臺灣博碩士論文加值系統

在自然界中，機率最高的狀態是能量最低的狀態，機率最高的過程是作用量最小的過程。利用作用量最小原理，Maier跟Stein建立了非保守力場的近零噪聲逃逸速率理論，其刻畫逃逸困難度的作用量即是種Wentzell-Freidlin理論的作用量泛函。最近Feng、Zhang跟Wang對有限小噪聲系統提出另一更廣義的逃逸理論，造成不少迴響，然而它對逃逸的條件跟Maier的有些不同。本篇論文尋求統一兩理論的可能性，將Wang的Hamiltonian嫁接到Maier的作用量泛函理論，使用相空間Hamiltonian方程、作用量Hessian矩陣方程以及WKB振幅傳輸方程，分析討論Maier理論與嫁接理論的結果，並以實例點出可能的問題，此工作結果提供未來檢視、證明、推廣、以及應用此嫁接理論很豐富的資訊。

In nature, the most probable state is the state with the lowest energy, and the most probable process is the process with the least action. Using the principle of least action, Maier and Stein established a escape rate theory for non-conservative force fields in the zero noise limit. Therein, the quantity to measure the difficulty of escape is the action functional of the Wentzell-Freidlin theory. Recently, Feng, Zhang and Wang proposed another more general rate theory for finite small noises under a different escape condition, causing a lot of attention. This thesis looks for the possibility of unifying both theories. We inplanted Wang’s Hamiltonian into Maier’s action functional theory, used the Hamiltonian equations for the phase space dynamics, the Hessian matrix equations for the action evolution, and the transport equation for the WKB amplitude to analyze the results of Maier’s theory and the implanted theory, and furthermore used examples to point out possible problems. The results of this work provided a wealth of information for inspecting, proving, generalizing, and applying this implanted theory in the future study.

摘　　要 I
ABSTRACT II
誌　　謝 III
一、緒論 1
二、噪聲誘導逃逸理論 3
2.1　KRAMERS的逃逸理論 3
2.2　MAIER-STEIN的逃逸理論 6
2.3　FENG-ZHANG-WANG的逃逸理論 18
三、最小作用量路徑 24
3.1　一維力場的相空間古典路徑 24
3.2　二維力場的實空間隨機路徑平均與輔助古典路徑投影 29
3.3　二維系統的相空間古典路徑 34
3.4　二維無旋度力場的古典路徑與作用量 37
3.5　二維有旋度力場的古典路徑與作用量變化量 48
四、LANGEVIN動力學逃逸實驗 52
4.1　逃逸機率實驗 53
4.2　平衡態實驗 56
4.3　非平衡態模擬 68
五、結論 73
附錄 75
參考文獻 78

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