Stationary Apollonian Packings
The notion of stationary Apollonian packings in the d -dimensional Euclidean space is introduced as a mathematical formalization of so-called random Apollonian packings and rotational random Apollonian packings, which constitute popular grain packing models in physics. Apart from dealing with issues...
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| Published in | Journal of statistical physics Vol. 161; no. 1; pp. 35 - 72 |
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| Main Authors | , , |
| Format | Journal Article |
| Language | English |
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01.10.2015
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| Abstract | The notion of stationary Apollonian packings in the
d
-dimensional Euclidean space is introduced as a mathematical formalization of so-called random Apollonian packings and rotational random Apollonian packings, which constitute popular grain packing models in physics. Apart from dealing with issues of existence and uniqueness in the entire Euclidean space, asymptotic results are provided for the growth durations and it is shown that the packing is space-filling with probability 1, in the sense that the Lebesgue measure of its complement is zero. Finally, the phenomenon is studied that grains arrange in clusters and properties related to percolation are investigated. |
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| AbstractList | The notion of stationary Apollonian packings in the d-dimensional Euclidean space is introduced as a mathematical formalization of so-called random Apollonian packings and rotational random Apollonian packings, which constitute popular grain packing models in physics. Apart from dealing with issues of existence and uniqueness in the entire Euclidean space, asymptotic results are provided for the growth durations and it is shown that the packing is space-filling with probability 1, in the sense that the Lebesgue measure of its complement is zero. Finally, the phenomenon is studied that grains arrange in clusters and properties related to percolation are investigated. Keywords Dense packing * Fractal germ-grain model * Continuum percolation * Growth model Mathematics Subject Classification 60D05 * 05C70 * 28A80 The notion of stationary Apollonian packings in the d -dimensional Euclidean space is introduced as a mathematical formalization of so-called random Apollonian packings and rotational random Apollonian packings, which constitute popular grain packing models in physics. Apart from dealing with issues of existence and uniqueness in the entire Euclidean space, asymptotic results are provided for the growth durations and it is shown that the packing is space-filling with probability 1, in the sense that the Lebesgue measure of its complement is zero. Finally, the phenomenon is studied that grains arrange in clusters and properties related to percolation are investigated. |
| Audience | Academic |
| Author | Hirsch, Christian Delaney, Gary Schmidt, Volker |
| Author_xml | – sequence: 1 givenname: Christian surname: Hirsch fullname: Hirsch, Christian email: christian.hirsch@wias-berlin.de organization: Weierstrass Institute for Applied Analysis and Stochastics – sequence: 2 givenname: Gary surname: Delaney fullname: Delaney, Gary organization: CSIRO Digital Productivity Flagship – sequence: 3 givenname: Volker surname: Schmidt fullname: Schmidt, Volker organization: Institute of Stochastics, Ulm University |
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| Cites_doi | 10.1007/978-3-540-78859-1 10.1239/aap/1363354101 10.1017/CBO9780511750489 10.1112/S0025579300004745 10.1239/aap/1029955194 10.1214/08-AOP415 10.1239/aap/1127483738 10.1006/anbo.1996.0018 10.1017/CBO9781139174695 10.1002/rsa.20410 10.1007/PL00001103 10.1007/s00440-005-0459-y 10.1007/BF01048305 10.1214/aop/1024404279 10.1239/aap/1086957574 10.1007/s10955-008-9523-1 10.1002/rsa.20099 10.1103/PhysRevLett.77.1785 10.1007/s10955-009-9722-4 10.1002/(SICI)1098-2418(199610)9:3<295::AID-RSA3>3.0.CO;2-S 10.1080/15326349.2015.973722 10.1515/9780691209296 |
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| Keywords | 60D05 Growth model Fractal germ-grain model 28A80 Dense packing 05C70 Continuum percolation |
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| References | Dodds, Weitz (CR8) 2002; 65 Hug, Last, Weil (CR16) 2006; 135 Hecht (CR12) 2000; 157 Last, Penrose (CR17) 2013; 42 Daley, Last (CR6) 2005; 37 Delaney, Hutzler, Aste (CR7) 2008; 101 Hirsch, Neuhäuser, Schmidt (CR14) 2013; 45 Boyd (CR2) 1973; 20 Cotar, Volkov (CR3) 2004; 36 Fey, Meester, Redig (CR10) 2009; 37 Liggett, Schonmann, Stacey (CR19) 1997; 25 Schneider, Weil (CR23) 2008 Daley, Stoyan, Stoyan (CR5) 1999; 31 Lawler, Révész, Tóth (CR18) 1999 CR4 Heveling, Last (CR13) 2006; 29 Häggström, Meester (CR11) 1996; 9 CR9 Takahashi (CR24) 1996; 77 Manna, Vicsek (CR20) 1991; 64 Horn (CR15) 1971 Radin (CR22) 2008; 131 Mörters, Peres (CR21) 2010 Turcotte (CR25) 1997 van der Marck (CR26) 1996; 77 Aristoff, Radin (CR1) 2009; 135 PS Dodds (1326_CR8) 2002; 65 H Horn (1326_CR15) 1971 GW Delaney (1326_CR7) 2008; 101 D Turcotte (1326_CR25) 1997 C Hecht (1326_CR12) 2000; 157 TM Liggett (1326_CR19) 1997; 25 D Aristoff (1326_CR1) 2009; 135 DJ Daley (1326_CR5) 1999; 31 M Heveling (1326_CR13) 2006; 29 S Manna (1326_CR20) 1991; 64 C Radin (1326_CR22) 2008; 131 GF Lawler (1326_CR18) 1999 DW Boyd (1326_CR2) 1973; 20 P Mörters (1326_CR21) 2010 1326_CR4 C Hirsch (1326_CR14) 2013; 45 A Fey (1326_CR10) 2009; 37 K Takahashi (1326_CR24) 1996; 77 DJ Daley (1326_CR6) 2005; 37 C Cotar (1326_CR3) 2004; 36 O Häggström (1326_CR11) 1996; 9 D Hug (1326_CR16) 2006; 135 R Schneider (1326_CR23) 2008 G Last (1326_CR17) 2013; 42 SC Marck van der (1326_CR26) 1996; 77 1326_CR9 |
| References_xml | – volume: 101 start-page: 602 issue: 120 year: 2008 ident: CR7 article-title: Relation between grain shape and fractal properties in random Apollonian packing with grain rotation publication-title: Phys. Rev. Lett. – year: 2008 ident: CR23 publication-title: Stochastic and Integral Geometry doi: 10.1007/978-3-540-78859-1 – volume: 45 start-page: 20 year: 2013 end-page: 36 ident: CR14 article-title: Connectivity of random geometric graphs related to minimal spanning forests publication-title: Adv. Appl. Probab. doi: 10.1239/aap/1363354101 – ident: CR4 – year: 2010 ident: CR21 publication-title: Brownian Motion doi: 10.1017/CBO9780511750489 – volume: 20 start-page: 170 year: 1973 end-page: 174 ident: CR2 article-title: The residual set dimension of the Apollonian packing publication-title: Mathematika doi: 10.1112/S0025579300004745 – volume: 31 start-page: 610 year: 1999 end-page: 624 ident: CR5 article-title: The volume fraction of a Poisson germ model with maximally non-overlapping spherical grains publication-title: Adv. Appl. Probab. doi: 10.1239/aap/1029955194 – volume: 37 start-page: 654 year: 2009 end-page: 675 ident: CR10 article-title: Stabilizability and percolation in the infinite volume sandpile model publication-title: Ann. Probab. doi: 10.1214/08-AOP415 – volume: 65 start-page: 108 issue: 056 year: 2002 ident: CR8 article-title: Packing-limited growth publication-title: Phys. Rev. E – volume: 37 start-page: 604 year: 2005 end-page: 628 ident: CR6 article-title: Descending chains, the lilypond model, and mutual-nearest-neighbour matching publication-title: Adv. Appl. Probab. doi: 10.1239/aap/1127483738 – volume: 77 start-page: 159 year: 1996 end-page: 164 ident: CR24 article-title: Plastic response of crown architecture to crowding in understorey trees of two co-dominating conifers publication-title: Ann. Bot. doi: 10.1006/anbo.1996.0018 – year: 1997 ident: CR25 publication-title: Fractals and Chaos in Geology and Geophysics doi: 10.1017/CBO9781139174695 – volume: 42 start-page: 226 year: 2013 end-page: 249 ident: CR17 article-title: Percolation and limit theory for the Poisson lilypond model publication-title: Random Struct. Algorithms doi: 10.1002/rsa.20410 – volume: 157 start-page: 487 year: 2000 end-page: 504 ident: CR12 article-title: Appolonian packing and fractal shape of grains improving geomechanical properties in engineering geology publication-title: Pure Appl. Geophys. doi: 10.1007/PL00001103 – volume: 135 start-page: 169 year: 2006 end-page: 200 ident: CR16 article-title: Polynomial parallel volume, convexity and contact distributions of random sets publication-title: Probab. Theory Relat. Fields doi: 10.1007/s00440-005-0459-y – volume: 64 start-page: 525 year: 1991 end-page: 539 ident: CR20 article-title: Multifractality of space-filling bearings and Apollonian packings publication-title: J. Stat. Phys. doi: 10.1007/BF01048305 – volume: 25 start-page: 71 year: 1997 end-page: 95 ident: CR19 article-title: Domination by product measures publication-title: Ann. Probab. doi: 10.1214/aop/1024404279 – year: 1971 ident: CR15 publication-title: Adaptive Geometry of Trees – volume: 36 start-page: 325 year: 2004 end-page: 339 ident: CR3 article-title: A note on the lilypond model publication-title: Adv. Appl. Probab. doi: 10.1239/aap/1086957574 – volume: 131 start-page: 567 year: 2008 end-page: 573 ident: CR22 article-title: Random close packing of granular matter publication-title: J. Stat. Phys. doi: 10.1007/s10955-008-9523-1 – volume: 29 start-page: 338 year: 2006 end-page: 350 ident: CR13 article-title: Existence, uniqueness, and algorithmic computation of general lilypond systems publication-title: Random Struct. Algorithms doi: 10.1002/rsa.20099 – volume: 77 start-page: 1785 year: 1996 end-page: 1788 ident: CR26 article-title: Network approach to void percolation in a pack of unequal spheres publication-title: Phys. Rev. Lett. doi: 10.1103/PhysRevLett.77.1785 – ident: CR9 – volume: 135 start-page: 1 year: 2009 end-page: 23 ident: CR1 article-title: Random loose packing in granular matter publication-title: J. Stat. Phys. doi: 10.1007/s10955-009-9722-4 – volume: 9 start-page: 295 year: 1996 end-page: 315 ident: CR11 article-title: Nearest neighbor and hard sphere models in continuum percolation publication-title: Random Struct. Algorithms doi: 10.1002/(SICI)1098-2418(199610)9:3<295::AID-RSA3>3.0.CO;2-S – start-page: 219 year: 1999 end-page: 258 ident: CR18 article-title: Geometric and fractal properties of Brownian motion and random walk paths in two and three dimensions publication-title: Random Walks (Budapest, 1998) – volume-title: Fractals and Chaos in Geology and Geophysics year: 1997 ident: 1326_CR25 doi: 10.1017/CBO9781139174695 – volume: 135 start-page: 169 year: 2006 ident: 1326_CR16 publication-title: Probab. Theory Relat. Fields doi: 10.1007/s00440-005-0459-y – volume: 36 start-page: 325 year: 2004 ident: 1326_CR3 publication-title: Adv. Appl. Probab. doi: 10.1239/aap/1086957574 – start-page: 219 volume-title: Random Walks (Budapest, 1998) year: 1999 ident: 1326_CR18 – ident: 1326_CR9 doi: 10.1080/15326349.2015.973722 – volume: 64 start-page: 525 year: 1991 ident: 1326_CR20 publication-title: J. Stat. Phys. doi: 10.1007/BF01048305 – volume: 77 start-page: 1785 year: 1996 ident: 1326_CR26 publication-title: Phys. Rev. Lett. doi: 10.1103/PhysRevLett.77.1785 – volume: 31 start-page: 610 year: 1999 ident: 1326_CR5 publication-title: Adv. Appl. Probab. doi: 10.1239/aap/1029955194 – ident: 1326_CR4 – volume: 9 start-page: 295 year: 1996 ident: 1326_CR11 publication-title: Random Struct. Algorithms doi: 10.1002/(SICI)1098-2418(199610)9:3<295::AID-RSA3>3.0.CO;2-S – volume: 29 start-page: 338 year: 2006 ident: 1326_CR13 publication-title: Random Struct. Algorithms doi: 10.1002/rsa.20099 – volume: 37 start-page: 654 year: 2009 ident: 1326_CR10 publication-title: Ann. Probab. doi: 10.1214/08-AOP415 – volume-title: Stochastic and Integral Geometry year: 2008 ident: 1326_CR23 doi: 10.1007/978-3-540-78859-1 – volume: 25 start-page: 71 year: 1997 ident: 1326_CR19 publication-title: Ann. Probab. doi: 10.1214/aop/1024404279 – volume: 45 start-page: 20 year: 2013 ident: 1326_CR14 publication-title: Adv. Appl. Probab. doi: 10.1239/aap/1363354101 – volume-title: Brownian Motion year: 2010 ident: 1326_CR21 doi: 10.1017/CBO9780511750489 – volume: 65 start-page: 108 issue: 056 year: 2002 ident: 1326_CR8 publication-title: Phys. Rev. E – volume: 101 start-page: 602 issue: 120 year: 2008 ident: 1326_CR7 publication-title: Phys. Rev. Lett. – volume-title: Adaptive Geometry of Trees year: 1971 ident: 1326_CR15 doi: 10.1515/9780691209296 – volume: 135 start-page: 1 year: 2009 ident: 1326_CR1 publication-title: J. Stat. 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d
-dimensional Euclidean space is introduced as a mathematical formalization of so-called random Apollonian... The notion of stationary Apollonian packings in the d-dimensional Euclidean space is introduced as a mathematical formalization of so-called random Apollonian... |
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| Title | Stationary Apollonian Packings |
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