Real collective density variables $C\left(\mathbf{k}\right)$ [cf. Eq. (1.3)] in many-particle systems arise from nonlinear transformations of particle positions, and determine the structure factor $S\left(\mathbf{k}\right)$, where $\mathbf{k}$ denotes the wave vector. Our objective is to prescribe $C\left(\mathbf{k}\right)$ and then to find many-particle configurations that correspond to such a target $C\left(\mathbf{k}\right)$ 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 $S\left(\mathbf{k}\right)$ in the neighborhood of $\mid \mathbf{k}\mid =0$. The optimization method employed generates multiparticle configurations for which $S\left(\mathbf{k}\right)\propto {\mid \mathbf{k}\mid }^{\alpha }$, $\mid \mathbf{k}\mid ⩽K$, 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 ${}^4\mathrm{He}$, and for jammed amorphous sphere packings. The analysis also provides specific examples of interaction potentials whose classical ground states are configurationally degenerate and disordered.

• Received 30 March 2006

DOI:https://doi.org/10.1103/PhysRevE.74.031104