Ising模型的数值模拟及FPGA实现
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摘要
相变是自然界中广泛存在的一类突变现象，是物质系统由一种稳定状态向另一种稳定状态的跃迁过程，是物理学中极为重要的研究领域。Ising模型是从磁性系统中抽象出来的，描述物质相变与临界现象的一种简单模型。用软件实现Ising模型的模拟速度慢，但用硬件实现具有速度快的特点，因此采用硬件实现Ising模型的模拟具有重要价值和意义。本文主要应用蒙特卡罗方法和元胞自动机方法对Ising模型的研究结果进行分析和总结，用FPGA设计和实现了Ising模型的元胞自动机模拟，并对实验结果进行了对比分析。
论文首先介绍了近几年国内外Ising模型的研究现状和一维、二维模型的精确解，分析了在研究三维Ising模型精确解过程中遇到的困难，并用蒙特卡罗数值模拟方法和重整化群方法对Ising模型研究结果进行了总结。
其次研究了Ising模型的元胞自动机数值模拟。系统阐述了元胞自动机及对物理系统的模拟，详细介绍了Ising模型的元胞自动机模拟的原理和方法，对用软件实现的二维Ising模型的元胞自动机模拟结果进行了分析和总结，得到了物理系统发生相变及形成宏观磁化的系统能量临界值。
最后用原理图输入法和Verilog HDL语言相结合的设计方法实现了一维和二维Ising模型的元胞自动机模拟，并用QuartusII软件进行综合仿真，下载到开发板中。二维仿真结果与用软件方法得到的结果进行比较，结果表明FPGA并行实现Ising模型的速度比用软件方法的速度快。
Phase transition, is a kind of break phenomena which exits in the nature widely, is akind of transition process in which substance system changes from one steady state toanother, and is an important research area in Physics. Ising model is abstracted from themagnetic system,which describes a simple model of the material phase transitions andcritical phenomena.Compared to software implementation, implementation throughhardware is faster.Therefore, adopting the method of hardware to complete the simulationof Ising model has a more important value and significance. This paper mainly analyzesthe results of Ising model which is generated by Monte Carlo's method and cellularautomata method,then designs and realizes the simulation of cellular automata of Isingmodel by FPGA. At last,it compares and analyzes the experiment result.
Firstly,it presents domestic and overseas research on Ising model at present and alsointroduces the exact solution to one-dimensional and two-dimensional models,thenanalyzes the difficulties in the study on the exact solution to three-dimensional model.Moreover, numerical simulation method set by Monte Carlo and renormalization groupmethod are used to summarize the results of work on Ising model.
Secondly, the numerical simulation of cellular automata of Ising model are discussedin this paper. The cellular automata and simulation of physical systems are formulated.Furthermore, the principles and methods of cellular automata of Ising model are describedin detail. Based on the information obtained from the cellular automata's simulation resultsof two-dimensional Ising model which achieved through software,this article gets thecritical value of system energy when the phase transformations and macroscopicmagnetization are generated.
Finally,with the method combining schematic input method and Verilog HDLlanguage,it achieves the cellular automata's simulation to one-dimensional andtwo-dimensional models,then uses QuartusⅡ software to realize the comprehensivesimulation and download to the evaluation board. Through the comparison of the resultsbetween two-dimensional simulation and software method,it demonstrates thatimplementation of parallel method is superior to that of software method.
论文首先介绍了近几年国内外Ising模型的研究现状和一维、二维模型的精确解，分析了在研究三维Ising模型精确解过程中遇到的困难，并用蒙特卡罗数值模拟方法和重整化群方法对Ising模型研究结果进行了总结。
其次研究了Ising模型的元胞自动机数值模拟。系统阐述了元胞自动机及对物理系统的模拟，详细介绍了Ising模型的元胞自动机模拟的原理和方法，对用软件实现的二维Ising模型的元胞自动机模拟结果进行了分析和总结，得到了物理系统发生相变及形成宏观磁化的系统能量临界值。
最后用原理图输入法和Verilog HDL语言相结合的设计方法实现了一维和二维Ising模型的元胞自动机模拟，并用QuartusII软件进行综合仿真，下载到开发板中。二维仿真结果与用软件方法得到的结果进行比较，结果表明FPGA并行实现Ising模型的速度比用软件方法的速度快。
Phase transition, is a kind of break phenomena which exits in the nature widely, is akind of transition process in which substance system changes from one steady state toanother, and is an important research area in Physics. Ising model is abstracted from themagnetic system,which describes a simple model of the material phase transitions andcritical phenomena.Compared to software implementation, implementation throughhardware is faster.Therefore, adopting the method of hardware to complete the simulationof Ising model has a more important value and significance. This paper mainly analyzesthe results of Ising model which is generated by Monte Carlo's method and cellularautomata method,then designs and realizes the simulation of cellular automata of Isingmodel by FPGA. At last,it compares and analyzes the experiment result.
Firstly,it presents domestic and overseas research on Ising model at present and alsointroduces the exact solution to one-dimensional and two-dimensional models,thenanalyzes the difficulties in the study on the exact solution to three-dimensional model.Moreover, numerical simulation method set by Monte Carlo and renormalization groupmethod are used to summarize the results of work on Ising model.
Secondly, the numerical simulation of cellular automata of Ising model are discussedin this paper. The cellular automata and simulation of physical systems are formulated.Furthermore, the principles and methods of cellular automata of Ising model are describedin detail. Based on the information obtained from the cellular automata's simulation resultsof two-dimensional Ising model which achieved through software,this article gets thecritical value of system energy when the phase transformations and macroscopicmagnetization are generated.
Finally,with the method combining schematic input method and Verilog HDLlanguage,it achieves the cellular automata's simulation to one-dimensional andtwo-dimensional models,then uses QuartusⅡ software to realize the comprehensivesimulation and download to the evaluation board. Through the comparison of the resultsbetween two-dimensional simulation and software method,it demonstrates thatimplementation of parallel method is superior to that of software method.
引文
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[3]冯丽雅，辛子华，王吴韬.一维钻石链反铁磁Ising模型磁化的模拟[J].计算物理，2010，4(27):613-618.
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[16] Stefan Kirchner, Qimiao Si, Kevin Ingersent. Finite-Size Scaling of Classical Long-RangedIsing Chains and the Criticality of Dissipative Quantum Impurity Models[J]. Physical ReviewLetters,2009,16(102):166405-1-166405-4.
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[3]冯丽雅，辛子华，王吴韬.一维钻石链反铁磁Ising模型磁化的模拟[J].计算物理，2010，4(27):613-618.
[4]陈小余.三维Ising模型矩阵解法的简化与近似解[J].物理学报，1995，9(44):1484-1488.
[5] Bastien Chopard, Michel Droz.物理系统的元胞自动机模拟[M].祝玉学，赵学龙译.北京：清华大学出版社，2003:1-46.
[6]张志东.伊辛模型的研究进展简介[J].自然杂志(科技进展)，2008,2(30):94-101.
[7]郑荣森，谭惠丽，秦继民.二维伊辛模型相变临界现象的元胞自动机模拟[J].大学物理2005，11(24):10-13.
[8]李晓寒，王宗笠，马显光，等.二维Ising模型元胞自动机(Q2R)模拟[J].安徽大学学报（自然科学版），2007，1(31):57-60.
[9]李立奇，王宗笠.二维微正则Ising模型临界相变的Q2R方法[J].重庆大学学报(自然学版)，2006,11(29):115-118.
[10]雷晓蔚，郑波，应和平.二维自旋系统aging现象的数值模拟研究[J].物理学报，2007，3(56):1713-1718.
[11]邵元智，钟伟荣，林光明，等.随机外磁场作用下Ising自旋体系的随机共振[J].物理学报，2004，9(53):3157-3164.
[12] Bastien Chopard, Alexandre Masselot. Cellular automata and lattice Boltzmann methods: anew approach to computational fluid dynamics and particle transport[J]. Future GenerationComputer Systems,1999,16(2-3):249-257.
[13] Zorzenon dos Santos RM, Coutinho S. A Cellular Automata Approch[J]. Phys. Rev. Lett.,2001,87(16):102-104.
[14] W. M. Lang, D. Stauffer. Test of three-dimensional Q2R Ising algorithm[J]. J. Phys. A.:Math.Gen.1987,20:5413-5415.
[15] Zheng-jian Bai. Inexact Numerical Methods for Inverse Eigenvalue Problems[J]. Math.Comput.2006,172:682-689.
[16] Stefan Kirchner, Qimiao Si, Kevin Ingersent. Finite-Size Scaling of Classical Long-RangedIsing Chains and the Criticality of Dissipative Quantum Impurity Models[J]. Physical ReviewLetters,2009,16(102):166405-1-166405-4.
[17] H. J. Herrmann, H. O. Carmesin, D. Stauffer. Periods and Clusters in Ising cellula automata[J].J. Phys. A: Math. Gen.,1987,20:4939-4948.
[18]张祥，陈冬保，陈武鸣.二维伊辛模型蒙特卡罗模拟[J].南京大学学报，1997,1(33):137-141.
[19] Martin Weigel. Ising, Heisenberg and spin glass model simulations on GPU[J]. TheoretischePhysik.2009,11:1-39.
[20] Ken,ichi Takano, Hidenori Suzuki, Kazuo Hida, et al. Exact spin-cluster ground states in amixed diamond chain[J]. Physical Review B.2009,10(80):1-4.
[21] T. Hofsass, W. Janke, H. Kleinert. From Self-avoiding Random Loops to Ising Systems anInterpolating Spin Model and its Monte Carlo Study[J]. Physics Letters.1994,9(105A):463-467.
[22] Tai Tsun Wu.Concept of nonintegrable phase factors and global formulation of gague fields[J].Physical Review D.1975,12(12):3845-3857.
[23] Iwo Bialynicki-Birula. Weyl, Dirac, and Maxwell equations on a lattice as unitary cellularautomata[J]. Physical Review D.1994,12(49):6920-6927.
[24] B. Hedet, H. J. Hemnann. Fast simulation of the Ising model using cellular automata[J].J.Phys. A.: Math. Gen.1991,24: L691-L697.
[25] Carlos P. Herrero. Ising model in small-world networks[J]. Phy.Rev.E.,2002,6(65):1103-1113.
[26] Kazuo Hida,Kenichi Takano and Hidenori Suzuki. Finite Temperature Properties ofMixedDiamond Chain with Spins1and1/2[J]. J.Phys.Soc.Jpn.,2009,78(084716):1-23.
[27] J. A. De Sales, M. L. Martins, D. A. Stariolo. Cellular automata model for genenetwork[J].Physical Review E,1997,55(3):3262-3270.
[28] Zhang Zhi Dong. Conjectures on exact solution of three-dimensional(3D) simpleorthorhombic Ising lattices[J]. Philosophical Magazine,2007,87:5309-5419.
[29] R.R.Montenegro-Filho and M.D.Coutinho-Filho. Frustration-induced quantum phasetransitions in quasi-one-dimensional ferrimagnet Hard-core boson map and theTonks-Girardeau limit[J]. Phys. Rev.B,2008,1(78):1-19.
[30] Weng Zhenzhen, Feng Qian, Xiao Yan, et al. Micromagnetism and MonteCarlo simulation ofmagnetism of a two-dimension diluted system[J]. Chinese Journal of Computational Physis,2004,21(3):339-345.
[31] F.Belletti, M.Cotallo, A.Cruz, et al. Simulating spin systems on IANUS, an FPGA_basedcomputer[J]. Computer Physics Communications,2008,178(3):208-216.
[32]吴大鹏.元胞自动机在电磁场模拟中的应用及硬件实现[D].秦皇岛：燕山大学电路与系统学科硕士学位论文，2008:15-25.
[33]王仲君，王能超，冯飞，等.元胞自动机的演化行为研究[J].计算机应用研究，2007，24(8):38-41.
[34] JavierValls，MartinKuhlmann and Keshab K.Parhi. Evaluation of CORDIC Algorithms forFPGA Design[J]. The Journal of VLSI Signal Processing.,2002,32(3):207-222.
[35] Ken Arnold. Embedded Controller Hardware Design[M]. LLH Technology Publishing,2000:13-93.
[36]王诚，吴继华，范丽珍，等. Altera FPGA/CPLD设计（基础篇）[M].北京：人民邮电出版社，2011:1-45.
[37]王诚，吴继华. Altera FPGA/CPLD设计(高级篇)[M].北京：人民邮电出版社，2011：70-115.
[38]潘松，黄继业. EDA技术与VHDL[M].北京：清华大学出版社，2009:69-89.
[39]王超，张军青，徐传进.基于FPGA的EMT系统设计[J].天津大学学报，2011，44(2):95-100.
[40] Bhasker, Jayaram. Verilog HDL入门[M].夏宇闻，甘伟译.北京：北京航空航天大学出版社，2008:2-265.
[41] Peter J. Ashenden. Verilog嵌入式数字系统设计教程：an embedded systems approach usingVerilog/(澳)[M].夏宇闻，夏嘉宁，等译.北京：北京航空航天大学出版社，2009:40-120.
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