Analytical Study on the State-Evolution Operator and Nonequilibrium States for Nonlinear Dynamical Systems including Chaotic Systems

我们主要研究了非线性动力系统和无限扩展量子系统的非平衡态。结果总结如下：(1)非线性动力系统的统计行为(a)双曲系统：对于无限扩展双曲系统的非平衡稳态，称为多贝克映射，比较了Gaspard等人的相对熵产和粗粒度熵产，并在适当的标度极限下与热力学一致。(b)非双曲系统：对于单位区间上的分段线性间歇映射，相关的多项式衰减表现为Frobenius-Perron算子的连续谱。另一方面，对于具有超扩散的空间扩展分段线性间歇映射，发现Frobenius-Perron算子1附近的特征值控制了异常扩散。(2)无限扩展量子系统的非平衡态(a) C*动力系统：将C*代数方法应用于一维晶格导体，严格构造了非平衡稳态，并证明了Landauer公式的有效性和相对熵产生的正性。另一方面，对于渐近阿贝尔C*动力系统，我们证明了非平衡稳态的存在性、Gallavotti-Cohen涨落定理的有效性以及与Zubarev-MacLennan系综的等价性。(b)退相干控制：在开放系统中，与环境的相互作用降低了量子相干性（退相干）。对于耦合环境场的三能级系统，研究了动态解耦控制和量子芝诺控制，非理想控制可以增强退相干。

We have mainly investigated nonequilibrium states for nonlinear dynamical systems and, infinitely extended quantum systems. The results are summarized as follows :(1)STATISTICAL BEHAVIOR OF NONLINEAR DYNAMICAL SYSTEMS (a)Hyperbolic Systems : For nonequilibrium steady states of an infinitely extended hyperbolic system, called a multibaker map, the relative entropy production and a coarse-grained entropy production of Gaspard et al. are is compared and are shown to be consistent with each other and with thermodynamics in an appropriate scaling limit. (b)Nonhyperbolic Systems : For a piecewise linear intermittent map on the unit interval, the polynomial decay of correlation is shown to be characterized by a continuous spectrum of the Frobenius-Perron operator. On the other hand, for a spatially extended piecewise linear intermittent map exhibiting super diffusion, eigenvalues near 1 of the Frobenius-Perron operator are found to control the anomalous diffusion. (2)NONEQUILIBRIUM STATES OF INFINITELY EXTENDED QUANTUM SYSTEMS (a) C* Dynamical Systems : Applying the C* algebraic method to a 1-d lattice conductor, nonequilibrium steady states are rigorously constructed and the validity of Landauer formula & the positivity of the relative entropy production are shown. On the other hand, for asymptotically abelian C* dynamical systems, we have shown the existence of nonequilibrium steady states, the validity of Gallavotti-Cohen fluctuation theorem and the equivalence with the Zubarev-MacLennan ensembles, (b)Decoherence Control : In open systems, the interaction with environment reduces quantum coherence (decoherence). For a three-level system coupled with an environmental field, the dynamical decoupling control and quantum Zeno control are investigated and non-ideal controls are shown to enhance decoherence.