移動邊界截斷格點法(TG)，先前開發用於計算相空間中分佈函數的時間演化。此篇論文將TG用於斯莫魯霍夫斯基方程式(Smoluchowski equation)，計算位置空間中機率密度分佈函數隨時間演化。隨著機率密度分佈函數隨時間的演化而適當地決定TG的邊界，在每個時部都使用不同數量的位置空間網格點用於對機率密度分佈函數的運動方程進行積分。並且TG不需要與動力學有關的預先信息，就能適當地啟用和停用網格點來演化分佈函數。全網格計算結果與TG計算結果的比較顯示，TG不僅顯著減少了計算量，而且精確模擬位置空間中機率密度分佈函數隨時間演化。
在本研究中，我們探討了有解析解的系統，可以直接檢驗TG演算法的準確度，如純擴散系統、線性位能系統、簡諧震盪位能系統，接著再應用在沒有解析解的系統，如雙位能井系統、質子轉移系統，分別計算有無量子修正的情形，在雙位能井系統裡，斯莫魯霍夫斯基方程式相當於是克萊茵－克喇莫方程式降維度後的算法，配合TG後可以更加有效率且準確地計算出系統在過阻尼時的速率常數，在量子修正的案例中，雖然量子修正的影響不大，但是TG仍可以相當準確地捕捉這微小的差異，同時省去時間。這些計算結果表明，TG方法不僅大大減少了精確計算所需的網格點數量，而且比全網格計算的計算效率更高。

Moving Boundary Truncated Grid (TG) method, previously developed for the computation of the time evolution of distribution functions in phase space. This thesis applies TG to the Smoluchowski equation, calculating the temporal evolution of probability density distribution functions in position space. By appropriately determining the boundaries of TG in accordance with the temporal evolution of the probability density distribution function, a varying number of grid points in position space are used at each time step to integrate the motion equation of the probability density distribution function. Furthermore, TG enables the activation and deactivation of grid points for the evolution of the distribution function without requiring any a priori information related to the dynamics. A comparison between the results obtained from full-grid calculations and TG calculations shows that TG not only significantly reduces computational effort but also accurately simulates the temporal evolution of probability density distribution functions in position space.
In this study, we investigate systems with analytically solutions, which allow for a direct assessment of the accuracy of the TG algorithm. Examples include pure diffusion systems, linear potential systems, and harmonic oscillator potential systems. We then apply the method to systems without analytically solutions, such as double-well potential systems and proton transfer systems, considering both quantum-corrected and non-quantum-corrected conditions. In the case of the double-well potential system, the Smoluchowski equation corresponds to an algorithm derived from the Klein-Kramers equation after dimensionality reduction. When combined with TG, it becomes more efficient and accurate in calculating the rate constant of the system under over-damping conditions. In the quantum-corrected cases, although the impact of quantum corrections is small, TG still captures these slight differences with high accuracy while saving time. These computational results demonstrate that the TG method not only significantly reduces the number of grid points required for accurate calculations but also exhibits higher computational efficiency compared to full-grid calculations.

摘要 1
Abstract 2
目錄 4
圖表目錄 6
圖目錄 6
表目錄 8
第一章 緒論 10
第二章 理論介紹和原理 11
2.1 相空間 (Phase space) 11
2.2 哈密頓正則方程式 (Hamiltonian canonical equations) 11
2.3 牛頓力學 (Newtonian mechanics) 12
2.4 劉維爾方程 (Liouville equation) 13
2.5 朗之萬動力學 (Langevin dynamics) 15
2.6 克萊茵－克喇莫方程式 (Klein-Kramers equation) 18
2.7 斯莫魯霍夫斯基方程式 (Smoluchowski equation) 19
2.8 量子斯莫魯霍夫斯基方程式 (Quantum Smoluchowski equation) 21
第三章 計算方法 23
3.1 機率密度分布演化之演算法 23
3.2 移動邊界截斷格點法(moving boundary truncated grid ,MBTG) 26
第四章 研究成果 28
4.1 純擴散系統 28
4.2 線性位能系統 33
4.3 簡諧震盪位能系統 40
4.4 雙位能井系統 48
4.5 質子轉移系統 54
第五章 結論 59
第六章 參考文獻 60

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