Random Wave Functions with boundary and normalization constraints: Quantum statistical physics meets quantum chaos

当混沌遇见计算机

When Chaos Meets Computers

本文发现数字计算机即使在浮点高精度下也无法正确模拟某些混沌系统（如帐篷映射、伯努利移位映射）的长期动力学，揭示了用计算机数值研究连续动力系统的风险。

Record statistics in random vectors and quantum chaos

The record statistics of complex random states are analytically calculated, and shown that the probability of a record intensity is a Bernoulli process. The correlation due to normalization leads to a probability distribution of the records that is non-universal but tends to the Gumbel distribution asymptotically. The quantum standard map is used to study these statistics for the effect of correlations apart from normalization. It is seen that in the mixed phase space regime the number of intensity records is a power law in the dimensionality of the state as opposed to the logarithmic growth f…

混沌遇见注意力：用于大规模动力学预测的Transformer

Chaos Meets Attention: Transformers for Large-Scale Dynamical Prediction

针对耗散混沌系统长期预测的可扩展性问题，提出基于Transformer的框架，通过修改注意力机制并利用遍历性保持长期统计，在五个指标上优于现有方法，并引入140k快照的湍流通道流基准数据集。

Wave function correlations on the ballistic scale: Exploring quantum chaos by quantum disorder

We study the statistics of wave functions in a ballistic chaotic system. The statistical ensemble is generated by adding weak smooth disorder. The conjecture of Gaussian fluctuations of wave functions put forward by Berry and generalized by Hortikar and Srednicki is proven to hold on sufficiently short distances, while it is found to be strongly violated on larger scales. This also resolves the conflict between the above conjecture and the wave function normalization. The method is further used to study ballistic correlations of wave functions in a random magnetic field.

Extreme statistics of complex random and quantum chaotic states

An exact analytical description of extreme intensity statistics in complex random states is derived. These states have the statistical properties of the Gaussian and Circular Unitary Ensemble eigenstates of random matrix theory. Although the components are correlated by the normalization constraint, it is still possible to derive compact formulae for all values of the dimensionality N. The maximum intensity result slowly approaches the Gumbel distribution even though the variables are bounded, whereas the minimum intensity result rapidly approaches the Weibull distribution. Since random matrix…

Numerical aspects of eigenvalue and eigenfunction computations for chaotic quantum systems

We give an introduction to some of the numerical aspects in quantum chaos. The classical dynamics of two--dimensional area--preserving maps on the torus is illustrated using the standard map and a perturbed cat map. The quantization of area--preserving maps given by their generating function is discussed and for the computation of the eigenvalues a computer program in Python is presented. We illustrate the eigenvalue distribution for two types of perturbed cat maps, one leading to COE and the other to CUE statistics. For the eigenfunctions of quantum maps we study the distribution of the eigen…