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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. We derive new duality relations for a class of soft potentials, including three-body and higher-order functions, that can be applied to ordered and disordered classical ground states. These duality relations link the energy of configurations associated with a real-space potential to the corresponding energy of the dual (Fourier-transformed) potential. We apply the duality relations by demonstrating 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 relate low- and high-density ground-state energies. We analyze the ground state configurations and thermodynamic properties of a one-dimensional system previously thought 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.