From clusters of particles to 2D bubble clusters Edwin Flikkema, Simon Cox IMAPS, Aberystwyth University, UK

Introduction and overview 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

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.

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

Energy landscapes Energy vs coordinate 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 Local optimisation

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

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

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

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

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

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 optimisers have difficulty with polygonal potential due to discontinuous gradient Smoothing needed? Use gradient-less optimisers (e.g. Powell)?

Results: bubble clusters: Free, circle, hexagon

Results: bubble clusters: pentagon, square, triangle Elec. J. Combinatorics 17:R45 (2010)

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

Acknowledgements Simon Cox Adil Mughal

Energy landscapes Energy vs coordinate 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. Local optimisation Energy vs coordinate

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