Realizability of iso-g₂ processes via effective pair interactions

Area of Science:

• Statistical Mechanics
• Condensed Matter Physics
• Materials Science

Background:

• A key challenge is determining if prescribed pair correlation functions (g₂(r) or S(k)) are achievable by many-body systems.
• The Zhang-Torquato conjecture posits that any realizable pair statistics can be achieved by equilibrium systems with two-body interactions.

Purpose of the Study:

• To test the Zhang-Torquato conjecture by investigating the realizability of the nonequilibrium iso-g₂(r) process.
• To determine density-dependent effective potentials for equilibrium states with invariant pair correlation functions across a range of densities.

Main Methods:

• Employed a precise inverse algorithm to determine effective potentials matching hypothesized g₂(r) and S(k) functional forms.
• Studied the unit-step function g₂, the zero-density limit of the hard-sphere potential, for realizability in d=1, 2, and 3 dimensions.

Main Results:

• The unit-step function g₂ is realizable up to packing fraction ϕ=0.49 (d=1) and ϕc=1/2ᵈ (d=2, 3), where systems become hyperuniform.
• Effective potentials exhibit specific large-r behaviors (e.g., Yukawa form in 3D) at densities near ϕc, consistent with Coulombic forms at ϕc.
• The inverse methodology successfully identified effective potentials for realizable targets and failed for a known non-realizable target.

Conclusions:

• The iso-g₂(r) process and inverse methodology provide a robust approach to the realizability problem in statistical mechanics.
• Findings support the Zhang-Torquato conjecture and pave the way for designing novel nanoparticle systems with tailored effective potentials, including hyperuniform states.