臺灣博碩士論文加值系統

沙堆模型是描述自組臨界現象理論的經典模型。 本論文主要為藉由探討 Manna 沙堆模型的尺度 (scaling) 行為, 希望更進一步了解沙堆模型中耗散率所代表的物理意義。 本研究主要包含了三個部份: (1) 研究具有內部耗散的 Manna 模型之尺度行為。 (2) 研究改變不同的系統形狀與驅策模式後, 具有內部耗散的 Manna 模型所表現的尺度行為。 (3) 研究重整群 (renormalization-group) 趨近法對 Manna 模型之應用。 經由數值模擬與理論分析, 我們發現 Manna 模型中耗散率的角色與一般臨界現象理論中的調控參數有很大的相似性, 此發現將有助於了解自組臨界現象理論。

The sandpile model is a prototype to describe the concept of self-organized criticality. In this thesis, we investigate the scaling behaviors of the Manna sandpile and further discuss the role of dissipation playing in self-organized criticality. The research mainly includes three parts (1) The scaling behavior of the Manna sandpile with bulk dissipation, (2) The scaling behavior of the dissipative Manna sandpile with different driving ways and geometic shapes, and (3) The study of renormalization-group approach to the Manna sandpile. Through the numerical simulations and analytical treatment, we find that dissipation of the Manna sandpile in self-organized criticality plays the similar role as a tuning parameter in critical phenomena. This finding may advance in the understanding of self-organized criticality.

1 相變, 臨界現象與自組臨界現象. . . . . . . . . . . . . . . . . . . . 1
1.1 相變. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.2 臨界現象. . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
1.3 自組臨界現象. . . . . . . . . . . . . . . . . . . . . . . . . . 7
1.4 自組臨界現象模型. . . . . . . . . . . . . . . . . . . . . . . . 8
1.4.1 BTW 沙堆模型. . . . . . . . . . . . . . . . . . . . . . . 10
1.4.2 Manna 模型. . . . . . . . . . . . . . . . . . . . . . . . 11
1.4.3 變形的 Manna 模型 . . . . . . . . . . . . . . . . . . . . 12
1.4.4 沙堆演化與機率分布函數. . . . . . . . . . . . . . . . . . 13
2 具內部耗散之 Manna 模型的尺度行為研究 . . . . . . . . . . . . . . . 16
2.1 具內部耗散之 Manna 模型 . . . . . . . . . . . . . . . . . . . . 18
2.2 特徵長度的定義. . . . . . . . . . . . . . . . . . . . . . . . . 18
2.3 機率分布的尺度函數. . . . . . . . . . . . . . . . . . . . . . . 23
2.4 聯合機率分布的尺度函數. . . . . . . . . . . . . . . . . . . . . 27
3 具內部耗散之 Manna 模型在不同幾何形狀與驅策模式下的尺度行為研究 . . 31
3.1 具不同幾何形狀之內部耗散 Manna 模型 . . . . . . . . . . . . . . 33
3.1.1 非普適量測因子的定義. . . . . . . . . . . . . . . . . . . 33
3.1.2 機率分布的尺度函數和普適尺度函數. . . . . . . . . . . . . 39
3.2 具不同驅策模式之內部耗散 Manna 模型 . . . . . . . . . . . . . . 44
3.2.1 特徵長度的定義. . . . . . . . . . . . . . . . . . . . . . 44
3.2.2 機率分布的尺度函數. . . . . . . . . . . . . . . . . . . . 50
4 q 態 Manna 模型的重整群研究 . . . . . . . . . . . . . . . . . . . . 55
4.1 q 態 Manna 模型 . . . . . . . . . . . . . . . . . . . . . . . . 57
4.2 粗粒化過程與重整群參數. . . . . . . . . . . . . . . . . . . . . 60
4.3 2 x 2 重整群晶胞中的演化機率. . . . . . . . . . . . . . . . . . 63
4.4 重整群轉換. . . . . . . . . . . . . . . . . . . . . . . . . . . 72
4.5 重整群方程式. . . . . . . . . . . . . . . . . . . . . . . . . . 75
4.6 固定點分析之結果. . . . . . . . . . . . . . . . . . . . . . . . 84
4.7 蒙地卡羅重整群法之取樣技術. . . . . . . . . . . . . . . . . . . 85
4.8 重整群法對指數之計算與應用. . . . . . . . . . . . . . . . . . . 88
4.8.1 崩落次數的指數. . . . . . . . . . . . . . . . . . . . . . 88
4.8.2 動力指數. . . . . . . . . . . . . . . . . . . . . . . . . 99
5 結論. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .103
參考文獻. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .107

[1] H. E. Stanley, Introduction to Phase Transitions and Critical Phenomena (Oxford Univ. Press, New York, 1971).
[2] P. Bak, How Nature works: The Science of Self-Organized Criticality (Copernicus, New York, 1996).
[3] H. J. Jensen, Self-Organized Criticality (Cambridge Univ. Press, New York, 1998).
[4] P. Bak, C. Tang, and K. Wiesenfeld, Phys. Rev. Lett. 59, 381 (1987).
[5] P. Bak, C. Tang, and K. Wiesenfeld, Phys. Rev. A 38, 364 (1988).
[6] V. Frette, K. Christensen, A. Malthe-Sorenssen, J. Feder, T. Jossang, and P. Meakin, Nature 379, 49 (1995).
[7] M. D. Menech, A. L. Stella, and C. Tebaldi, Phys. Rev. E 58, R2677 (1998); C. Tebaldi, M. De Menech, and A. L. Stella, Phys. Rev. Lett. 83, 3952 (1999).
[8] S. S. Manna, J. Phys. A 24, L363 (1991).
[9] Z. Olami, H. J. S. Feder, and K. Christensen, Phys. Rev. Lett. 68, 1244 (1992).
[10] A. Chessa, H. E. Stanley, A. Vespignani, and S. Zapperi, Phys. Rev. E 59, R12 (1999).
[11] S. Lubeck, Phys. Rev. E 61, 204 (2000).
[12] D. Dhar, Phys. Rev. Lett. 64, 1613 (1990); S. N. Majumdar, and D. Dhar, Physica A 185, 129 (1992).
[13] D. Dhar, Physica A 224, 162 (1996).
[14] O. Malcai, Y. Shilo, and O. Biham, Phys. Rev. E 73, 056125 (2006).
[15] A. Chessa, A. Vespignani, and S. Zapperi, Computer Phys. Communications 121, 299 (1999).
[16] G. Pruessner, Ph.D. thesis, University of London (2004).
[17] G. Grinstein, Generic scale invariance and self-organized criticality, in Scale Invariance, Interfaces, and Non-Equilibrium Dynamics, eds. A. McKane., Nato ASI Series B: Physics Vol. 344 (Plenum Press, London, 1995).
[18] D. Sornette, A. Johansen, and I. Dornic, J. Phys. I France 5, 325 (1995).
[19] A. Vespignani, S. Zapperi, and V. Loreto, Phys. Rev. Lett. 77, 4560 (1996).
[20] A. Vespignani, and S. Zapperi, Phys. Rev. Lett. 78, 4793 (1997); Phys. Rev. E 57, 6345 (1998).
[21] R. Dickman, A. Vespignani, and S. Zapperi, Phys. Rev. E 57, 5095 (1998); A. Vespignani, R. Dickman, M. A. Munoz, and S. Zapperi, Phys. Rev. Lett. 81, 5676 (1998).
[22] M. A. Mu~ noz, R. Dickman, A. Vespignani, and S. Zapperi, Phys. Rev. E 59, 6175 (1999).
[23] C.-Y. Lin, C.-F. Chen, C.-N. Chen, C.-S. Yang, and I.-M. Jiang, Phys. Rev. E 74, 031304 (2006).
[24] S. Lubeck, Int. J. Mod. Phys. B 18, 3977 (2004).
[25] R. Dickman, M. A. Munoz, A. Vespignani, and S. Zapperi, Brazilian Journal of Physics 30, 27 (2000).
[26] J. B. Brankov, D. M. Danchev, and N. S. Tonchev, Theory of Critical Phenomena in Finite-size System: Scaling and Quantum Effects (World Scientific Publishing Co., Singapore, 2000).
[27] C.-F. Chen, A.-C. Cheng, Y.-D. Wang, and C.-Y. Lin, Computer Physics Communications (accepted).
[28] H. Nakanishi, and K. Sneppen, Phys. Rev. E 55, 4012 (1997).
[29] L. Pietronero, A. Vespignani, and S. Zapperi, Phys. Rev. Lett. 72, 1690 (1994).
[30] A. Vespignani, S. Zapperi, and L. Pietronero, Phys. Rev. E 51, 1711 (1995).
[31] V. Loreto, L. Pietronero, A. Vespignani, and S. Zapperi, Phys. Rev. Lett. 75, 465 (1995).
[32] E. V. Ivashkevich, Phys. Rev. Lett. 76, 3368 (1996).
[33] Vl. V. Papoyan, and A. M. Povolotsky, Physica A 246, 241 (1997).
[34] J. Hasty, and K. Wiesenfeld, Phys. Rev. Lett. 81, 1722 (1998).
[35] C.-Y. Lin, A.-C. Cheng, and T.-M. Liaw, Phys. Rev. E 76, 041114 (2007).
[36] V. B. Priezzhev, J. Stat. Phys. 74, 955 (1994).
[37] R. J. Creswick, H. A. Farach, and C. P. Poole, JR, Introduction to Renormalization Group Methods in Physics (John Wiley & Sons,Inc., 1992).
[38] C.-Y. Lin, Phys. Rev. E 81, 021112 (2010).
[39] C.-F. Chen, and C.-Y. Lin, Int. J. Mod. Phys. C 20, 273 (2009).
[40] A. A. Middleton, and C. Tang, Phys. Rev. Lett. 74, 742 (1995).