Manos Thanos, Robnik Marko
CAMTP - Center for Applied Mathematics and Theoretical Physics, University of Maribor, Krekova 2, SI-2000 Maribor, Slovenia.
Phys Rev E Stat Nonlin Soft Matter Phys. 2013 Jun;87(6):062905. doi: 10.1103/PhysRevE.87.062905. Epub 2013 Jun 11.
We study the kicked rotator in the classically fully chaotic regime using Izrailev's N-dimensional model for various N≤4000, which in the limit N→∞ tends to the quantized kicked rotator. We do treat not only the case K=5, as studied previously, but also many different values of the classical kick parameter 5≤K≤35 and many different values of the quantum parameter kε[5,60]. We describe the features of dynamical localization of chaotic eigenstates as a paradigm for other both time-periodic and time-independent (autonomous) fully chaotic or/and mixed-type Hamilton systems. We generalize the scaling variable Λ=l(∞)/N to the case of anomalous diffusion in the classical phase space by deriving the localization length l(∞) for the case of generalized classical diffusion. We greatly improve the accuracy and statistical significance of the numerical calculations, giving rise to the following conclusions: (1) The level-spacing distribution of the eigenphases (or quasienergies) is very well described by the Brody distribution, systematically better than by other proposed models, for various Brody exponents β(BR). (2) We study the eigenfunctions of the Floquet operator and characterize their localization properties using the information entropy measure, which after normalization is given by β(loc) in the interval [0,1]. The level repulsion parameters β(BR) and β(loc) are almost linearly related, close to the identity line. (3) We show the existence of a scaling law between β(loc) and the relative localization length Λ, now including the regimes of anomalous diffusion. The above findings are important also for chaotic eigenstates in time-independent systems [Batistić and Robnik, J. Phys. A: Math. Gen. 43, 215101 (2010); arXiv:1302.7174 (2013)], where the Brody distribution is confirmed to a very high degree of precision for dynamically localized chaotic eigenstates, even in the mixed-type systems (after separation of regular and chaotic eigenstates).
我们使用伊兹拉列夫的N维模型，在经典完全混沌区域研究了受踢转子，其中N≤4000，在N→∞的极限情况下趋向于量子化受踢转子。我们不仅处理了如先前研究的K = 5的情况，还处理了经典踢参数5≤K≤35的许多不同值以及量子参数k∈[5,60]的许多不同值。我们将混沌本征态的动力学局域化特征描述为其他时间周期和时间无关（自治）的完全混沌或/和混合型哈密顿系统的一个范例。通过推导广义经典扩散情况下的局域化长度l(∞)，我们将标度变量Λ = l(∞)/N推广到经典相空间中反常扩散的情况。我们极大地提高了数值计算的精度和统计显著性，得出以下结论：(1) 对于各种布罗迪指数β(BR)，本征相位（或准能量）的能级间距分布能很好地用布罗迪分布描述，系统地比其他提出的模型更好。(2) 我们研究了弗洛凯算子的本征函数，并使用信息熵测度来表征它们的局域化性质，归一化后在区间[0,1]中由β(loc)给出。能级排斥参数β(BR)和β(loc)几乎线性相关，接近恒等线。(3) 我们展示了β(loc)与相对局域化长度Λ之间存在标度律，现在包括反常扩散区域。上述发现对于时间无关系统中的混沌本征态也很重要[巴蒂斯蒂奇和罗布尼克，《物理学报A：数学一般》43, 215101 (2010); arXiv:1302.7174 (2013)]，在那里，即使在混合型系统中（在分离正则和混沌本征态之后），对于动态局域化的混沌本征态，布罗迪分布也被高精度地证实。
