Lubachevsky–Stillinger algorithm: Difference between revisions

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{{Short description|computationalComputational physics simulation algorithm}}

'''Lubachevsky-Stillinger (compression) algorithm''' (LS algorithm, LSA, or LS protocol) is a numerical procedure suggested by [[F. H. Stillinger]] and B.Boris D. Lubachevsky that simulates or imitates a physical process of compressing an assembly of hard particles.<ref name="StillingerLubachevskyJStat">{{cite journal|doi=10.1007/bf01025983|url=https://www.princeton.edu/~fhs/geodisk/geodisk.pdf|title=Geometric properties of random disk packings|journal=Journal of Statistical Physics|volume=60|issue=5–6|pages=561–583|year=1990|last1=Lubachevsky|first1=Boris D.|last2=Stillinger|first2=Frank H.|bibcode=1990JSP....60..561L|s2cid=15485746 }}</ref> As the LSA may need thousands of arithmetic operations even for a few particles, it is usually carried out on a computer.[[File:1000 triangles packed in rectangle.png|thumb|Using a variant of Lubachevsky-Stillinger algorithm, 1000 congruent isosceles triangles are randomly packed by compression in a rectangle with periodic (wrap-around) boundary. The rectangle which is the period of pattern repetition in both directions is shown. Packing density is 0.8776]]

==Phenomenology==

A physical process of compression often involves a contracting hard boundary of the container, such as a piston pressing against the particles. The LSA is able to simulate such a scenario.<ref>{{cite journal|doi=10.1007/bf01054337|title=Crystalline-amorphous interface packings for disks and spheres|journal=Journal of Statistical Physics|volume=73|issue=3–4|pages=497–514|year=1993|last1=Stillinger|first1=Frank H.|last2=Lubachevsky|first2=Boris D.|s2cid=59429012 }}</ref> However, the LSA was originally introduced in the setting without a hard boundary<ref name="StillingerLubachevskyJStat"/><ref>{{cite journal|doi=10.1016/0021-9991(91)90222-7|title=How to simulate billiards and similar systems|journal=Journal of Computational Physics|volume=94|issue=2|pages=255–283|year=1991|last1=Lubachevsky|first1=Boris D.|bibcode=1991JCoPh..94..255L|arxiv=cond-mat/0503627|s2cid=6215418 }}</ref> where the virtual particles were "swelling" or expanding in a fixed, finite virtual volume with [[periodic boundary conditions]]. The absolute sizes of the particles were increasing but particle-to-particle relative sizes remained constant. In general, the LSA can handle an external compression and an internal particle expansion, both occurring simultaneously and possibly, but not necessarily, combined with a hard boundary. In addition, the boundary can be mobile.

In a final, compressed, or "jammed" state, some particles are not jammed, they are able to move within "cages" formed by their immobile, jammed neighbors and the hard boundary, if any. These free-to-move particles are not an artifact, or pre-designed, or target feature of the LSA, but rather a real phenomenon. The simulation revealed this phenomenon, somewhat unexpectedly for the authors of the LSA. Frank H. Stillinger coined the term "rattlers" for the free-to-move particles, because if one physically shakes a compressed bunch of hard particles, the rattlers will be rattling.

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short range particle-particle force interaction. External [[force field (physics)|force fields]], such as [[gravitation]], can be also introduced, as long as the inter-collision motion of each particle can be represented by a simple one-step calculation.

Using LSA for spherical particles of different sizes and/or for jamming in a non-commeasureable size container proved to be a useful technique for generating and studying micro-structures formed under conditions of a [[crystallographic defect]]<ref>{{cite journal|doi=10.1007/bf02183698|title=Patterns of broken symmetry in the impurity-perturbed rigid-disk crystal|journal=Journal of Statistical Physics|volume=78|issue=3–4|pages=1011–1026|year=1995|last1=Stillinger|first1=Frank H.|last2=Lubachevsky|first2=Boris D.|bibcode=1995JSP....78.1011S|s2cid=55943037 }}</ref> or a [[geometrical frustration]]<ref>{{cite journal|doi=10.1103/physreve.70.041604|pmid=15600418|title=Epitaxial frustration in deposited packings of rigid disks and spheres|journal=Physical Review E|volume=70|issue=4|pages=041604|year=2004|last1=Lubachevsky|first1=Boris D.|last2=Stillinger|first2=Frank H.|bibcode=2004PhRvE..70d1604L|arxiv=cond-mat/0405650|s2cid=1309789 }}</ref><ref>{{cite journal|last1=Lubachevsky|first1=Boris D.|last2=Graham|first2=Ron L.|last3=Stillinger|first3=Frank H.|title=Spontaneous Patterns in Disk Packings|journal=Visual Mathematics|year=1995|url=http://vismath5.tripod.com/lub/}}</ref> It should be added that the original LS protocol was designed primarily for spheres of same or different sizes.<ref>{{cite journal|doi=10.1063/1.1511510|title=Computer generation of dense polydisperse sphere packings|journal=The Journal of Chemical Physics|volume=117|issue=18|pages=8212–8218|year=2002|last1=Kansal|first1=Anuraag R.|last2=Torquato|first2=Salvatore|last3=Stillinger|first3=Frank H.|bibcode=2002JChPh.117.8212K}}</ref>

Any deviation from the spherical (or circular in two dimensions) shape, even a simplest one, when spheres are replaced with ellipsoids (or ellipses in two dimensions),<ref>{{cite journal|doi=10.1103/physrevlett.92.255506|pmid=15245027|title=Unusually Dense Crystal Packings of Ellipsoids|journal=Physical Review Letters|volume=92|issue=25|pages=255506|year=2004|last1=Donev|first1=Aleksandar|last2=Stillinger|first2=Frank H.|last3=Chaikin|first3=P. M.|last4=Torquato|first4=Salvatore|bibcode=2004PhRvL..92y5506D|arxiv=cond-mat/0403286|s2cid=7982407 }}</ref> causes thus modified LSA to slow down substantially.

But as long as the shape is spherical, the LSA is able to handle particle assemblies in tens to hundreds of thousands

on today's (2011) standard [[personal computers]]. Only a very limited experience was reported<ref>{{cite journal|doi=10.1103/physreve.74.041127|pmid=17155042|title=Packing hyperspheres in high-dimensional Euclidean spaces|journal=Physical Review E|volume=74|issue=4|pages=041127|year=2006|last1=Skoge|first1=Monica|last2=Donev|first2=Aleksandar|last3=Stillinger|first3=Frank H.|last4=Torquato|first4=Salvatore|bibcode=2006PhRvE..74d1127S|arxiv=cond-mat/0608362|s2cid=18639889 }}</ref>

in using the LSA in dimensions higher than 3.

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* [http://onlinelibrary.wiley.com/doi/10.1002/pamm.200610180/pdf LSA used for production of representative volumes of microscale failures in packed granular materials]

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