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中文論文名稱 一維Fokker-Planck方程式之微擾近似解法
英文論文名稱 On perturbative approach to the one-dimensional Fokker-Planck equation
校院名稱 淡江大學
系所名稱(中) 物理學系碩士班
系所名稱(英) Department of Physics
學年度 95
學期 2
出版年 96
研究生中文姓名 戴彥民
研究生英文姓名 Yan-Min Dai
學號 694180224
學位類別 碩士
語文別 英文
口試日期 2007-06-15
論文頁數 38頁
口試委員 指導教授-何俊麟
委員-何俊麟
委員-曾文哲
委員-施華強
中文關鍵字 布朗運動  Fokker-Planck方程式  微擾近似解法 
英文關鍵字 Brownian motion  Fokker-Planck equation  perturbative approach 
學科別分類 學科別自然科學物理
中文摘要 本篇論文應用微擾法來解某些Fokker-Planck方程式,而這種Fokker- Planck方程式具有擴散係數是常數以及漂移係數是很小的參數的特徵。我們在第二章敘述了Einstein如何用隨機的方法解釋布朗運動;在Einstein之後,Fokker和Planck又加上了外加力場的條件來推廣這個擴散方程式。在這章中,我們舉了一個例子說明外加力場如何影響布朗粒子的擴散行為。在第三章,我們以本徵值展開法來解Fokker-Planck方程式,同時並舉了兩個例子來說明這種方法的實用性。

上述處理問題的假設,都是在時間條件獨立的情況下達成,而Fokker-Planck方程式的解也並不容易得到。很明顯的如果我們討論的問題加入了時間變數,則要得到精確解更是困難。所以我們在論文最後介紹一種新的方法來處理某一種Fokker-Planck方程式,這個方法就是微擾近似法。我們發現,當外加力場具有某些時間的形式時,應用微擾近似法來解Fokker-Planck方程式可以得到精確解,同樣地,我們也用兩個例子來說明微擾近似法的實用性。
英文摘要 The purpose of this thesis is to present a direct perturbative method to solving certain Fokker-Planck equation, which have constant diffusion coefficients and some small parameters in the drift coefficients. In the second chapter, we describe how Einstein explained Brownian movement by stochastic method. After Einstein, Fokker and Planck distributed this diffusion equation with the condition of extra force field. In this chapter, we have an example to illustrate how the extra force field influences the diffusion of Brown particles. In chapter three, we solve the Fokker- Plank equation by the method of eigen- value expansion. We have two examples to describe the practicability of this method.

It is not always easy to solve Fokker-Planck equations with time-independent diffusion and drift coefficients. Needless to say, finding exact solution of the Fokker- Planck equation with time-dependent drift coefficients is also very difficult in general. In final chapter, we will discuss a perturbation method to solve the one-dimension Fokker-Planck equation with a constant diffusion coefficient and time-dependent drift coefficient. Two examples are used to show that such perturbative method can be a useful way to obtain the approximate solutions of the Fokker-Planck equation.
論文目次 1 Introduction..........................................................................................................1
2 Brownian motion and Fokker-Planck equation...................................................................3
§ 2.1 Introduction..............................................................................................................3
§ 2.2 Diffusion equation................... ................................................................................3
§ 2.3 Fokker-Planck equation..........................................................................................10
§ 2.4 Uhlenbeck-Ornstein process...................................................................................12
3 Schrodinger-like equation associated with the Fokker-Planck equation........................15
§ 3.1 Introduction............................................................................................................15
§ 3.2 Stationary solution......................................... ...... ................................................16
§ 3.3 Transformation of the Fokker-Planck equation......... ......... .................................18
§ 3.4 Example 1: Parabolic potential for the Schrodinger potential......... .... ................21
§ 3.5 Example 2: Infinite square Well for the Schrodinger potential......... .... ..............23
4 On a perturbative approach to the one-dimensional Fokker-Planck equation........ .... ....26
§ 4.1 Introduction......... .... ................. ......... .... ...... .. .... ...................... ......... .... .......26
§ 4.2 Hermitian operator ...... .... ................. ......... .... ...... ......... .... ........................26
§ 4.3 Perturbative approximation...... .... ................. ......... .... ...... ......... .. ...................28
§ 4.4 Example1: ...... .... ................. ......... ......... ......... .... ...............29
§ 4.5 Example 2: A perturbative approach to the Uhlenbeck-Ornstein process. ...........35
5 Summary........ .... ................. ......... .... ...... .. .... ........................ ......... .... .....................37
References........ .... ................. ......... .... ...... .. .... ........................ ......... .... ...................38
參考文獻 [1] R. Brown, Phil. Mag. 4, 171 (1828).
[2] A. Einstein, Ann. Phys. (Leipzig), 17, 549, (1905) ; ibid. 19, 371 (1906).
[3] Perrin, J., Comptes Rendus, 158, 1168-1171, 1914.
[4] P. Langevin: Comptes rendus 146, 530 (1908).
[5] A. D. Fokker: Ann. Phys. 43, 810 (1914).
[6] M. Planck: Sitzber. PreuB. Akad. Wiss. p. 324 (1917).
[7] H. Risken, The Fokker-Planck Equation (2nd. ed.) (Springer-Verlag, Berlin, 1996).
[8] G. E. Uhlenbeck and L. S. Ornstein: Phys. Rev. 36, 823 (1930).
[9] Deb Shankar Ray, Notes on Brownian motion and related phenomena, arXiv:9903033.
[10] C.-L. Ho and Y.-M. Dai, A perturbative approach to a class of Fokker-Planck
equations, arXiv:0705.2798.
[11] C.-L. Ho and R. Sasaki, Quasi-exactly solvable Fokker-Planck equations, arXiv:0705.0863.
[12] B. Caroli, C. Caroli, and B. Roulet. J. Stat. Phys. 21, 415( 1979).
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