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周期受击陀螺系统随时间演化波函数的多重分形

周洁 杨双波

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周期受击陀螺系统随时间演化波函数的多重分形

周洁, 杨双波

Multifractal behaviors of the wave function for the periodically kicked free top

Zhou Jie, Yang Shuang-Bo
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  • 研究了周期受击陀螺系统波函数的多重分形. 发现: 1)在打击次数较小时, 周期受击陀螺系统波包的扩散速度、扩散方向与打击强度相关, 打击强度越大扩散越混乱、扩散速度也越大; 2)波函数在相空间的精细结构的分布范围随着打击强度的增大而扩大, 最后充满整个相空间; 3)局部分维a的分布范围对应波函数在相空间的分布, 规则态时a 的分布范围最宽, 过渡态的a的分布范围较窄, 而混沌态的a的分布范围则最狭窄且稳定.
    Starting from time evolution of wave function, quantum dynamics for a periodically kicked free top system is studied in this paper. For an initial spherical coherent state wave packet (localized) we find that 1) as the number of kicking is small, the speed and the direction of the diffusion for a time-evolving wave packet on a periodically kicked free top is related to the kicking strength: the stronger the kicking strength, the more chaotic for the diffusion (which means the more randomized in direction) is and the faster the speed of diffusion is, and then more quickly the full phase space is filled up; 2) as the kicking number is large, the time-evolving wave function will take on fine structure distribution in phase space, and the scope of the distribution for the fine structure will expand with the increase of the kicking strength, and the whole phase space will be filled up finally, and then the wave function will show multifractal property in phase space.#br#We study the multifractal behavior for a time-evolving wave function by partition function method: 1) for different kicking strengths and different q values, we study the scaling properties of partition function X(q), and find the power law relation between the partition function and the scaling L, i.e., X(q)-Lτ(q); 2) at different kicking strength, for a time-evolving wave function we calculate the singularity spectrum f(a)-a, and find that a maximum value of f(a) is 2.0 independent of the kicking strength, but the width of the singularity spectrum becomes narrow with the increase of the kicking strength, which means that the scope of the distribution for a is widest for regular state (localized), and is narrower for transition state from regular to chaotic, and is narrowest for chaotic state; 3) in the time-evolving process, the fluctuation for the width of the singular spectrum is smallest for chaotic state, intermediate for transition state of regular to chaotic, and the largest for regular state; 4) we calculate the generalized fractal dimension Dq-q for different kicking strengths, and find D0 = 2 independent of the kicking strength.#br#We study the mutifractal behaviors for the mean propbability amplitude distribution for a sequence of time-evolving wave functions and find that the result is similar to that of the single wave function type but has the difference: the width of the spectrum is reduced for each kicking strength.
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    Martin J, Giraud O, Georgeot B 2008 Phys. Rev. E 77 035201

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    Sorensen C M 2001 Aerosol Sci. Tech. 35 648

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    Qiao W, Sun J, Liu S T 2015 Chin. Phys. B 24 050504

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    [22]

    Huang L, Lai Y C, Grebogi C 2011 Chaos 21 013102

    [23]

    Qin C C, Yang S B 2014 Acta Phys. Sin. 63 140507 (in Chinese) [秦陈陈, 杨双波 2014 物理学报 63 140507]

    [24]

    Liu D K, Yang S B 2014 J. Nanjing Normal University (Natural Science Edition) 37 2 (in Chinese) [刘达可, 杨双波 2014 南京师大学报(自然科学版) 37 2]

  • [1]

    Enns R H, McGuire G C 2001 Nonlinear Physics with Mathematica for Scientists and Engineers (Boston: Birkhäuser)

    [2]

    Kono M, Skoric M 2010 Nonlinear Physics of Plasmas (Berlin: Springer)

    [3]

    Mandelbrot B B 1983 The Fractal Geometry of Nature (New York: Freeman)

    [4]

    Zhang J Z 1997 Fractal (Beijing: Qing Hua University Press)

    [5]

    Zhou J, Yang S B 2014 Acta Phys Sin. 63 22 (in Chinese) [周洁, 杨双波 2014 物理学报 63 22]

    [6]

    Heller E J 1984 Phys. Rev. Lett. 53 1515

    [7]

    Nakamura K, Okazaki Y, Bishop A R 1986 Phys. Rev. Lett. 57 5

    [8]

    Halsey T C, Jensen M H, Kadanoff L P, Procaccia I, Shraimant B I 1986 Phys. Rev. A 33 1141

    [9]

    Nakamura K, Bishop A R, Shudo A 1989 Phys. Rev. B 39 12422

    [10]

    Chhabra A, Jensen R V 1989 Phys. Rev. Lett. 62 1328

    [11]

    Martin J, Giraud O, Georgeot B 2008 Phys. Rev. E 77 035201

    [12]

    Goldberger A L, Amaral L A N, Hausdorff J M, Ivanov P C, Peng C K, Stanley H E 2009 Proc. Natl. Acad. Sci. USA 99 2466

    [13]

    Sorensen C M 2001 Aerosol Sci. Tech. 35 648

    [14]

    Evers F, Mirlin A D 2008 Rev. Mod. Phys. 80 1355

    [15]

    Albrecht C, Smet J H, von Klitzing K, Weiss D, Umansky V, Schweizer H 2001 Phys. Rev. Lett. 86 147

    [16]

    Gu G F, Zhou W X 2010 Phys. Rev. E 82 011136

    [17]

    Qiao W, Sun J, Liu S T 2015 Chin. Phys. B 24 050504

    [18]

    Cai J C 2014 Chin. Phys. B 23 044701

    [19]

    Gu G F, Zhou W X 2006 Phys. Rev. E 74 061104

    [20]

    Gutzwiller M C1990 Chaos in Classical and Quantum Mechanics (New York: Springer)

    [21]

    Hönig A, Wintgen D 1989 Phys. Rev. A 39 5642

    [22]

    Huang L, Lai Y C, Grebogi C 2011 Chaos 21 013102

    [23]

    Qin C C, Yang S B 2014 Acta Phys. Sin. 63 140507 (in Chinese) [秦陈陈, 杨双波 2014 物理学报 63 140507]

    [24]

    Liu D K, Yang S B 2014 J. Nanjing Normal University (Natural Science Edition) 37 2 (in Chinese) [刘达可, 杨双波 2014 南京师大学报(自然科学版) 37 2]

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出版历程
  • 收稿日期:  2015-04-03
  • 修回日期:  2015-05-22
  • 刊出日期:  2015-10-05

周期受击陀螺系统随时间演化波函数的多重分形

  • 1. 南京师范大学物理科学与技术学院, 大规模复杂系统数值模拟江苏省重点实验室, 南京 210023

摘要: 研究了周期受击陀螺系统波函数的多重分形. 发现: 1)在打击次数较小时, 周期受击陀螺系统波包的扩散速度、扩散方向与打击强度相关, 打击强度越大扩散越混乱、扩散速度也越大; 2)波函数在相空间的精细结构的分布范围随着打击强度的增大而扩大, 最后充满整个相空间; 3)局部分维a的分布范围对应波函数在相空间的分布, 规则态时a 的分布范围最宽, 过渡态的a的分布范围较窄, 而混沌态的a的分布范围则最狭窄且稳定.

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