随机矩阵理论中的超统计量
Superstatistics in Random Matrix Theory
摘要：
随机矩阵理论（RMT）为量子系统提供了一个成功的模型，其经典对应物具有混沌动力学。它基于两个假设：（1）矩阵元素独立性，（2）基不变性。在过去的十年里，我们看到了一些尝试，将RMT扩展到描述具有混合正则-混沌动力学的量子系统。大多数提出的推广保留了第一个假设，违反了第二个。最近，几位作者提出了其他版本的理论，保持基础上允许矩阵元素之间的相关性为代价的不变性。这是通过从非广延熵而不是标准的香农熵开始，或者遵循最近提出的超统计概念的基本规定来实现的。后一个概念是作为平衡热力学的推广引入的，通过允许温度波动来描述非平衡系统。在这里，我们回顾超统计推广的RMT和说明其价值，通过计算最近邻间距分布和比较的计算结果与实验台球模拟系统从有序过渡到混沌。
英文摘要：
Random matrix theory (RMT) provides a successful model for quantum systems, whose classical counterpart has a chaotic dynamics. It is based on two assumptions: (1) matrix-element independence, and (2) base invariance. Last decade witnessed several attempts to extend RMT to describe quantum systems with mixed regular-chaotic dynamics. Most of the proposed generalizations keep the first assumption and violate the second. Recently, several authors presented other versions of the theory that keep base invariance on the expense of allowing correlations between matrix elements. This is achieved by starting from non-extensive entropies rather than the standard Shannon entropy, or following the basic prescription of the recently suggested concept of superstatistics. The latter concept was introduced as a generalization of equilibrium thermodynamics to describe non-equilibrium systems by allowing the temperature to fluctuate. We here review the superstatistical generalizations of RMT and illustrate their value by calculating the nearest-neighbor-spacing distributions and comparing the results of calculation with experiments on billiards modeling systems in transition from order to chaos.
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