经典和量子遍历柠檬台球的粘性效应
Effects of stickiness in the classical and quantum ergodic lemon billiard
摘要：
本文研究了Heller和Tomsovic在Phys. Today 46（7），38（1993）中提出的经典和量子遍历柠檬台球，其中$B=1/2$是一个经典遍历系统（没有严格的证明），在零测度弹跳球模周围表现出强粘性区域.经典粘性区域的结构在Lozej引入的S图中揭示[Phys. Rev. E 101，052204（2020）]。唯一的经典输运或扩散时间不能被定义。因此，量子态具有以下非普适性质：（1）所有本征态都是混沌的，但局域化，如庞加莱-胡西米（PH）函数所示。(ii)熵局部化测量A（也是归一化的逆参与比）具有非普适分布，通常为双峰分布，因此偏离β分布，后者是没有粘性区域的均匀混沌系统的特征。(iii)能级间距分布是Berry-Robnik-Brody（BRB），捕获两个效应：量子分相空间（因为大多数PH函数要么是内部函数，要么是外部函数，由经典粘性决定，有效参数$μ_1$测量以粘性不变对象（即cantorus）为边界的内部区域的大小），以及由能级排斥（Brody）参数$β$表征的PH函数的局部化。(iv)在所考虑的能量范围内（在基态以上20 000个状态到400 000个状态之间），图像（本征态的结构和能谱的统计数据）没有发生质的变化，因为$β$在0.8附近波动，而$μ_1$几乎单调地随着能量的增加而减少。
英文摘要：
We study the classical and quantum ergodic lemon billiard introduced by Heller and Tomsovic in Phys. Today 46(7), 38 (1993), for the case $B=1/2$, which is a classically ergodic system (without a rigorous proof) exhibiting strong stickiness regions around a zero-measure bouncing ball modes. The structure of the classical stickiness regions is uncovered in the S-plots introduced by Lozej [Phys. Rev. E 101, 052204 (2020)]. A unique classical transport or diffusion time cannot be defined. As a consequence the quantum states are characterized by the following nonuniversal properties: (i) All eigenstates are chaotic but localized as exhibited in the Poincaré-Husimi (PH) functions. (ii) The entropy localization measure A (also the normalized inverse participation ratio) has a nonuniversal distribution, typically bimodal, thus deviating from the beta distribution, the latter one being characteristic of uniformly chaotic systems with no stickiness regions. (iii) The energy-level spacing distribution is Berry-Robnik-Brody (BRB), capturing two effects: the quantally divided phase space (because most of the PH functions are either the inner-ones or the outer-ones, dictated by the classical stickiness, with an effective parameter $μ_1$ measuring the size of the inner region bordered by the sticky invariant object, namely, a cantorus), and the localization of PH functions characterized by the level repulsion (Brody) parameter $β$. (iv) In the energy range considered (between 20 000 states to 400 000 states above the ground state) the picture (the structure of the eigenstates and the statistics of the energy spectra) is not changing qualitatively, as $β$ fluctuates around 0.8, while $μ_1$ decreases almost monotonically, with increasing energy.
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