Collective coordinate control of density distributions

Real collective density variables C (k) [cf. Eq. 1 3] in many-particle systems arise from nonlinear transformations of particle positions, and determine the structure factor S (k), where k denotes the wave vector. Our objective is to prescribe C (k) and then to find many-particle configurations that correspond to such a target C (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 (k) in the neighborhood of | k | =0. The optimization method employed generates multiparticle configurations for which S (k) | ∝ | k α, k | ≤K, and α=1, 2, 4, 6, 8, and 10. The case α=1 is relevant for the Harrison-Zeldovich model of the early universe, for superfluid He4, and for jammed amorphous sphere packings. The analysis also provides specific examples of interaction potentials whose classical ground states are configurationally degenerate and disordered.