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研究生：	湯偉佑 Wei-You Tang
論文名稱：	Entropy production and Information rates in non-equilibrium network dynamics
指導教授：	黎璧賢 Pik-Yin Lai
口試委員:
學位類別：	碩士 Master
系所名稱：	理學院 - 物理學系 Department of Physics
論文出版年：	2024
畢業學年度：	112
語文別：	英文
論文頁數：	73
中文關鍵詞：	熵生成、信息速率、噪聲網絡動力學、非平衡動力學、布朗粒子
外文關鍵詞：	Entropy production, Information rate, Noisy network dynamics, Non-equilibrium dynamics, Brownian particles
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我們採用隨機力推斷（SFI）方法來研究布朗粒子在非線性力場下的噪聲網絡動力學。然後我們考慮了噪聲網絡動力學的情況，在這種情況下，每個節點的固有動力學是上述的一維布朗粒子，並且節點之間通過有向加權連接進行相互作用。我們分析了熵產生和信息速率，以揭示網絡屬性如何影響網絡的耗散，並探討非平衡集體動態與網絡結構的關聯。

We employ the stochastic force inference (SFI) method to investigate the noisy network dynamics of Brownian particles under a non-linear force field. We then consider the case of noisy network dynamics in which each node’s intrinsic dynamics is the above one-dimensional Brownian particle and the nodes are interacting with directed and weighted connections. The entropy production and information rates are analyzed to reveal how network properties affect the dissipation of the network and relate the non-equilibrium collective dynamics in terms of the network structures.

Abstract iii
1 Introduction 1
1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Theoretical Background . . . . . . . . . . . . . . . . . . . . . . . . 2
1.2.1 Brownian Dynamics . . . . . . . . . . . . . . . . . . . . . . 2
Free Brownian Motion . . . . . . . . . . . . . . . . . . . . . 2
Einstein Relation . . . . . . . . . . . . . . . . . . . . . . . . 3
Langevin Equation . . . . . . . . . . . . . . . . . . . . . . . 4
1.2.2 Entropy Production Rate . . . . . . . . . . . . . . . . . . . 5
1.2.3 Information Rate . . . . . . . . . . . . . . . . . . . . . . . . 6
1.3 Network . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
1.3.1 Network Property [11] . . . . . . . . . . . . . . . . . . . . 7
Weighted Mean degree . . . . . . . . . . . . . . . . . . . . . 8
1.3.2 Modularity [12, 13] . . . . . . . . . . . . . . . . . . . . . . . 9
1.3.3 Network Structure . . . . . . . . . . . . . . . . . . . . . . . 10
Erd˝os-Rényi random network . . . . . . . . . . . . . . . . . 10
Hierarchical network . . . . . . . . . . . . . . . . . . . . . . 10
Hyper-cubic structure . . . . . . . . . . . . . . . . . . . . . 11
Bethe lattice . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
Triangular Lattice . . . . . . . . . . . . . . . . . . . . . . . . 11
BCC Lattice . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
2 Method 13
2.1 SFI [20] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
2.1.1 Information in Brownian trajectory . . . . . . . . . . . . . . 13
2.1.2 Stochastic Force Inference . . . . . . . . . . . . . . . . . . . 15
2.2 Numerical simulation . . . . . . . . . . . . . . . . . . . . . . . . . 17
3 Result 19
3.1 Analytic result for a single Brownian particle in two-dimension . 19
Pure anti-symmetric case . . . . . . . . . . . . . . . . . . . 21
General case . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
3.2 Inference F and D for Network Dynamics(2.12) . . . . . . . . . . 23
3.3 Heterogeneous Diffusion coefficients . . . . . . . . . . . . . . . . . 24
3.4 Mean degree ¯k & Weighted Mean degree ¯W . . . . . . . . . . . . . 27
3.5 Symmetric level μ & Anti-Symmetric level ν . . . . . . . . . . . . 29
3.6 Modularity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
3.7 Different Network structures . . . . . . . . . . . . . . . . . . . . . 32
4 Conclusion 35
4.1 Entropy Production Rate . . . . . . . . . . . . . . . . . . . . . . . . 35
4.2 Information Rate . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
A Appendix 37
A.1 Analytic result for a single Brownian particle in two-dimension . 37
A.1.1 Calculation of Entropy Production . . . . . . . . . . . . . . 38
A.1.2 Calculation of Information Rate . . . . . . . . . . . . . . . 40
A.2 Program . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
A.2.1 Generating Network . . . . . . . . . . . . . . . . . . . . . . 40
A.2.2 Using SFI to calculate the ˙S and I˙ by solving Langevin
equation in network dynamics . . . . . . . . . . . . . . . . 48
Bibliography 57
                                

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