Lubachevsky–Stillinger algorithm: Difference between revisions

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'''Lubachevsky-Stillinger (compression) algorithm''' (LS algorithm, LSA, or LS protocol) is a numerical procedure suggested by [[F. H. Stillinger]] and B.D. Lubachevsky that simulates or imitates a physical process of compressing an assembly of hard particles<ref name="StillingerLubachevskyJStat">B. D. Lubachevsky and F. H. Stillinger, Geometric properties of random disk packings, J. Statistical Physics 60 (1990), 561-583 http://www.princeton.edu/~fhs/geodisk/geodisk.pdf</ref>. As the LSA may need thousands of arithmetic operations even for a few particles, it is usually carried out on a [[digital 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]]

It was suggested by [[F. H. Stillinger]] and B.D. Lubachevcky

==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>F. H. Stillinger and B. D. Lubachevsky, Crystalline-Amorphous Interface Packings for Disks and Spheres, J. Stat. Phys. 73, 497-514 (1993)</ref> However, the LSA was originally introduced in the setting without a hard boundary<ref>B. D. Lubachevsky and F. H. Stillinger, Geometric properties of random disk packings, J. Statistical Physics 60 (1990), 561-583 http:name="StillingerLubachevskyJStat"//www.princeton.edu/~fhs/geodisk/geodisk.pdf</ref><ref>B.D. Lubachevsky, How to Simulate Billiards and Similar Systems, Journal of Computational Physics Volume 94 Issue 2, May 1991 https://arxiv.org/PS_cache/cond-mat/pdf/0503/0503627v2.pdf</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 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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it is possible for a few particles, even just for a single particle, to experience a very high collision rate along the approach to a certain simulated time. The rate will be increasing without a bound in proportion to the rates of collisions in the rest of the particle ensemble. If this happens, then the simulation will be stuck in time, it won't be able to progress toward the state of jamming.

The stuck-in-time failure can also occur when simulating a granular flow, without the particle compression or expansion. This failure mode was recognized by the practitioners of granular flow simulations as an "inelastic collapse" <ref>S. McNamara, and W.R. Young, Inelastic collapse in two dimensions, Physical Review E 50: pp. R28-R31, 1994</ref> because it often occurs in such simulations when the [[restitution coefficient]] atin collisions is low (and hence the collisions arei.e. inelastic). The failure is not specific to only the LSA algorithm. Techniques to avoid the failure have been proposed.<ref>John J. Drozd, Computer Simulation of Granular Matter: A Study of An Industrial Grinding Mill, Thesis, Univ. Western Ontario, Canada, 2004. {{cite web |url=http://imaging.robarts.ca/~jdrozd/thesisjd.pdf |title=Archived copy |accessdate=2011-05-25 |deadurl=yes |archiveurl=https://web.archive.org/web/20110818102135/http://imaging.robarts.ca/~jdrozd/thesisjd.pdf |archivedate=2011-08-18 |df= }}</ref>

== History ==