Designer Pair Statistics for Many-Body Systems

Knowledge of exact analytical functional forms for the pair correlation function $g_2(r)$ and its corresponding structure factor $S(k)$ of disordered many-body systems is limited. For fundamental and practical reasons, it is highly desirable to add to the existing data base of analytical functional forms for such pair statistics. In this investigation, we design a plethora of such pair functions in direct and Fourier spaces across the first three Euclidean space dimensions that are realizable by diverse many-body systems with varying degrees of correlated disorder across length scales, spanning a wide spectrum of hyperuniform and nonhyperuniform ones. This is accomplished by utilizing an efficient inverse algorithm that determines equilibrium states with up to pair interactions at positive temperature that precisely match targeted forms for both $g_2(r)$ and $S(k)$. Among other results, we discover an example that possesses a stronger hyperuniform property than any other positive-temperature equilibrium state known previously. We show how one of our functional designs may be experimentally achieved by polymer systems. To illustrate how our pair functions enable one to achieve a wide range of structural order and physical properties, we show that the translational order metric $ au$ and the self-diffusion coefficient $mathcal{D}$ are inversely related for our three-dimensional designs. Our pair functions can be used as basis functions to achieve many other realizable designs at different densities and temperatures. Our approach provides a general means to tune many-body systems with prescribed pair statistics to facilitate the inverse design of materials with desirable mechanical, optical and chemical properties.