Exploring Analogies Between Bubble and Particle Clusters

1. Exploring analogies between bubble clusters and particle clusters Edwin Flikkema, Simon Cox IMAPS, Aberystwyth University, UK

2. Introduction and overview • Introduction: • The minimal perimeter problem for 2D equal area bubble clusters. • Systems of interacting particles • Global optimisation • 2D particle clusters to 2D bubble clusters • Voronoi construction • 2D particle systems: • -log(r) or 1/rp repulsive potential • Harmonic or polygonal confining potentials • Results

3. 2D bubble clusters • Minimal perimeter problem: • 2D cluster of N bubbles. • All bubbles have equal area. • Free or confined to the interior of a circle or polygon. • Minimize total perimeter (internal + external). • Objective: apply techniques used in interacting particle clusters to this minimal perimeter problem.

4. Systems of interacting particles • System energy: • Usually: • Example: Lennard-Jones potential: LJ13: Ar13 • Used in: Molecular Dynamics, Monte Carlo, Energy landscapes

5. Local optimisation Energy landscapes • Stationary points of U: zero net force on each particle • Minima of U correspond to (meta-)stable states. • Global minimum is the most stable state. • Local optimisation (finding a nearby minimum) relatively easy: • Steepest descent, L-BFGS, Powell, etc. • Global optimisation: hard. Energy vs coordinate

6. Global optimisation methods • Inspired by simulated annealing: • Basin hopping • Minima hopping • Evolutionary algorithms: • Genetic algorithm • Other: • Covariance matrix adaption • Simply starting from many random geometries

7. 2D particle systems • Energy: • Repulsive inter-particle potential: • Confining potential: or harmonic polygonal

8. 2D particle clusters • Pictures of particle clusters: e.g. N=41, bottom 3 in energy -945.419508 -945.419781 -945.421319

9. Particles to bubbles Qhull Surface Evolver particle cluster Voronoi cells optimized perimeter

10. 2D particle clusters • Polygonal confining potential: e.g. triangular unit vectors contour lines discontinuous gradient: smoothing needed?

11. Technical details • List of unique 2D geometries produced • Problem: permutational isomers. • Distinguishing by energy U not sufficient: • Spectrum of inter-particle distances compared. • Gradient-based local optimisation algorithms have difficulty with polygonal potential due to discontinuous gradient • Smoothing needed? • Use gradient-less optimisation algorithms (e.g. Powell)?

12. Results: bubble clusters: Free, circle, hexagon N=31-37

13. Results: bubble clusters: pentagon, square, triangle N=31-37 Elec. J. Combinatorics 17:R45 (2010)

14. Conclusions • Optimal geometries of clusters of interacting particles can be used as candidates for the minimal perimeter problem. • Various potentials have been tried. 1/r seems to work slightly better than –log(r). • Using multiple potentials is recommended. • Polygonal potentials have been introduced to represent confinement to a polygon • Outlook: bi-disperse bubble clusters.

15. Acknowledgements • Simon Cox • Adil Mughal

16. Local optimisation Energy landscapes • Stationary points of U: zero net force on each particle • Minima of U correspond to (meta-)stable states. • Global minimum is the most stable state. • Saddle points (first order): transition states • Network of minima connected by transition states • Local optimisation (finding a nearby minimum) relatively easy: • L-BFGS, Powell, etc. • Global optimisation: hard. Energy vs coordinate

17. 2D clusters: perimeter is fit to data for free clusters