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Lubachevsky-Stillinger (compression) algorithm (LS algorithm, LSA, or LS protocol) is a numerical procedure that simulates or imitates a physical process of compressing an assembly of hard particles. As the LSA may need thousands of arithmetic operations even for a few particles, it is usually carried out on a digital computer. ==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.〔F. H. Stillinger and B. D. Lubachevsky, Crystalline-Amorphous Interface Packings for Disks and Spheres, J. Stat. Phys. 73, 497-514 (1993)〕 However, the LSA was originally introduced in the setting without a hard boundary〔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〕〔B.D. Lubachevsky, How to Simulate Billiards and Similar Systems, Journal of Computational Physics Volume 94 Issue 2, May 1991 http://arxiv.org/PS_cache/cond-mat/pdf/0503/0503627v2.pdf〕 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. In the "pre-jammed" mode when the density of the configuration is low and when the particles are mobile, the compression and expansion can be stopped, if so desired. Then the LSA, in effect, would be simulating a granular flow. Various dynamics of the instantaneous collisions can be simulated such as: with or without a full restitution, with or without tangential friction. Differences in masses of the particles can be taken into account. It is also easy and sometimes proves useful to "fluidize" a jammed configuration, by decreasing the sizes of all or some of the particles. Another possible extension of the LSA is replacing the hard collision force potential (zero outside the particle, infinity at or inside) with a piece-wise constant force potential. The LSA thus modified would approximately simulate molecular dynamics with continuous short range particle-particle force interaction. External 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〔F. H. Stillinger and B. D. Lubachevsky. Patterns of Broken Symmetry in the Impurity-Perturbed Rigid Disk Crystal, J. Stat. Phys. 78, 1011-1026 (1995)〕 or a geometrical frustration〔Boris D. Lubachevsky and Frank H. Stillinger, Epitaxial frustration in deposited packings of rigid disks and spheres. Physical Review E 70:44, 41604 (2004) http://arxiv.org/PS_cache/cond-mat/pdf/0405/0405650v5.pdf〕〔Boris D. Lubachevsky, Ron L. Graham, and Frank H. Stillinger, Spontaneous Patterns in Disk Packings. Visual Mathematics, 1995. http://vismath5.tripod.com/lub/〕 It should be added that the original LS protocol was designed primarily for spheres of same or different sizes.〔A.R. Kansal, S. Torquato, and F.H. Stillinger, Computer Generation of Dense Polydisperse Sphere Packings, J. Chem. Phys. 117, 8212-8218 (2002) http://cat.inist.fr/?aModele=afficheN&cpsidt=13990882〕 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),〔A. Donev, F.H. Stillinger, P.M. Chaikin, and S. Torquato, Unusually Dense Crystal Packings of Ellipsoids, Phys. Rev. Letters 92, 255506 (2004)〕 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〔M. Skoge, A. Donev, F.H. Stillinger, and S. Torquato, Packing Hyperspheres in High-Dimensional Euclidean Spaces, Phys. Rev. E 74, 041127 (2006)〕 in using the LSA in dimensions higher than 3. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Lubachevsky–Stillinger algorithm」の詳細全文を読む スポンサード リンク
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