TY - JOUR
T1 - Realizability of iso- g 2processes via effective pair interactions
AU - Wang, Haina
AU - Stillinger, Frank H.
AU - Torquato, Salvatore
N1 - Publisher Copyright:
© 2022 Author(s).
PY - 2022/12/14
Y1 - 2022/12/14
N2 - An outstanding problem in statistical mechanics is the determination of whether prescribed functional forms of the pair correlation function g2(r) [or equivalently, structure factor S(k)] at some number density ρ can be achieved by many-body systems in d-dimensional Euclidean space. The Zhang-Torquato conjecture states that any realizable set of pair statistics, whether from a nonequilibrium or equilibrium system, can be achieved by equilibrium systems involving up to two-body interactions. To further test this conjecture, we study the realizability problem of the nonequilibrium iso-g2 process, i.e., the determination of density-dependent effective potentials that yield equilibrium states in which g2 remains invariant for a positive range of densities. Using a precise inverse algorithm that determines effective potentials that match hypothesized functional forms of g2(r) for all r and S(k) for all k, we show that the unit-step function g2, which is the zero-density limit of the hard-sphere potential, is remarkably realizable up to the packing fraction φ = 0.49 for d = 1. For d = 2 and 3, it is realizable up to the maximum "terminal"packing fraction φc = 1/2d, at which the systems are hyperuniform, implying that the explicitly known necessary conditions for realizability are sufficient up through φc. For φ near but below φc, the large-r behaviors of the effective potentials are given exactly by the functional forms exp [- κ(φ)r] for d = 1, r-1/2 exp [- κ(φ)r] for d = 2, and r-1 exp [- κ(φ)r] (Yukawa form) for d = 3, where κ-1(φ) is a screening length, and for φ = φc, the potentials at large r are given by the pure Coulomb forms in the respective dimensions as predicted by Torquato and Stillinger [Phys. Rev. E 68, 041113 (2003)]. We also find that the effective potential for the pair statistics of the 3D "ghost"random sequential addition at the maximum packing fraction φc = 1/8 is much shorter ranged than that for the 3D unit-step function g2 at φc; thus, it does not constrain the realizability of the unit-step function g2. Our inverse methodology yields effective potentials for realizable targets, and, as expected, it does not reach convergence for a target that is known to be non-realizable, despite the fact that it satisfies all known explicit necessary conditions. Our findings demonstrate that exploring the iso-g2 process via our inverse methodology is an effective and robust means to tackle the realizability problem and is expected to facilitate the design of novel nanoparticle systems with density-dependent effective potentials, including exotic hyperuniform states of matter.
AB - An outstanding problem in statistical mechanics is the determination of whether prescribed functional forms of the pair correlation function g2(r) [or equivalently, structure factor S(k)] at some number density ρ can be achieved by many-body systems in d-dimensional Euclidean space. The Zhang-Torquato conjecture states that any realizable set of pair statistics, whether from a nonequilibrium or equilibrium system, can be achieved by equilibrium systems involving up to two-body interactions. To further test this conjecture, we study the realizability problem of the nonequilibrium iso-g2 process, i.e., the determination of density-dependent effective potentials that yield equilibrium states in which g2 remains invariant for a positive range of densities. Using a precise inverse algorithm that determines effective potentials that match hypothesized functional forms of g2(r) for all r and S(k) for all k, we show that the unit-step function g2, which is the zero-density limit of the hard-sphere potential, is remarkably realizable up to the packing fraction φ = 0.49 for d = 1. For d = 2 and 3, it is realizable up to the maximum "terminal"packing fraction φc = 1/2d, at which the systems are hyperuniform, implying that the explicitly known necessary conditions for realizability are sufficient up through φc. For φ near but below φc, the large-r behaviors of the effective potentials are given exactly by the functional forms exp [- κ(φ)r] for d = 1, r-1/2 exp [- κ(φ)r] for d = 2, and r-1 exp [- κ(φ)r] (Yukawa form) for d = 3, where κ-1(φ) is a screening length, and for φ = φc, the potentials at large r are given by the pure Coulomb forms in the respective dimensions as predicted by Torquato and Stillinger [Phys. Rev. E 68, 041113 (2003)]. We also find that the effective potential for the pair statistics of the 3D "ghost"random sequential addition at the maximum packing fraction φc = 1/8 is much shorter ranged than that for the 3D unit-step function g2 at φc; thus, it does not constrain the realizability of the unit-step function g2. Our inverse methodology yields effective potentials for realizable targets, and, as expected, it does not reach convergence for a target that is known to be non-realizable, despite the fact that it satisfies all known explicit necessary conditions. Our findings demonstrate that exploring the iso-g2 process via our inverse methodology is an effective and robust means to tackle the realizability problem and is expected to facilitate the design of novel nanoparticle systems with density-dependent effective potentials, including exotic hyperuniform states of matter.
UR - http://www.scopus.com/inward/record.url?scp=85144166387&partnerID=8YFLogxK
UR - http://www.scopus.com/inward/citedby.url?scp=85144166387&partnerID=8YFLogxK
U2 - 10.1063/5.0130679
DO - 10.1063/5.0130679
M3 - Article
C2 - 36546822
AN - SCOPUS:85144166387
SN - 0021-9606
VL - 157
JO - Journal of Chemical Physics
JF - Journal of Chemical Physics
IS - 22
M1 - 224106
ER -