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对按膨胀规律A→AB和B→A生成的Fibonacci序列，采用一维随机行走模型数值计算了序列的自相关函数以及自行定义的准标准偏差.利用Hurst分析法研究了序列的再标度范围函数及其Hurst指数，并将结果与一维随机二元序列进行了对比.发现这些统计量有奇特的准周期振荡行为以及小于05的Hurst指数，直接论证了Fibonacci序列具有关联、标度不变及自相似等性质.从Anderson紧束缚模型出发，采用传输矩阵方法研究了Fibonacci序列的电子输运特性，讨论了输运系数对能量及其序列长度的依赖关系.研究
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关键词:
- Fibonacci序列 /
- 统计属性 /
- 电子输运系数
For the Fibonacci sequence constructed by following the inflation rule A→AB and B→A, using the one-dimensional random walk model and Hurst’ analysis, we calculate numerically the auto-correlation function, the pseudo standard deviation of displacement defined by ourselves and the rescaled range function and investigate systematically the statistical properties. The results are compared with that of one-dimensional random binary sequence. We show that the Fibonacci sequence presents correlated behavior as well as scaling invariability and self-similarity. In addition, basing on the tight-binding model of the single electron and transfer matrix method, we study the charge transfer properties of Fibonacci sequence and discuss specially the dependence of electron transmission on energy and the length of the sequence. We find some resonant peaks can survive in relatively longer Fibonacci sequences than in random sequences, which also implies that there are long-range correlations in Fibonacci sequences.[1] ［1］Randic M, Morales D A, Araujo O 1996 J. Math. Chem. 20 79
[2] ［2］Jean R V 1984 Mathematical Approach to Patterns and Form in Plant Growth (New York: Wiley Press)
[3] ［3］He L X, Li X Z, Zhang Z 1988 Phys. Rev. Lett. 61 1116
[4] ［4］Merlin R, Bajima K, Charke R 1985 Phys. Rev. Lett. 55 1768
[5] ［5］Yan X H, Yan J R, Zhong J X, You J Q 1992 Acta Phys. Sin. 41 1652 (in Chinese) ［颜晓红、颜家壬、钟建新、游建强 1992 41 1652］
[6] ［6］Huang X Q, Gong C D 1998 Phys. Rev. B 58 739
[7] ［7］Li P F, Yan X H, Wang R Z 2002 Acta Phys. Sin. 51 2139 (in Chinese) ［李鹏飞、颜晓红、王如志 2002 51 2139］
[8] ［8］Cao Y J, Yang X 2008 Acta Phys. Sin. 57 3620 (in Chinese) ［曹永军、杨旭 2008 57 3620］
[9] ［9］Enrique M, Francisco D A 1996 Phys. Rev. Lett. 76 2957
[10] ］Atsushi N, Shinkichi H 2007 Phys. Rev. B 76 235113
[11] ］Kohmoto M, Banavar J R 1986 Phys. Rev. B 34 563
[12] ］Oh G Y, Choi H Y 1996 Phys. Rev. B 54 6043
[13] ］You J Q, Zhang L D, Yang Q B 1997 Phys. Rev. B 55 1314
[14] ］Stephan R, Dominique B, Enrique M, Kats E 2003 Phys. Rev. Lett. 91 228101
[15] ］Albuquerque E L, Vasconcelos M S, Lyra M L, de Moura F A B F 2005 Phys. Rev. E 71 021910
[16] ］Peng C K, Buldyrev S V, Goldberger A L 1992 Nature 356 168
[17] ］Liu X L, Xu H, Deng C S, Ma S S 2006 Physica B 383 226
[18] ］Roche S 2003 Phys. Rev. Lett. 91 108101
[19] ］Carpena P, Bernaola-Galvan P, Ivanov P C 2002 Nature 418 955
[20] ］Kramer B, MacKinnon A 1993 Rep. Prog. Phys. 56 1469
[21] ］Meng X L, Gao X T, Qu Z, Kang D W, Liu D S, Xie S J 2008 Acta Phys. Sin. 57 5316 (in Chinese) ［孟宪兰、高绪团、渠朕、康大伟、刘德胜、解士杰 2008 57 5316］
[22] ］Liu X L, Xu H, Deng C S, Ma S S 2007 Physica B 392 107
[23] ］Liu X L, Xu H, Li Y F, Li M J 2008 Chin. J. Comp. Phys. 25 358 (in Chinese) ［刘小良、徐慧、李燕峰、李明君 2008 计算物理 25 358］
[24] ］Zhang W, Ulloa S E 2006 Phys. Rev. B 74 115304
[25] ］Hurst H E, Black R, Sinaika Y M 1965 Long-Term Storage in Reservior: An Experimental Study (London: Constable)
[26] ］Guo A M, Xiong S J 2009 Phys. Rev. B 80 035115
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[1] ［1］Randic M, Morales D A, Araujo O 1996 J. Math. Chem. 20 79
[2] ［2］Jean R V 1984 Mathematical Approach to Patterns and Form in Plant Growth (New York: Wiley Press)
[3] ［3］He L X, Li X Z, Zhang Z 1988 Phys. Rev. Lett. 61 1116
[4] ［4］Merlin R, Bajima K, Charke R 1985 Phys. Rev. Lett. 55 1768
[5] ［5］Yan X H, Yan J R, Zhong J X, You J Q 1992 Acta Phys. Sin. 41 1652 (in Chinese) ［颜晓红、颜家壬、钟建新、游建强 1992 41 1652］
[6] ［6］Huang X Q, Gong C D 1998 Phys. Rev. B 58 739
[7] ［7］Li P F, Yan X H, Wang R Z 2002 Acta Phys. Sin. 51 2139 (in Chinese) ［李鹏飞、颜晓红、王如志 2002 51 2139］
[8] ［8］Cao Y J, Yang X 2008 Acta Phys. Sin. 57 3620 (in Chinese) ［曹永军、杨旭 2008 57 3620］
[9] ［9］Enrique M, Francisco D A 1996 Phys. Rev. Lett. 76 2957
[10] ］Atsushi N, Shinkichi H 2007 Phys. Rev. B 76 235113
[11] ］Kohmoto M, Banavar J R 1986 Phys. Rev. B 34 563
[12] ］Oh G Y, Choi H Y 1996 Phys. Rev. B 54 6043
[13] ］You J Q, Zhang L D, Yang Q B 1997 Phys. Rev. B 55 1314
[14] ］Stephan R, Dominique B, Enrique M, Kats E 2003 Phys. Rev. Lett. 91 228101
[15] ］Albuquerque E L, Vasconcelos M S, Lyra M L, de Moura F A B F 2005 Phys. Rev. E 71 021910
[16] ］Peng C K, Buldyrev S V, Goldberger A L 1992 Nature 356 168
[17] ］Liu X L, Xu H, Deng C S, Ma S S 2006 Physica B 383 226
[18] ］Roche S 2003 Phys. Rev. Lett. 91 108101
[19] ］Carpena P, Bernaola-Galvan P, Ivanov P C 2002 Nature 418 955
[20] ］Kramer B, MacKinnon A 1993 Rep. Prog. Phys. 56 1469
[21] ］Meng X L, Gao X T, Qu Z, Kang D W, Liu D S, Xie S J 2008 Acta Phys. Sin. 57 5316 (in Chinese) ［孟宪兰、高绪团、渠朕、康大伟、刘德胜、解士杰 2008 57 5316］
[22] ］Liu X L, Xu H, Deng C S, Ma S S 2007 Physica B 392 107
[23] ］Liu X L, Xu H, Li Y F, Li M J 2008 Chin. J. Comp. Phys. 25 358 (in Chinese) ［刘小良、徐慧、李燕峰、李明君 2008 计算物理 25 358］
[24] ］Zhang W, Ulloa S E 2006 Phys. Rev. B 74 115304
[25] ］Hurst H E, Black R, Sinaika Y M 1965 Long-Term Storage in Reservior: An Experimental Study (London: Constable)
[26] ］Guo A M, Xiong S J 2009 Phys. Rev. B 80 035115
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