Asymptotic Properties of Collective-Rearrangement Algorithms
Hirsch, Christian1 ; Gaiselmann, Gerd1 ; Schmidt, Volker1
1 Institute of Stochastics, Ulm University, 89069 Ulm, Germany.
ESAIM: Probability and Statistics, Tome 19 (2015), pp. 236-250.

We analyze asymptotic properties of collective-rearrangement algorithms being a class of dense packing algorithms. Traditionally, they transform finite systems of (possibly overlapping) particles into non-overlapping configurations by collective rearrangement of particles in finitely many steps. We consider the convergence of such algorithms for not necessarily finite input data, which means that the configuration of particles in any bounded sampling window remains unchanged after finitely many rearrangement steps. More precisely, we derive sufficient conditions implying the convergence of such algorithms when a stationary process of particles is used as input. We also provide numerical results and present an application in computational materials science.

Reçu le :
DOI : 10.1051/ps/2014026
Classification : 60K35, 60D05, 82C22
Mots clés : Asymptotics, force-biased algorithm, collective rearrangement, random sphere-packing, stationary particle system
Hirsch, Christian 1 ; Gaiselmann, Gerd 1 ; Schmidt, Volker 1

1 Institute of Stochastics, Ulm University, 89069 Ulm, Germany. 
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     title = {Asymptotic {Properties} of {Collective-Rearrangement} {Algorithms}},
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Hirsch, Christian; Gaiselmann, Gerd; Schmidt, Volker. Asymptotic Properties of Collective-Rearrangement Algorithms. ESAIM: Probability and Statistics, Tome 19 (2015), pp. 236-250. doi : 10.1051/ps/2014026. http://www.numdam.org/articles/10.1051/ps/2014026/

H. Altendorf and D. Jeulin, Random-walk-based stochastic modeling of three-dimensional fiber systems. Phys. Rev. E 83 (2011) 041804.

B. Baumeier, O. Stenzel, C. Poelking, D. Andrienko and V. Schmidt, Stochastic modeling of molecular charge transport networks. Phys. Rev. B 86 (2012) 184202.

J. D. Bernal and J. Mason, Packing of spheres: co-ordination of randomly packed spheres. Nature 188 (1960) 910–911.

A. Bezrukov, M. Bargieł and D. Stoyan, Statistical analysis of simulated random packings of spheres. Particle & Particle Systems Characterization 19 (2002) 111–118.

S.N. Chiu, D. Stoyan, W.S. Kendall and J. Mecke, Stochastic Geometry and its Applications. J. Wiley and Sons, Chichester, third edition (2013). | MR

E.G. Coffman, Jr., L. Flatto, P. Jelenković and B. Poonen, Packing random intervals on-line. Algorithmica 22 (1998) 448–476. | MR | Zbl

D.J. Cumberland and R.J. Crawford, The Packing of Particles. Elsevier, New York (1987).

D.J. Daley and D.D. Vere-Jones, An Introduction to the Theory of Point Processes I/II. Springer, New York (2005/2008). | MR | Zbl

J. W. Evans, Random and cooperative sequential adsorption. Rev. Modern Phys. 65 (1993) 1281.

P.J. Flory, Intramolecular reaction between neighboring substituents of vinyl polymers. J. Amer. Chem. Soc. 61 (1939) 1518–1521.

G. Gaiselmann, D. Froning, C. Tötzke, C. Quick, I. Manke, W. Lehnert and V. Schmidt, Stochastic 3D modeling of non-woven materials with wet-proofing agent. International J. Hydrogen Energy 38 (2013) 8448–8460.

J. J. Gonzalez, P. C. Hemmer and J. S. Høye. Cooperative effects in random sequential polymer reactions. Chem. Phys. 3 (1974) 228–238.

D. Illian, P. Penttinen, H. Stoyan and D. Stoyan, Statistical Analysis and Modelling of Spatial Point Patterns. J. Wiley and Sons, Chichester (2008). | MR | Zbl

J. Mościński, M. Bargieł, Z. Rycerz and P. Jacobs, The force-biased algorithm for the irregular close packing of equal hard spheres. Molecular Simulation 3 (1989) 201–212.

M.D. Penrose, Limit theorems for monolayer ballistic deposition in the continuum. J. Stat. Phys. 105 (2001) 561–583. | MR | Zbl

M.D. Penrose, Random Geometric Graphs. Oxford University Press, Oxford (2003). | MR | Zbl

M.D. Penrose, Existence and spatial limit theorems for lattice and continuum particle systems. Probab. Surveys 5 (2008) 1–36. | MR | Zbl

M.D. Penrose and J.E. Yukich, Limit theory for random sequential packing and deposition. Ann. Appl. Probab. 12 (2002) 272–301. | MR | Zbl

V. Rühle, A. Lukyanov, F. May, M. Schrader, T. Vehoff, J. Kirkpatrick, B. Baumeier and D. Andrienko, Microscopic simulations of charge transport in disordered organic semiconductors. J. Chem. Theory Comput. 7 (2011) 3335–3345.

G. Scott, Packing of spheres: packing of equal spheres. Nature 188 (1960) 908–909. | Zbl

D. Stoyan, Simulation and characterization of random systems of hard particles. Image Anal. Stereol. 21 (2002) S41–S48.

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