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The statistical properties and electronic transfer coefficients of Fibonacci sequence

Huang Xiao-Mei Xu Hui Ren Yi Liu Xiao-Liang

The statistical properties and electronic transfer coefficients of Fibonacci sequence

Huang Xiao-Mei, Xu Hui, Ren Yi, Liu Xiao-Liang
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  • 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.
    • Funds:
    [1]

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    [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 物理学报 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 物理学报 51 2139]

    [8]

    [8]Cao Y J, Yang X 2008 Acta Phys. Sin. 57 3620 (in Chinese) [曹永军、杨旭 2008 物理学报 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 物理学报 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]

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    ]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

  • [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 物理学报 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 物理学报 51 2139]

    [8]

    [8]Cao Y J, Yang X 2008 Acta Phys. Sin. 57 3620 (in Chinese) [曹永军、杨旭 2008 物理学报 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 物理学报 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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  • Received Date:  09 August 2009
  • Accepted Date:  02 November 2009
  • Published Online:  15 June 2010

The statistical properties and electronic transfer coefficients of Fibonacci sequence

  • 1. (1)长沙电力职业技术学院电力工程系,长沙 410131; (2)中南大学物理科学与技术学院,长沙 410083; (3)中南大学物理科学与技术学院,长沙 410083;中南大学冶金科学与工程学院,长沙 410083

Abstract: 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.

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