  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.
