Packing hyperspheres in high-dimensional Euclidean spaces

Area of Science:

• Physics
• Materials Science
• Statistical Mechanics

Background:

• Disordered jammed hard-sphere packings are fundamental in statistical mechanics.
• Understanding their behavior in higher dimensions is crucial for theoretical and practical applications.

Purpose of the Study:

• To investigate disordered jammed hard-sphere packings in four-, five-, and six-dimensional Euclidean spaces.
• To determine the packing fractions of maximally random jammed (MRJ) states and analyze their dimensional dependence.
• To examine structural properties like pair correlation and structure factor, and their relation to dimensionality.

Main Methods:

• Utilized a collision-driven packing generation algorithm.
• Calculated packing fractions for MRJ states in dimensions d=4, 5, and 6.
• Analyzed pair correlation functions (g2(r)) and structure factors (S(k)).

Main Results:

• Estimated MRJ packing fractions: phi(MRJ) ≈ 0.46 (d=4), 0.31 (d=5), 0.20 (d=6).
• Observed a decrease in short-range ordering with increasing dimension, consistent with the decorrelation principle.
• Found packings to be isostatic with no crystallization signs, and a power-law divergence in g2(r) at contact.

Conclusions:

• The MRJ density scales with dimension, approaching a high-dimensional limit.
• Increasing dimensionality suppresses crystallization and reduces short-range order.
• Freezing and melting densities were estimated, indicating a suppressed nucleation of crystalline solids in higher dimensions.