

References in corpus (2)

• Exactly Solvable Disordered Sphere-Packing Model in Arbitrary-Dimension Euclidean Spaces
• Crystalline ground states for classical particles

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