Precise determination of pair interactions from pair statistics of many-body systems in and out of equilibrium

The determination of the pair potential $v\left(\mathbf{r}\right)$ that accurately yields an equilibrium state at positive temperature $T$ with a prescribed pair correlation function $g_2\left(\mathbf{r}\right)$ or corresponding structure factor $S\left(\mathbf{k}\right)$ in $d$-dimensional Euclidean space ${\mathbb{R}}^d$ is an outstanding inverse statistical mechanics problem with far-reaching implications. Recently, Zhang and Torquato [Phys. Rev. E 101, 032124 (2020)] conjectured that any realizable $g_2\left(\mathbf{r}\right)$ or $S\left(\mathbf{k}\right)$ 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 an optimization algorithm to precisely determine 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-, intermediate- 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 $g_2\left(\mathbf{r}\right)$ and structure factor $S\left(\mathbf{k}\right)$ so that the small intermediate- 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: (1) Lennard-Jones system in the vicinity of its critical point, (2) liquid under the Dzugutov potential, (3) nonequilibrium random sequential addition packing, and (4) a nonequilibrium hyperuniform “cloaked” uniformly randomized lattice. 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. The results of our investigation lend further support to the Zhang-Torquato conjecture. Furthermore, our algorithm will enable one to probe systems with identical pair statistics but different higher-body statistics, which will shed light on the well-known degeneracy problem of statistical mechanics.

Figure 1

Snapshots of the configurations corresponding to the four target systems considered in this study. (a) A representative 2D 1000-particle configuration of the target Lennard-Jones system near its critical point. (b) A representative 3D 512-particle configuration of the target liquid under the Dzugutov potential. (c) A portion of a representative 2D 10 000-particle RSA configuration very near the saturation state. Only 1000 particles are displayed. (d) A portion of a representative 3D 2744-particle configuration of a cloaked URL system. Only 512 particles are displayed.

Figure 2

Flowchart of the optimization algorithm for our inverse procedure, assuming homogeneous target systems.

Figure 3

(a) A snapshot of a 2D configuration 1000-particle system that is equilibrated under the optimized potential [Eq. (53)] for the target LJ fluid in the vicinity of its critical point. (b) The target-generating potential $v_{\text{LJ}}\left(r\right)$ and the optimized pair potential $v_F(r;\mathbf{a})$. (c) Targeted and optimized pair correlation functions. (d) Targeted and optimized structure factors.

Figure 4

(a) A snapshot of a 3D configuration 512-particle system that is equilibrated under the optimized potential [Eq. (55)] for the Dzugutov liquid. (b) The target-generating potential $v_{\text{D}}\left(r\right)$ (54) and the optimized pair potential (55). (c) Targeted and optimized pair correlation functions. (d) Targeted and optimized structure factors.

Figure 5

(a) A snapshot a 2D configuration 1000-particle system that is equilibrated under the optimized potential [Eq. (56)] for the target nonequilibrium RSA packing near its saturation state. (b) Optimized pair potential. (c) Targeted and optimized pair correlation functions. (d) Targeted and optimized structure factors.

Figure 6

(a) A portion of a 3D configuration of a 2744-particle system that is equilibrated under the optimized effective one- and two-body potential for the target 3D cloaked URL. Only 512 particles are displayed. (b) Optimized pair potential [Eq. (57)] minus its long-ranged repulsive part $0.940/r$. (c) Targeted and optimized pair correlation functions. (d) Targeted and optimized structure factors.

Figure 7

Schematic illustration of a statistically homogeneous but anisotropic nematic liquid-crystal configuration.