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Localization and Pattern Formation in Quantum Physics. Ⅱ. Waveletons in Quantum Ensembles

机译:量子物理学中的定位和模式形成。 Ⅱ。量子集成中的小波

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In this second part we present a set of methods, analytical and numerical, which can describe behaviour in (non) equilibrium ensembles, both classical and quantum, especially in the complex systems, where the standard approaches cannot be applied. The key points demonstrating advantages of this approach are: (ⅰ) effects of localization of possible quantum states; (ⅱ) effects of non-perturbative multiscales which cannot be calculated by means of perturbation approaches; (ⅲ) effects of formation of complex/collective quantum patterns from localized modes and classification and possible control of the full zoo of quantum states, including (meta) stable localized patterns (waveletons). We demonstrate the appearance of nontrivial localized (meta) stable states/patterns in a number of collective models covered by the (quantum)/(master) hierarchy of Wigner-von Neumann-Moyal-Lindblad equations, which are the result of "wignerization" procedure (Weyl-Wigner-Moyal quantization) of classical BBGKY kinetic hierarchy, and present the explicit constructions for exact analyticalumerical computations. Our fast and efficient approach is based on variational and multiresolution representations in the bases of polynomial tensor algebras of generalized localized states (fast convergent variational-wavelet representation). We construct the representations for hierarchy/algebra of observables (symbols)/distribution functions via the complete multiscale decompositions, which allow to consider the polynomial and rational type of nonlinearities. The solutions are represented via the exact decomposition in nonlinear high-localized eigenmodes, which correspond to the full multiresolution expansion in all underlying hidden time/space or phase space scales. In contrast with different approaches we do not use perturbation technique or linearization procedures. Numerical modeling shows the creation of different internal structures from localized modes, which are related to the localized (meta) stable patterns (waveletons), entangled ensembles (with subsequent decoherence) and/or chaotic-like type of behaviour.
机译:在第二部分中,我们介绍了一组分析和数值方法,它们可以描述(非)平衡合奏中的行为,包括经典和量子行为,尤其是在无法应用标准方法的复杂系统中。证明这种方法优势的关键是:(ⅰ)可能的量子态局部化的影响; (ⅱ)无法通过摄动方法计算的非摄动多尺度的影响; (ⅲ)从局部模式形成复杂/集体量子模式的影响以及对量子态完整动物园的分类和可能的控制,包括(元)稳定的局部模式(小波)。我们在Wigner-von Neumann-Moyal-Lindblad方程的(量子)/(主)层次结构所涵盖的许多集体模型中,证明了非平凡的局部(元)稳定状态/模式的出现,这是“假想”的结果BBGKY动力学层次的过程(Weyl-Wigner-Moyal量化),并给出用于精确分析/数值计算的显式构造。我们的快速有效方法是基于广义局部状态的多项式张量代数的变分和多分辨率表示(快速收敛的变分小波表示)。通过完整的多尺度分解,我们构造了可观察性(符号)/分布函数的层次/代数的表示形式,从而可以考虑非线性的多项式和有理类型。通过非线性高局部本征模的精确分解来表示解,这对应于所有潜在的隐藏时间/空间或相空间标度中的完整多分辨率展开。与不同的方法相反,我们不使用扰动技术或线性化程序。数值建模显示了与局部模式不同的内部结构的产生,这些内部结构与局部(元)稳定模式(小波),纠缠的合奏(具有随后的去相干性)和/或类似混沌的行为类型有关。

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