以數值模擬探討一維電漿的熱鬆弛過程

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蔡承宸(Cheng-Chen Tsai) 查詢紙本館藏

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物理學系

論文名稱

以數值模擬探討一維電漿的熱鬆弛過程
(Numerical Study of Thermal Relaxation in One-Dimensional Plasmas)

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摘要(中)

一維的電漿模型常被用作研究真實電漿的熱力學性質或遍歷行為；一維的電漿薄板模型 (plasma sheet model, PSM) 即為一個多粒子系統的典型例子。此篇論文所論及的電漿薄板模型，將薄板視作電子的類比，而離子則被簡化成固定不動的均勻背景，所有的薄板只能在該背景之中活動。

熱鬆弛是熱力學系統由非熱平衡狀態趨向熱平衡的過程。倘若僅考慮雙粒子之間的相關性，由若干相同的一維粒子所構成的系統，並不具有熱鬆弛的現象。然如先前所述的電漿薄板模型，由於薄板與薄板之間或薄板與背景之間的庫倫作用力的緣故，當薄板的初始速度的機率密度分布，呈現均勻狀或拋物線形，其速度分布均會演變成馬士威爾分布，即該系統會趨向熱平衡。此庫倫作用力可當作三體粒子之間的相關性。
此篇論文涵蓋的議題如下：

一、演算法對系統總能量的影響。
程式採取的演算法是將時間先行離散化，再計算各個時間區段的薄板軌跡。
因為解出的軌跡是近似解，因此有必要分析區段的長度與總能量變化的關
係，結果是總能量變化正比於區段長度的三次方。

二、達成熱平衡時，電位能與動能的比值。

三、由薄板的速度計算而得的卡方統計量，隨時間的變化。
卡方統計量除了針對薄板的速度分布會演變成馬士威爾分布的論點給出
統計學的依據之外，其隨時間的變化尚有助於測量熱鬆弛的特徵時間。卡方
統計量的漸進行為是指數衰減。

四、熱鬆弛的特徵時間與nD的相依性。
利用卡方統計量所測得的熱鬆弛的特徵時間與nD 2 大致成正比，同Dawson
的結論相吻合。鑒於兩種初始速度分布，不同於Dawson僅使用均勻分布，
遂採用次方擬合以解析特徵時間與nD 的某次方的正比關係，與初始速度分
布之間的關聯。拋物線形對應的次方比均勻狀的更加接近2，原因或許為前
者較後者更像馬士威爾分布。

摘要(英)

One-dimensional plasma models exhibit some thermodynamic properties or ergodic behavior of real plasmas; of many-body systems, a one-dimensional plasma model, consisting of a number of identical charged sheets embedded in a uniform fixed neutralizing background, is such a representative one.

Thermal relaxation is an evolution during which a thermodynamic system approaches to thermal equilibrium. If a system of identical one-dimensional particles takes into account only two-particle correlations, no thermal relaxation takes place. Rather, due to the long-range forces among the background and the sheets themselves, as the initial velocity of sheets is either square or quadratic, the velocity of sheets will relax to Maxwellian; the system will attain to thermal equilibrium.

The topics in this thesis:

1. Energy conservation in the numerical scheme
The program deals with the discretized time to solve the approximated traces of sheets; therefore, it is necessary to examine the influence of the period t of a time interval on the total energy per sheet. It turns out that the total energy per sheet is dependent on the cube of the time increment.

2. The ratio of the electric energy per sheet to the kinetic energy per sheet in thermal equilibrium

3. Time development of the 2 statistic constructed by the velocities of sheets
In addition to provide statistical evidence that the velocity distribution of sheets relaxes to Maxwellian, the chi-squared statistic measures the thermal relaxation time, decaying exponentially asymptotically.

4. The dependence of the thermal relaxation time R on nD
By use of the quadratic fitting, R seems to depend on the square of nD, and it is consistent with Dawson’s conclusion. With the power fitting, it is found that the exponent for the quadratic profile is closer to 2 than that for the square profile.

關鍵字(中)

★ 一維電漿薄板模型
★ 熱鬆弛

關鍵字(英)

★ plasma sheet model
★ thermal relaxation

論文目次

Chapter I Introduction
I.1 Thermal relaxation…2
I.1.1 The phase space of a classically-mechanical system…2
I.1.2 Ergodic hypothesis…3
I.1.3 Thermal equilibrium…5
I.2 Thermalization in the systems of particles through short-range forces…6
I.2.1 A system of two-dimensional hard balls through short-range forces…6
I.2.2 A system of point particles through short-range forces…7
I.2.3 An application of the mean ergodic theorem…8
I.3 Main goals of the thesis…12

Chapter II One-Species, One-Dimensional Plasma Sheet Model
II.1 Equations of motion…13
II.2 Plasma frequency…15
II.3 Behavior of sheets…16
II.4 Debye length…17
II.5 Thermal velocity…18
II.6 Normalized units…20
II.7 Hypothesis testing to thermalization…20
II.7.1 Dawson’s approach…21
II.7.2 Chi-squared goodness-of-fit test……21

Chapter III Algorithms
III.1 The crossing between two adjacent sheets…25
III.2 The approximated crossing time…27
III.3 Correlations of the crossings…28
III.4 The detection to the crossings…30
III.4.1 The pair-wise method…30
III.4.2 The two-length-scale method…31
III.4.3 The divide-and-conquer method…31
III.5 Updates of the states of sheets…34
III.6 The initial velocity distribution of sheets…35
III.7 Pseudo codes…36

Chapter IV Numerical Results and Conclusions
IV.1 Energy conservation in the numerical scheme…43
IV.2 The ratio of the electric energy to the kinetic energy in thermal equilibrium…45
IV.3 Time development of the velocity distribution of sheets…46
IV.4 The thermal relaxation time…48
IV.5 Conclusions…50

References…54

參考文獻

1. John M. Dawson, “One-Dimensional Plasma Model,” Phys. Fluids, Vol. 5, No. 4, 445 (1962)
2. John M. Dawson, “Thermal Relaxation in a One-Species, One-Dimensional Plasma,” Phys. Fluids, Vol. 7, No. 3, 419 (1964)
3. Wackerly, Mendenhall, Scheaffer, “Mathematical Statistics with Applications, 7th Ed.,” Thomson Learning, Inc. (2008)
4. Kerson Huang, “STATISTICAL MECHANICS, 2nd Ed.,” John Wiley & Sons, Inc. (1987)
5. L. D. Landau, and E. M. Lifshitz, “Statistical Physics,” Course of Theoretical Physics, Vol. 5 (1958)
6. Dwight R. Nicholson, “Introduction to Plasma Theory,” Wiley (1983)
7. Francis F. Chen, “INTRODUCTION TO PLASMA PHYSICS AND CONTROLLED FUSION, Vol. 1: Plasma Physics, 2nd Ed,” Springer Science + Business Media, LLC (2006)
8. R. P. Feynman, R. B. Leighton, and M. Sands, “The Feynman Lectures on Physics,” California Institute of Technology, Vol. 2 (1968)
9. O. C. Eldridge, and M. Feix, “One-dimensional Plasma Model at Thermodynamic Equilibrium,” Phys. Fluids, Vol. 5, No. 9, 1076 (1962)
10. William H. Press, Saul A. Teukolsky, William T. Vetterling, Brian P. Flannery, “NUMERICAL RECIPES: The Art of Scientific Computing, 3rd Ed,” Cambridge University press (2007)
11. J. M. Dawson, “The Electrostatic Sheet Model for a Plasma and its Modification to Finite-Size Particles,” METHODS IN COMPUTATIONAL PHYSICS, Vol. 9, pp. 1—28
12. James R. Munkres, “TOPOLOGY, 2nd Ed.,” Prentice Hall, Inc. (2000)
13. Jouni Smed, and Harri Hakonen, “Algorithms and Networking for Computer Games,” John Wiley & Sons Ltd (2006)
14. David Montgomery, and C. W. Nielson, “Thermal Relaxation in One- and Two- Dimensional Plasma Models”
15. William L. Kruer, “The Physics of Laser Plasma Interactions,” Addison-Wesley Publishing Company, Inc. (1988)
16. M. M. Turner, “Kinetic properties of particle-in-cell simulations compared by Monte Carlo collisions,” Physics of Plasma 13, 033506 (2006)

指導教授

林留玉仁(Y. R. Lin-Liu)

審核日期

2012-11-8

推文