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

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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.