From clusters of particles to 2D bubble clusters

1
From clusters of particles to 2D bubble clusters

• Edwin Flikkema, Simon Cox
• IMAPS, Aberystwyth University, UK

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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

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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.

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Systems of interacting particles

• System energy
• Usually
• Example

Lennard-Jones potential
LJ13 Ar13

• Used in Molecular Dynamics, Monte Carlo, Energy
  landscapes

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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
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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

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2D particle systems

• Energy
• Repulsive inter-particle potential
• Confining potential

or
harmonic
polygonal
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2D particle clusters

• Pictures of particle clusters e.g. N41, bottom
  3 in energy

-945.419508
-945.419781
-945.421319
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Particles to bubbles
Qhull
Surface Evolver
particle cluster
Voronoi cells
optimized perimeter
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2D particle clusters

• Polygonal confining potential e.g. triangular

unit vectors
contour lines
discontinuous gradient smoothing needed?
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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)?

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Results bubble clusters Free, circle, hexagon
N31-37
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Results bubble clusters pentagon, square,
triangle
N31-37
Elec. J. Combinatorics 17R45 (2010)
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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

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Acknowledgements

• Simon Cox
• Adil Mughal

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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
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2D clusters perimeter
is fit to data for free clusters