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Three simple scenarios for high-dimensional sphere packings.

Patrick Charbonneau1,2, Peter K Morse1, Will Perkins3

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Summary
This summary is machine-generated.

This study explores hard sphere crystallization in high dimensions, proposing three scenarios. Results suggest crystallization is plausible before dynamical arrest, supporting Scenario C for densest packing.

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Area of Science:

  • Physics
  • Mathematics
  • Materials Science

Background:

  • Understanding hard sphere crystallization is crucial for dense packing.
  • High-dimensional systems present unique challenges for crystallization.
  • Existing conjectures suggest different outcomes for crystallization in high dimensions.

Purpose of the Study:

  • To formulate and assess scenarios for hard sphere crystallization in high dimensions.
  • To determine the most plausible scenario for crystallization based on physical and mathematical evidence.
  • To investigate the densest sphere packings in dimensions 3 through 10.

Main Methods:

  • Formulation of three distinct crystallization scenarios (A, B, C).
  • Utilizing cell-cluster expansions to analyze high-dimensional sphere packings.
  • Employing numerical simulations to estimate crystal entropy near close packing.

Main Results:

  • Estimates of crystal entropy near close packing favor Scenario C.
  • Scenario C posits that crystallization is possible before or shortly after dynamical arrest.
  • The crystal equation of state is primarily influenced by free-volume expansion.

Conclusions:

  • The findings support Scenario C as the most likely outcome for hard sphere crystallization in high dimensions.
  • Crystallization appears dynamically plausible from the liquid state.
  • A polynomial correction to the crystal equation of state is feasible.