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.

CHAPTER 1. 前言 1
CHAPTER 2. 理論、方法及儀器 3
2.1 結構分析 3
2.1.A. Monomer-monomer Pair Distribution Function g(r) 3
2.1.B. The Velocity Correlation Distribution Function vc(r) 4
2.1.C. The Structure Factor 5
2.2 隨機矩陣理論 6
2.2.A. 特徵值理論 6
2.2.B. Wishart matrix 10
2.2.C. Wigner Semi-circle distribution 12
2.2.D. Normal distribution 13
CHAPTER 3. 模擬及物理分析 14
3.1 模擬物理系統 14
3.2 物理分析 20
3.2.A. 結構分析 20
3.2.B. 統計分析 27
3.3 物理分析總結 30
CHAPTER 4. 隨機矩陣數值分析 31
4.1 方向關係矩陣建構 31
4.2 方向關係矩陣分析 32
4.2.A. 單步矩陣元素分佈 32
4.2.B. 單步特徵值、特徵向量分析 34
4.2.C. 步數演化特徵值、特徵向量分析 42
4.3 比較系統 49
4.3.A. 純流體粒子系統 49
4.3.B. 通道流體粒子 51
4.3.C. 綜合比較 53
CHAPTER 5. 結論 62
CHAPTER 6. 引用文獻 64
CHAPTER 7. 附錄 66
7.1 MPI平行運算 66
7.2 通道流體粒子補充 69

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