Configuration spaces of hard spheres.

Area of Science:

• Physics
• Statistical Mechanics
• Computational Physics

Background:

• Hard sphere systems model simple fluids and their phase transitions.
• Configuration spaces of hard spheres in a 3D torus modulo symmetry groups offer insights into phase transitions.
• Topological changes in configuration space are conjectured to relate to first-order phase transitions.

Purpose of the Study:

• To investigate the relationship between topological changes in hard sphere configuration spaces and phase transitions.
• To sample critical configurations for 1 to 12 spheres using Morse theory.
• To analyze the topological and geometric properties of configuration spaces for hard sphere systems.

Main Methods:

• Morse-theoretic approach to sample critical configurations.
• Construction of explicit triangulations for configuration spaces.
• Analysis of configuration space diameter using commute time and diffusion distances.

Main Results:

• Critical configurations associated with geometric changes that connect distant regions and reduce configuration space diameter.
• Number of critical configurations increases exponentially with the number of spheres.
• A database of critical configurations for 1 to 12 spheres is available online.

Conclusions:

• The onset of first-order phase transitions in hard sphere systems is linked to geometric changes in configuration space.
• The exponential increase in critical configurations suggests a discontinuity in configuration space diameter at the thermodynamic limit.
• This study provides a computational framework for understanding fluid phase transitions through configuration space topology.