Precise Determination of Pair Interactions from Pair Statistics of Many-Body Systems In and Out of Equilibrium
The determination of the pair potential $v({\bf r})$ that accurately yields an equilibrium state at positive temperature $T$ with a prescribed pair correlation function $g_2({\bf r})$ or corresponding structure factor $S({\bf k})$ in $d$-dimensional Euclidean space $\mathbb{R}^d$ is an outstanding inverse statistical mechanics problem with far-reaching implications. Recently, Zhang and Torquato conjectured that any realizable $g_2({\bf r})$ or $S({\bf k})$ corresponding to a translationally invariant nonequilibrium system can be attained by a classical equilibrium ensemble involving only (up to) effective pair interactions. Testing this conjecture for nonequilibrium systems as well as for nontrivial equilibrium states requires improved inverse methodologies. We have devised a novel optimization algorithm to find effective pair potentials that correspond to pair statistics of general translationally invariant disordered many-body equilibrium or nonequilibrium systems at positive temperatures. This methodology utilizes a parameterized family of pointwise basis functions for the potential function whose initial form is informed by small- and large-distance behaviors dictated by statistical-mechanical theory. Subsequently, a nonlinear optimization technique is utilized to minimize an objective function that incorporates both the target pair correlation function $g_2({\bf r})$ and structure factor $S({\bf k})$ so that the small- and large-distance correlations are very accurately captured. To illustrate the versatility and power of our methodology, we accurately determine the effective pair interactions of the following four diverse target systems. We found that the optimized pair potentials generate corresponding pair statistics that accurately match their corresponding targets with total $L_2$-norm errors that are an order of magnitude smaller than that of previous methods.