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題名 隨機矩陣方法-方向關係矩陣
Random Matrix Method of Correlation Matrix of Direction作者 黃崚瑋 貢獻者 馬文忠
Ma, Wen Jong
黃崚瑋關鍵詞 隨機矩陣
方向關係
分子動力系統日期 2014 上傳時間 1-Oct-2014 13:15:19 (UTC+8) 摘要 我們研究在三維的空間中，粒子與粒子間的方向關係矩陣，以時間長度為參數，在單一步數中，有三個極大的特徵值，將其三個極大的大小相當的特徵值所對應的特徵向量分別當作三維座標，並且將其繪製在三維圖上，並且用同樣的方法來分析其它複雜流體系統，其中包括純流體系統、通道流體系統以及聚合物鏈系統，我們發現當隨著時間長度的增加，各系統所顯示出來的資訊是不同的，尤其聚合物鏈系統，即使隨著時間序列的增加，其三個極大的特徵值仍獨立於其餘特徵值，而我們分析聚合物鏈粒子以及流體粒子的時間尺度發現是不一樣的，確定其三個極大來源來自於時間尺度不一樣。我們並針對C.M.D.及各個流體系統的特徵值標準差、矩陣元素標準差與時間長度的關係進行分析。
We study the principal components of the correlation matrix of direction cosines of motion between pairs of particles in three dimensions. We carry out a systematic analysis in change of the length of time sequences. In single time step, taking the eigenvector components of the three principle modes as coordinates, we obtain a collection of points of the particle number in a three dimensions mapped space. We analyze the pattern of those points and find that they are confined within a sphere for random matrix as well as in complex fluid systems, which include pure fluid, channel fluid and polymer-fluid mixture. Increasing the length of time sequence, the eigenvalue distribution for each complex fluid system under our study show different feature from that of random matrix. In particular, the three largest eigenvalues remain equally deviating from the rest eigenvalues in polymer-fluid mixture as soon as the fluid is so dilute that its direction relaxation time distinctly different from that of polymer chains and the time sequence is collected over a time regime in between the two time scales. We carried out a detailed analysis on the sequence-length dependence in deviations of entries and of eigenvalues for random matrix and for those complex fluid systems.參考文獻 [ 1] Wen-Jong MA , Chin-Kun HU, J.Phys.Soc.Jpn.Vol.79,NO.2,Feb.(2010) 024005[ 2] Wen-Jong MA , Chin-Kun HU, J.Phys.Soc.Jpn.Vol.79,NO.2,Feb.(2010) 024006[ 3] Wen-Jong MA , Chin-Kun HU,J.Phys.Soc.Jpn.Vol.79,NO.5,May(2010) 104002[ 4] Wen-Jong MA , Chin-Kun HU, J.Phys.Soc.Jpn.Vol.79,NO.10,Oct.(2011) 054001[ 5] Wen-Jong Ma, Lakshmanan K. Iyer,Saraswathi Vishveshwara, Joel Koplik, Jayanth R. Banavar,Phys.Rev. E Jan. vol.15 No.1 (1995)[ 6]N.W. Ashcroft,N.D. Mermin, Solid State Physics 1976[ 7] Jacob Schaefer , Robert Yaris, J. Chem. Phys. 51, 4469 (1969)[ 8] Eugene P. Wigner,Annals of Mathematics Vol.62,NO. 3,Nov. (1955)[ 9]Mehta, M. L. Random Matrices, Amsterdam ; San Diego, CA : Elsevier/Academic Press, 2004,3rd[ 10] Y. Malevergne, D. Sornette Phys. A 331 (2004) 660 – 668[ 11] C. W. J. Beenakker, Rev. Mod. Phys., Vol. 69, No. 3, July 1997[ 12] J. J. Sakurai, Modern Quantum Mechanics (2nd Edition)[ 13] Davey Alba,China`s Tianhe-2 Caps Top 10 Supercomputers, IEEE Spectrum, Jun. 17. 2013[ 14] James Jeffers，James Reinders, Intel Xeon Phi Coprocessor High Performance Programming, Feb.11. 2013[ 15] A. Vladimirov and V. Karpusenko,Heterogeneous Clustering With Homogeneous Code: Accelerate MPI Applications Without Code Surgeryusing Intel Xeon Phi Coprocessors, Oct. 17. 2013[ 16] Lennard-Jones, J. E. (1924), Proc. R. Soc. Lond. A 106 (738): 463–477[ 17] Pearson, K.. Nature. 72, 294. (1905)[ 18] P.K. Pathria,Statistical Mechanics, Third Edition (2007)[ 19] Marjorie G Hahn, Xinxin Jiang ,and Sabir Umarov, J. Phys. A: Math. Theor. 43 (2010) 165208[ 20] C Tsallis, Braz. J. Phys. vol.39 no.2a São Paulo Aug. 2009[ 21] John Wishart, Biometrika ,Vol. 20A, No. 1/2 (Jul., 1928), pp. 32-52[ 22] Silverstein, Jack W. "The smallest eigenvalue of a large dimensional Wishart matrix." The Annals of Probability (1985): 1364-1368.[ 23] Edelman, Alan. "Eigenvalues and condition numbers of random matrices." SIAM Journal on Matrix Analysis and Applications 9.4 (1988): 543-560.[ 24] Nadler, Boaz. "On the distribution of the ratio of the largest eigenvalue to the trace of a Wishart matrix." Journal of Multivariate Analysis 102.2 (2011): 363-371.[ 25] Nagar, Daya K., and Arjun K. Gupta. "Expectations of functions of complex Wishart matrix." Acta applicandae mathematicae 113.3 (2011): 265-288.[ 26] Sandoval Jr, Leonidas, Adriana Bruscato Bortoluzzo, and Maria Kelly Venezuela. Not all that glitters is RMT in the forecasting of risk of portfolios in the Brazilian stock market,Physica A: Statistical Mechanics and its Applications 410 (2014): 94-109.[ 27] Chen, Huan, Yong Mai, and Sai-Ping Li. Analysis of network clustering behavior of the Chinese stock market, Physica A: Statistical Mechanics and its Applications (2014).[ 28] The Oxford handbook of random matrix theory. Oxford University Press, 2011.[ 29] Laloux, L., Cizeau, P., Bouchaud, J. P., & Potters, M. (1999). Noise dressing of financial correlation matrices. Physical Review Letters, 83(7), 1467. 描述 碩士
國立政治大學
應用物理研究所
101755001
103資料來源 http://thesis.lib.nccu.edu.tw/record/#G0101755001 資料類型 thesis dc.contributor.advisor 馬文忠 zh_TW dc.contributor.advisor Ma, Wen Jong en_US dc.contributor.author (Authors) 黃崚瑋 zh_TW dc.creator (作者) 黃崚瑋 zh_TW dc.date (日期) 2014 en_US dc.date.accessioned 1-Oct-2014 13:15:19 (UTC+8) - dc.date.available 1-Oct-2014 13:15:19 (UTC+8) - dc.date.issued (上傳時間) 1-Oct-2014 13:15:19 (UTC+8) - dc.identifier (Other Identifiers) G0101755001 en_US dc.identifier.uri (URI) http://nccur.lib.nccu.edu.tw/handle/140.119/70247 - dc.description (描述) 碩士 zh_TW dc.description (描述) 國立政治大學 zh_TW dc.description (描述) 應用物理研究所 zh_TW dc.description (描述) 101755001 zh_TW dc.description (描述) 103 zh_TW dc.description.abstract (摘要) 我們研究在三維的空間中，粒子與粒子間的方向關係矩陣，以時間長度為參數，在單一步數中，有三個極大的特徵值，將其三個極大的大小相當的特徵值所對應的特徵向量分別當作三維座標，並且將其繪製在三維圖上，並且用同樣的方法來分析其它複雜流體系統，其中包括純流體系統、通道流體系統以及聚合物鏈系統，我們發現當隨著時間長度的增加，各系統所顯示出來的資訊是不同的，尤其聚合物鏈系統，即使隨著時間序列的增加，其三個極大的特徵值仍獨立於其餘特徵值，而我們分析聚合物鏈粒子以及流體粒子的時間尺度發現是不一樣的，確定其三個極大來源來自於時間尺度不一樣。我們並針對C.M.D.及各個流體系統的特徵值標準差、矩陣元素標準差與時間長度的關係進行分析。 zh_TW dc.description.abstract (摘要) We study the principal components of the correlation matrix of direction cosines of motion between pairs of particles in three dimensions. We carry out a systematic analysis in change of the length of time sequences. In single time step, taking the eigenvector components of the three principle modes as coordinates, we obtain a collection of points of the particle number in a three dimensions mapped space. We analyze the pattern of those points and find that they are confined within a sphere for random matrix as well as in complex fluid systems, which include pure fluid, channel fluid and polymer-fluid mixture. Increasing the length of time sequence, the eigenvalue distribution for each complex fluid system under our study show different feature from that of random matrix. In particular, the three largest eigenvalues remain equally deviating from the rest eigenvalues in polymer-fluid mixture as soon as the fluid is so dilute that its direction relaxation time distinctly different from that of polymer chains and the time sequence is collected over a time regime in between the two time scales. We carried out a detailed analysis on the sequence-length dependence in deviations of entries and of eigenvalues for random matrix and for those complex fluid systems. en_US dc.description.tableofcontents CHAPTER 1. 前言 1CHAPTER 2. 理論、方法及儀器 32.1 結構分析 32.1.A. Monomer-monomer Pair Distribution Function g(r) 32.1.B. The Velocity Correlation Distribution Function vc(r) 42.1.C. The Structure Factor 52.2 隨機矩陣理論 62.2.A. 特徵值理論 62.2.B. Wishart matrix 102.2.C. Wigner Semi-circle distribution 122.2.D. Normal distribution 13CHAPTER 3. 模擬及物理分析 143.1 模擬物理系統 143.2 物理分析 203.2.A. 結構分析 203.2.B. 統計分析 273.3 物理分析總結 30CHAPTER 4. 隨機矩陣數值分析 314.1 方向關係矩陣建構 314.2 方向關係矩陣分析 324.2.A. 單步矩陣元素分佈 324.2.B. 單步特徵值、特徵向量分析 344.2.C. 步數演化特徵值、特徵向量分析 424.3 比較系統 494.3.A. 純流體粒子系統 494.3.B. 通道流體粒子 514.3.C. 綜合比較 53CHAPTER 5. 結論 62CHAPTER 6. 引用文獻 64CHAPTER 7. 附錄 667.1 MPI平行運算 667.2 通道流體粒子補充 69 zh_TW dc.format.extent 5307452 bytes - dc.format.mimetype application/pdf - dc.language.iso en_US - dc.source.uri (資料來源) http://thesis.lib.nccu.edu.tw/record/#G0101755001 en_US dc.subject (關鍵詞) 隨機矩陣 zh_TW dc.subject (關鍵詞) 方向關係 zh_TW dc.subject (關鍵詞) 分子動力系統 zh_TW dc.title (題名) 隨機矩陣方法-方向關係矩陣 zh_TW dc.title (題名) Random Matrix Method of Correlation Matrix of Direction en_US dc.type (資料類型) thesis en dc.relation.reference (參考文獻) [ 1] Wen-Jong MA , Chin-Kun HU, J.Phys.Soc.Jpn.Vol.79,NO.2,Feb.(2010) 024005[ 2] Wen-Jong MA , Chin-Kun HU, J.Phys.Soc.Jpn.Vol.79,NO.2,Feb.(2010) 024006[ 3] Wen-Jong MA , Chin-Kun HU,J.Phys.Soc.Jpn.Vol.79,NO.5,May(2010) 104002[ 4] Wen-Jong MA , Chin-Kun HU, J.Phys.Soc.Jpn.Vol.79,NO.10,Oct.(2011) 054001[ 5] Wen-Jong Ma, Lakshmanan K. Iyer,Saraswathi Vishveshwara, Joel Koplik, Jayanth R. Banavar,Phys.Rev. E Jan. vol.15 No.1 (1995)[ 6]N.W. Ashcroft,N.D. Mermin, Solid State Physics 1976[ 7] Jacob Schaefer , Robert Yaris, J. Chem. Phys. 51, 4469 (1969)[ 8] Eugene P. Wigner,Annals of Mathematics Vol.62,NO. 3,Nov. (1955)[ 9]Mehta, M. L. Random Matrices, Amsterdam ; San Diego, CA : Elsevier/Academic Press, 2004,3rd[ 10] Y. Malevergne, D. Sornette Phys. A 331 (2004) 660 – 668[ 11] C. W. J. Beenakker, Rev. Mod. Phys., Vol. 69, No. 3, July 1997[ 12] J. J. Sakurai, Modern Quantum Mechanics (2nd Edition)[ 13] Davey Alba,China`s Tianhe-2 Caps Top 10 Supercomputers, IEEE Spectrum, Jun. 17. 2013[ 14] James Jeffers，James Reinders, Intel Xeon Phi Coprocessor High Performance Programming, Feb.11. 2013[ 15] A. Vladimirov and V. Karpusenko,Heterogeneous Clustering With Homogeneous Code: Accelerate MPI Applications Without Code Surgeryusing Intel Xeon Phi Coprocessors, Oct. 17. 2013[ 16] Lennard-Jones, J. E. (1924), Proc. R. Soc. Lond. A 106 (738): 463–477[ 17] Pearson, K.. Nature. 72, 294. (1905)[ 18] P.K. Pathria,Statistical Mechanics, Third Edition (2007)[ 19] Marjorie G Hahn, Xinxin Jiang ,and Sabir Umarov, J. Phys. A: Math. Theor. 43 (2010) 165208[ 20] C Tsallis, Braz. J. Phys. vol.39 no.2a São Paulo Aug. 2009[ 21] John Wishart, Biometrika ,Vol. 20A, No. 1/2 (Jul., 1928), pp. 32-52[ 22] Silverstein, Jack W. "The smallest eigenvalue of a large dimensional Wishart matrix." The Annals of Probability (1985): 1364-1368.[ 23] Edelman, Alan. "Eigenvalues and condition numbers of random matrices." SIAM Journal on Matrix Analysis and Applications 9.4 (1988): 543-560.[ 24] Nadler, Boaz. "On the distribution of the ratio of the largest eigenvalue to the trace of a Wishart matrix." Journal of Multivariate Analysis 102.2 (2011): 363-371.[ 25] Nagar, Daya K., and Arjun K. Gupta. "Expectations of functions of complex Wishart matrix." Acta applicandae mathematicae 113.3 (2011): 265-288.[ 26] Sandoval Jr, Leonidas, Adriana Bruscato Bortoluzzo, and Maria Kelly Venezuela. Not all that glitters is RMT in the forecasting of risk of portfolios in the Brazilian stock market,Physica A: Statistical Mechanics and its Applications 410 (2014): 94-109.[ 27] Chen, Huan, Yong Mai, and Sai-Ping Li. Analysis of network clustering behavior of the Chinese stock market, Physica A: Statistical Mechanics and its Applications (2014).[ 28] The Oxford handbook of random matrix theory. Oxford University Press, 2011.[ 29] Laloux, L., Cizeau, P., Bouchaud, J. P., & Potters, M. (1999). Noise dressing of financial correlation matrices. Physical Review Letters, 83(7), 1467. zh_TW
