Duality relations for the classical ground states of soft-matter systems

Bounded interactions are particularly important in soft-matter systems, such as colloids, microemulsions, and polymers. In this paper, we extend the results of a recent letter [S. Torquato and F. H. Stillinger, Phys. Rev. Lett., 2008, 100, 020602] on duality relations for ground states of pair interactions to include three-body and higher-order functions. Our novel and general relations link the energy of configurations associated with a real-space potential to the corresponding energy of the dual (Fourier-transformed) potential and can be applied to ordered and disordered classical ground states. We use the duality relations to demonstrate how information about the classical ground states of short-ranged potentials can be used to draw new conclusions about the ground states of long-ranged potentials and vice versa. The duality relations also lead to bounds on the T = 0 system energies in density intervals of phase coexistence. Additionally, we identify classes of "self- similar" potentials, for which one can rigorously relate low- and high-density ground-state energies. We analyze the ground state configurations and thermodynamic properties of a one-dimensional system expected to exhibit an infinite number of structural phase transitions and comment on the known ground states of purely repulsive monotonic potentials in the context of our duality relations.