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

摘要

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 analytical/numerical 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.