隨機過程的一些研究

Abstract

1.平均首度穿越時間 本章以在位能場中以及擴散係數隨座標改變的擴散方程式為始，求其adjoint equation，並討論吸收、反射以及輻射各種邊界條件下所對應的數學表示式。其次定義出平均首度穿越時間 (mean first passage time, MFPT)，並導出MFPT所滿足的方程式為上述的adjoint equation。 2.平均滯留時間 本章的目的在定義平均滯留時間 (mean residence time, MRT)。此外，依照Agmon的方式 [N. Agmon, J. Chem. Phys. 81, 3644 (1984)]，我們舉出兩個在線性位能場下擴散的例子，並運用格林函數與傳輸矩陣來求解MRT。其結果說明我們可利用傳輸矩陣來處理各種邊界與位能場的問題。 3.在Master Equation下之平均滯留時間，平均首度穿越時間與延遲時間 分子在各狀態間的分佈隨時間的改變可以master equation來表示。我們將master equation寫成矩陣的形式，經過Laplace轉換，可求得穿越時間、滯留時間與延遲時間之解，並以reciprocal bilinear form表示出來。 4.Kramers' Theory 在溶液中的化學反應可以沿著反應座標之一維位能曲線來表示。反應物獲得足夠的能量，翻越能障後在另一端形成穩定的生成物。Kramers [H. A. Kramers, Physica 7, 284 (1940)] 將分子越過能障的行為視作與布朗運動相同，可用Langevin equation來描述。本章以Langevin equation為始，推導出Kramers對反應速率的表示式，並以proton transfer為例，與由MFPT所得之反應速率常數值做比較。

1. Mean First Passage Time In this chapter we are concerned the diffusion processes under the influence of a forced field i.e., a Smoluchowski-type diffusion equation with a space-dependent diffusion constant. The corresponding adjoint equation is derived for absorbing, reflecting and radiative boundary con-ditions. Then we derive the equation for the mean first passage time (MFPT) is the adjoint equation just mentioned. 2. Mean Residence Time The subject of this chapter is to present the mean residence time (MRT) theory. Following Agmon's treatment [N. Agmon, J. Chem. Phys. 81, 3644 (1984)], we perform the calculations of MRT under the influ-ence of a linear potential field in two ways, Green's function and trans-mission matrix. The result demonstrates that diffusion subject to various kinds of boundary conditions and potential fields may be treated by transmission matrix. 3. The Mean Residence Time, Mean First Passage Time and Time Lag treated by Master Equation We write the master equation in the matrix form and obtain the solu-tions of mean passage time, residence time and time lag in reciprocal bi-linear form in Laplace domain. 4. Kramers' Theory The chemical reaction in solution can be visualized by the motion along the reaction coordinate over an energy barrier. The reactant species start in the well, crossover the barrier and finally form products on the other side. Kramers [H. A. Kramers, Physica 7, 284 (1940)] viewed a re-action as a barrier crossing influenced by interactions with surrounding solvent molecules. Passage across the barrier then appears similar to Brownian motion, which is described via the Langevin equation. The course of derivation of rate constant based on Kramers' theory is pre-sented. We take proton transfer for an example and compare the rate con-stant derived from Kramers' theory with the rate constant derived from MFPT.