Pub Date : 2003-10-03DOI: 10.1088/0305-4470/37/5/015
A. Kuniba, T. Takagi, A. Takenouchi
Factorized dynamics in soliton cellular automata with quantum group symmetry is identified with a motion of particles and anti-particles exhibiting pair creation and annihilation. An embedding scheme is presented showing that the D^{(1)}_n-automaton contains, as certain subsectors, the box-ball systems and all the other automata associated with the crystal bases of non-exceptional affine Lie algebras. The results extend the earlier ones to higher representations by a certain reduction and to a wider class of boundary conditions.
{"title":"Factorization, reduction and embedding in integrable cellular automata","authors":"A. Kuniba, T. Takagi, A. Takenouchi","doi":"10.1088/0305-4470/37/5/015","DOIUrl":"https://doi.org/10.1088/0305-4470/37/5/015","url":null,"abstract":"Factorized dynamics in soliton cellular automata with quantum group symmetry is identified with a motion of particles and anti-particles exhibiting pair creation and annihilation. An embedding scheme is presented showing that the D^{(1)}_n-automaton contains, as certain subsectors, the box-ball systems and all the other automata associated with the crystal bases of non-exceptional affine Lie algebras. The results extend the earlier ones to higher representations by a certain reduction and to a wider class of boundary conditions.","PeriodicalId":436460,"journal":{"name":"arXiv: Cellular Automata and Lattice Gases","volume":"44 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2003-10-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"128348375","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We demonstrate that a local mapping f in a space of bisequences over {0,1} which conserves the number of nonzero sites can be viewed as a deterministic particle system evolving according to a local mapping in a space of increasing bisequences over Z. We present an algorithm for determination of the local mapping in the space of particle coordinates corresponding to the local mapping f.
{"title":"A class of cellular automata equivalent to deterministic particle systems","authors":"H. Fuks","doi":"10.1090/fic/027/03","DOIUrl":"https://doi.org/10.1090/fic/027/03","url":null,"abstract":"We demonstrate that a local mapping f in a space of bisequences over {0,1} which conserves the number of nonzero sites can be viewed as a deterministic particle system evolving according to a local mapping in a space of increasing bisequences over Z. We present an algorithm for determination of the local mapping in the space of particle coordinates corresponding to the local mapping f.","PeriodicalId":436460,"journal":{"name":"arXiv: Cellular Automata and Lattice Gases","volume":"1 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2002-07-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"130250681","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2000-11-16DOI: 10.1088/0305-4470/34/18/301
J. Gier
An exact solution for a high speed deterministic traffic model with open boundaries and synchronous update rule is presented. Because of the strong correlations in the model, the qualitative structure of the stationary state can be described for general values of the maximum speed. It is shown in the case of $v_{rm max}=2$ that a detailed analysis of this structure leads to an exact solution. Explicit expressions for the stationary state probabilities are given in terms of products of $24times 24$ matrices. From this solution an exact expression for the correlation length is derived.
{"title":"Exact stationary state for a deterministic high speed traffic model with open boundaries","authors":"J. Gier","doi":"10.1088/0305-4470/34/18/301","DOIUrl":"https://doi.org/10.1088/0305-4470/34/18/301","url":null,"abstract":"An exact solution for a high speed deterministic traffic model with open boundaries and synchronous update rule is presented. Because of the strong correlations in the model, the qualitative structure of the stationary state can be described for general values of the maximum speed. It is shown in the case of $v_{rm max}=2$ that a detailed analysis of this structure leads to an exact solution. Explicit expressions for the stationary state probabilities are given in terms of products of $24times 24$ matrices. From this solution an exact expression for the correlation length is derived.","PeriodicalId":436460,"journal":{"name":"arXiv: Cellular Automata and Lattice Gases","volume":"19 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2000-11-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"114906593","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 1995-10-04DOI: 10.1142/9789814447188_0145
G. Ódor, F. Rohrbach, György Vesztergombi, G. Varga, F. Tatrai
In view of the tremendous computing power jump of modern RISC processors the interest in parallel computing seems to be thinning out. Why use a complicated system of parallel processors, if the problem can be solved by a single powerful micro-chip? It is a general law, however, that exponential growth will always end by some kind of a saturation, and then parallelism will again become a hot topic. We try to prepare ourselves for this eventuality. The MPPC project started in 1990 in the heydeys of parallelism and produced four ASTRA machines (presented at CHEP'92) with 4k processors (which are expandable to 16k) based on `yesterday's chip-technology' (chip presented at CHEP'91). These machines now provide excellent test-beds for algorithmic developments in a complete, real environment. We are developing for example fast-pattern recognition algorithms which could be used in high-energy physics experiments at the LHC (planned to be operational after 2004 at CERN) for triggering and data reduction. The basic feature of our ASP (Associative String Processor) approach is to use extremely simple (thus very cheap) processor elements but in huge quantities (up to millions of processors) connected together by a very simple string-like communication chain. In this paper we present powerful algorithms based on this architecture indicating the performance perspectives if the hardware (i.e. chip fabrication) quality reaches `present or even future technology levels'.
{"title":"Parallel Algorithms on the ASTRA SIMD Machine","authors":"G. Ódor, F. Rohrbach, György Vesztergombi, G. Varga, F. Tatrai","doi":"10.1142/9789814447188_0145","DOIUrl":"https://doi.org/10.1142/9789814447188_0145","url":null,"abstract":"In view of the tremendous computing power jump of modern RISC processors the interest in parallel computing seems to be thinning out. Why use a complicated system of parallel processors, if the problem can be solved by a single powerful micro-chip? It is a general law, however, that exponential growth will always end by some kind of a saturation, and then parallelism will again become a hot topic. We try to prepare ourselves for this eventuality. The MPPC project started in 1990 in the heydeys of parallelism and produced four ASTRA machines (presented at CHEP'92) with 4k processors (which are expandable to 16k) based on `yesterday's chip-technology' (chip presented at CHEP'91). These machines now provide excellent test-beds for algorithmic developments in a complete, real environment. We are developing for example fast-pattern recognition algorithms which could be used in high-energy physics experiments at the LHC (planned to be operational after 2004 at CERN) for triggering and data reduction. The basic feature of our ASP (Associative String Processor) approach is to use extremely simple (thus very cheap) processor elements but in huge quantities (up to millions of processors) connected together by a very simple string-like communication chain. In this paper we present powerful algorithms based on this architecture indicating the performance perspectives if the hardware (i.e. chip fabrication) quality reaches `present or even future technology levels'.","PeriodicalId":436460,"journal":{"name":"arXiv: Cellular Automata and Lattice Gases","volume":"106 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1995-10-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"131410166","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
The maximum computational density allowed by the laws of physics is available only in a format that mimics the basic spatial locality of physical law. Fine-grained uniform computations with this kind of local interconnectivity (Cellular Automata) are particularly good candidates for efficient and massive microphysical implementation.
{"title":"CAM-8: a computer architecture based on cellular automata","authors":"N. Margolus","doi":"10.1090/fic/006/13","DOIUrl":"https://doi.org/10.1090/fic/006/13","url":null,"abstract":"The maximum computational density allowed by the laws of physics is available only in a format that mimics the basic spatial locality of physical law. Fine-grained uniform computations with this kind of local interconnectivity (Cellular Automata) are particularly good candidates for efficient and massive microphysical implementation.","PeriodicalId":436460,"journal":{"name":"arXiv: Cellular Automata and Lattice Gases","volume":"14 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1995-09-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"123738369","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
A method is described for calculating corrections to the Boltzmann/Chapman-Enskog analysis of lattice gases due to the buildup of correlations. It is shown that renormalized transport coefficients can be calculated perturbatively by summing terms in an infinite series. A diagrammatic notation for the terms in this series is given, in analogy with the Feynman diagrams of quantum field theory. This theory is applied to an example lattice gas and shown to correctly predict experimental deviation from the Boltzmann prediction.
{"title":"Renormalization of Lattice Gas Transport Coefficients","authors":"B. Boghosian, W. Taylor","doi":"10.1090/fic/006/02","DOIUrl":"https://doi.org/10.1090/fic/006/02","url":null,"abstract":"A method is described for calculating corrections to the Boltzmann/Chapman-Enskog analysis of lattice gases due to the buildup of correlations. It is shown that renormalized transport coefficients can be calculated perturbatively by summing terms in an infinite series. A diagrammatic notation for the terms in this series is given, in analogy with the Feynman diagrams of quantum field theory. This theory is applied to an example lattice gas and shown to correctly predict experimental deviation from the Boltzmann prediction.","PeriodicalId":436460,"journal":{"name":"arXiv: Cellular Automata and Lattice Gases","volume":"18 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1994-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"131607013","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
A subgrid turbulence model for the lattice Boltzmann method is proposed for high Reynolds number fluid flow applications. The method, based on the standard Smagorinsky subgrid model and a single-time relaxation lattice Boltzmann method, incorporates the advantages of the lattice Boltzmann method for handling arbitrary boundaries and is easily implemented on parallel machines. The method is applied to a two-dimensional driven cavity flow for studying dynamics and the Reynolds number dependence of the flow structures. The substitution of other subgrid models, such as the dynamic subgrid model, in the framework of the LB method is discussed.
{"title":"A Lattice Boltzmann Subgrid Model for High Reynolds Number Flows","authors":"S. Hou, J. Sterling, Shiyi Chen, G. Doolen","doi":"10.1090/fic/006/12","DOIUrl":"https://doi.org/10.1090/fic/006/12","url":null,"abstract":"A subgrid turbulence model for the lattice Boltzmann method is proposed for high Reynolds number fluid flow applications. The method, based on the standard Smagorinsky subgrid model and a single-time relaxation lattice Boltzmann method, incorporates the advantages of the lattice Boltzmann method for handling arbitrary boundaries and is easily implemented on parallel machines. The method is applied to a two-dimensional driven cavity flow for studying dynamics and the Reynolds number dependence of the flow structures. The substitution of other subgrid models, such as the dynamic subgrid model, in the framework of the LB method is discussed.","PeriodicalId":436460,"journal":{"name":"arXiv: Cellular Automata and Lattice Gases","volume":"2 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1994-01-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"130588506","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Abstract : Introduced is a lattice-gas with long-range 2-body interactions. An effective inter-particle force is mediated by momentum exchanges. There exists the possibility of having both attractive and repulsive interactions using finite impact parameter collisions. There also exists an interesting possibility of coupling these long-range interactions to a heat bath. A fixed temperature heat bath induces a permanent net attractive interpartide potential, but at the expense of reversibility. Thus the long-range dynamics is a kind of a Monte Carlo Kawasaki updating scheme. The model has a PpT equation of state. Presented are analytical and numerical results for a lattice-gas fluid governed by a nonideal equation of state. The model's complexity is not much beyond that of the FHP lattice-gas. It is suitable for massively parallel processing and may be used to study critical phenomena in large systems.
{"title":"A Lattice-Gas with Long-Range Interactions Coupled to a Heat Bath","authors":"J. Yepez","doi":"10.1090/fic/006/20","DOIUrl":"https://doi.org/10.1090/fic/006/20","url":null,"abstract":"Abstract : Introduced is a lattice-gas with long-range 2-body interactions. An effective inter-particle force is mediated by momentum exchanges. There exists the possibility of having both attractive and repulsive interactions using finite impact parameter collisions. There also exists an interesting possibility of coupling these long-range interactions to a heat bath. A fixed temperature heat bath induces a permanent net attractive interpartide potential, but at the expense of reversibility. Thus the long-range dynamics is a kind of a Monte Carlo Kawasaki updating scheme. The model has a PpT equation of state. Presented are analytical and numerical results for a lattice-gas fluid governed by a nonideal equation of state. The model's complexity is not much beyond that of the FHP lattice-gas. It is suitable for massively parallel processing and may be used to study critical phenomena in large systems.","PeriodicalId":436460,"journal":{"name":"arXiv: Cellular Automata and Lattice Gases","volume":"45 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1993-09-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"123875406","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Intrinsic fluctuations around the solution of the lattice Boltzmann equation are described or modeled by addition of a white Gaussian noise source. For stationary states a fluctuation-dissipation theorem relates the variance of the fluctuations to the linearized Boltzmann collision operator and the pair correlation function.
{"title":"Lattice Boltzmann-Langevin Equations","authors":"J. Dufty, M. Ernst","doi":"10.1090/fic/006/08","DOIUrl":"https://doi.org/10.1090/fic/006/08","url":null,"abstract":"Intrinsic fluctuations around the solution of the lattice Boltzmann equation are described or modeled by addition of a white Gaussian noise source. For stationary states a fluctuation-dissipation theorem relates the variance of the fluctuations to the linearized Boltzmann collision operator and the pair correlation function.","PeriodicalId":436460,"journal":{"name":"arXiv: Cellular Automata and Lattice Gases","volume":"1 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1993-09-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"129175735","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 1993-04-22DOI: 10.1090/psapm/047/1267999
G. Beylkin
Wavelet based algorithms in numerical analysis are similar to other transform methods in that vectors and operators are expanded into a basis and the computations take place in this new system of coordinates. However, due to the recursive definition of wavelets, their controllable localization in both space and wave number (time and frequency) domains, and the vanishing moments property, wavelet based algorithms exhibit new and important properties. For example, the multiresolution structure of the wavelet expansions brings about an efficient organization of transformations on a given scale and of interactions between different neighbouring scales. Moreover, wide classes of operators which naively would require a full (dense) matrix for their numerical description, have sparse representations in wavelet bases. For these operators sparse representations lead to fast numerical algorithms, and thus address a critical numerical issue. We note that wavelet based algorithms provide a systematic generalization of the Fast Multipole Method (FMM) and its descendents. These topics will be the subject of the lecture. Starting from the notion of multiresolution analysis, we will consider the so-called non-standard form (which achieves decoupling among the scales) and the associated fast numerical algorithms. Examples of non-standard forms of several basic operators (e.g. derivatives) will be computed explicitly.
{"title":"Wavelets and Fast Numerical Algorithms","authors":"G. Beylkin","doi":"10.1090/psapm/047/1267999","DOIUrl":"https://doi.org/10.1090/psapm/047/1267999","url":null,"abstract":"Wavelet based algorithms in numerical analysis are similar to other transform methods in that vectors and operators are expanded into a basis and the computations take place in this new system of coordinates. However, due to the recursive definition of wavelets, their controllable localization in both space and wave number (time and frequency) domains, and the vanishing moments property, wavelet based algorithms exhibit new and important properties. \u0000For example, the multiresolution structure of the wavelet expansions brings about an efficient organization of transformations on a given scale and of interactions between different neighbouring scales. Moreover, wide classes of operators which naively would require a full (dense) matrix for their numerical description, have sparse representations in wavelet bases. For these operators sparse representations lead to fast numerical algorithms, and thus address a critical numerical issue. \u0000We note that wavelet based algorithms provide a systematic generalization of the Fast Multipole Method (FMM) and its descendents. \u0000These topics will be the subject of the lecture. Starting from the notion of multiresolution analysis, we will consider the so-called non-standard form (which achieves decoupling among the scales) and the associated fast numerical algorithms. Examples of non-standard forms of several basic operators (e.g. derivatives) will be computed explicitly.","PeriodicalId":436460,"journal":{"name":"arXiv: Cellular Automata and Lattice Gases","volume":"1 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1993-04-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"134221183","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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