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完整後設資料紀錄
DC 欄位 | 值 | 語言 |
---|---|---|
dc.contributor.advisor | 胡進錕(Chin-Kun Hu) | |
dc.contributor.author | Guan-Rong Huang | en |
dc.contributor.author | 黃冠榮 | zh_TW |
dc.date.accessioned | 2021-05-14T17:48:48Z | - |
dc.date.available | 2017-12-01 | |
dc.date.available | 2021-05-14T17:48:48Z | - |
dc.date.copyright | 2015-12-01 | |
dc.date.issued | 2015 | |
dc.date.submitted | 2015-09-22 | |
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dc.identifier.uri | http://tdr.lib.ntu.edu.tw/jspui/handle/123456789/4853 | - |
dc.description.abstract | In macro, the method and concept of statistical physics can be a powerful tool in analytical and numerical computation and applied to many fields such as chemical reaction, bio-evolution, and material science. In our work, the methods of statistical physics: chemical master equation (CME), Hamilton-Jacobi equation (HJE), and canonical ensemble are used to calculate various physical quantities. Our work is organized in three topics: the CME with the Gaussian and compound Poisson noise, bio-evolution of Eigen model, and energy conversion in the surface mechanical attrition treatment (SMAT) experiment.
In the CME part, the chemical reaction among DNA, mRNA, and protein can be regarded as a stochastic process. We consider the CME with compound Poisson and Gaussian noises and obtain the exact solution of steady state probability density function (PDF) verified by the algorithm of forward finite difference in large-scale time. Without Gaussian white noise, the solution of CME (set diffusion coefficient ϵ = 0) can be returned to that of CME derived by Long Cai, et al. In the bio-evolution part, we use the method of expansion in O(1/N) to obtain the HJE for probability distribution in Hamming class which is applied to calculate the correction of O(1/N) accuracy for the steady-state probability distribution in Hamming class and mean fitness in Eigen model. The steady-state distributions of O(1/N) correction are well-consistent with the Runge-Kutta simulation with relative errors less than 1 %, while the mean fitness of O( 1/N) is the same one derived by Michael Deem, et al. in quantum field theory. In the SMAT part, we consider the collisions among the 304-steel balls, motor top, and chamber bottom, where the chamber or motor can be treated as a hot reservoir. Since we assume that all the collisions among them are elastic except the ball-sample collisions, the balls with negligible potential among them can be regarded as the canonical ensemble. By this concept, we construct the link for energy conversion among the motor top, sample bottom, and balls, where the kinetic energy, heat energy, and internal energy are included in the energy conversion. We also introduce the one-dimensional heat equation with uniform-distributed heat source to obtain the temperature distribution of sample, and we use this temperature distribution of sample to connect the Zenner-Hollmann parameter and the heat energy and surface hardness of sample. | en |
dc.description.provenance | Made available in DSpace on 2021-05-14T17:48:48Z (GMT). No. of bitstreams: 1 ntu-104-D01222004-1.pdf: 1641391 bytes, checksum: b23c2b08f2fd81cac0ec6f5fe47e52fa (MD5) Previous issue date: 2015 | en |
dc.description.tableofcontents | Contents
致謝i Abstract ii Contents iv List of Figures vii List of Tables xi 1 Introduction 1 1.1 Chemical Master Equation . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.2 Bio-evolution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 1.3 SMAT modelling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 2 Basic theory for CME, Bio-evolution, and SMAT 10 2.1 Hamilton-Jacobi Equation . . . . . . . . . . . . . . . . . . . . . . . . . 10 2.1.1 Canonical Transformation . . . . . . . . . . . . . . . . . . . . . 11 2.1.2 Hamilton-Jacobi Equation . . . . . . . . . . . . . . . . . . . . . 12 2.1.3 HJE Application in CME . . . . . . . . . . . . . . . . . . . . . . 13 2.2 Diffusion Process . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 2.2.1 Brownian Motion . . . . . . . . . . . . . . . . . . . . . . . . . . 16 2.2.2 Itõ’s Lemma . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 2.2.3 Kolmogorov Forward Equation . . . . . . . . . . . . . . . . . . 21 2.3 Bio-evolution Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 2.3.1 Crow-Kimura Model . . . . . . . . . . . . . . . . . . . . . . . . 25 2.3.2 Eigen Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27 2.4 Collision and Heat Equation . . . . . . . . . . . . . . . . . . . . . . . . 28 2.4.1 Collision . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 2.4.2 Heat Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 3 Advanced Theory for CME, Bio-evolution, and SMAT 32 3.1 Formalism of Chemical Master Equation . . . . . . . . . . . . . . . . . . 33 3.1.1 Solution of Linear-Drift Process . . . . . . . . . . . . . . . . . . 33 3.1.2 Path Integral Formalism in Stochastic Process . . . . . . . . . . . 36 3.1.3 Formalism for CME with Compound Poisson Noise . . . . . . . 38 3.2 Formalism of Bio-evolution Model in Hamming Class . . . . . . . . . . 41 3.2.1 Crow-Kimura Model in Hamming Class . . . . . . . . . . . . . . 42 3.2.2 HJE Method in Crow-Kimura Model . . . . . . . . . . . . . . . 44 3.2.3 Modified Eigen Model in Hamming Class . . . . . . . . . . . . . 54 3.3 SMAT Modelling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55 3.3.1 The Kinetic Energy of balls . . . . . . . . . . . . . . . . . . . . 55 3.3.2 The Loss of Kinetic Energy for Balls . . . . . . . . . . . . . . . 60 4 Analytical and Numerical Solution 63 4.1 CME Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63 4.1.1 Finite Difference Method . . . . . . . . . . . . . . . . . . . . . . 64 4.1.2 Analytical Solution of Van Kampen CME without Diffusion Term 68 4.1.3 Asymptotic Solution of Van Kampen CME with Diffusion Term . 71 4.1.4 Analytical Solution of Van Kampen CME with Diffusion Term . . 77 4.1.5 Dynamic Simulation of Van Kampen CME with Diffusion Term . 85 4.2 Finite correction of Modified Eigen Model in Hamming Class . . . . . . 86 4.2.1 The derivation for Hamilton-Jacobi equation . . . . . . . . . . . 86 4.2.2 Comparison with Numerics . . . . . . . . . . . . . . . . . . . . 96 4.3 Solution of Heat Equation in SMAT . . . . . . . . . . . . . . . . . . . . 97 4.3.1 The Heat Source . . . . . . . . . . . . . . . . . . . . . . . . . . 97 4.3.2 The Temperature Distribution of Steady State . . . . . . . . . . . 98 4.3.3 The internal energy of sample . . . . . . . . . . . . . . . . . . . 101 4.3.4 Experimental Methods . . . . . . . . . . . . . . . . . . . . . . . 102 4.3.5 Relating strain rate and temperature with sample micro-structure in SMAT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103 5 Conclusion 105 5.1 Van Kampen CME with Gaussian White Noise . . . . . . . . . . . . . . 105 5.2 Finite Correction of Eigen Model . . . . . . . . . . . . . . . . . . . . . . 107 5.3 SMAT Modelling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107 .1 The Coefficient of Finite Difference . . . . . . . . . . . . . . . . . . . . 110 .2 The Power Series Expansion of Kummer’s Function . . . . . . . . . . . . 111 Bibliography 112 Figures and Tables 117 | |
dc.language.iso | en | |
dc.title | 統計物理在化學主方程式、艾根模型及表面機械研磨處理的應用 | zh_TW |
dc.title | Application of Statistical Physics in Chemical Master Equation, Eigen Model, and Surface Mechanical Attrition Treatment | en |
dc.type | Thesis | |
dc.date.schoolyear | 104-1 | |
dc.description.degree | 博士 | |
dc.contributor.oralexamcommittee | 黃志青,林財鈺,余瑞琳,陳企寧,馬文忠 | |
dc.subject.keyword | 化學主方程式,艾根模型,表面機械研磨處理, | zh_TW |
dc.subject.keyword | chemical master equation,Gaussian white noise,compound Poisson noise,bio-evolution,Eigen model,SMAT,collision,energy conversion, | en |
dc.relation.page | 133 | |
dc.rights.note | 同意授權(全球公開) | |
dc.date.accepted | 2015-09-23 | |
dc.contributor.author-college | 理學院 | zh_TW |
dc.contributor.author-dept | 物理研究所 | zh_TW |
顯示於系所單位: | 物理學系 |
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