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Analytical Research on the Eigenvalue Problem of the Evolution Operator of Distribution Functions for Nonlinear Systems exhibiting Chaos and Bifurcation

混沌分岔非线性系统分布函数演化算子特征值问题的解析研究

基本信息

批准号：
09640473
负责人：
TASAKI Shuichi
金额：
$ 1.54万
依托单位：
Nara Women's University
依托单位国家：
日本
项目类别：
Grant-in-Aid for Scientific Research (C)
财政年份：
1997
资助国家：
日本
起止时间：
1997 至 1999
项目状态：
已结题

项目摘要

We studied analytically and exactly the behaviors of the distribution functions for dynamical systems including chaotic systems. Following results are obtained : 1)SRB measures for dissipative Baker maps : Invariant distributions and physical measures are constructed with the aid of de Rham's functional equations and their properties are investigated. 2)Microscopic mechanism of dissipation for a MultiBaker map with energy : We introduced a multibaker map with a coordinate corresponding to kinetic energy and studied nonequilibrium stationary states, relaxation modes, scaling-limit behaviors, time evolution towards the past. We found that the dynamical reversibility is consistent with irreversible evolution of distribution functions. 3)Boundary element methods (BEM) for 2d quantum billiards and Fredholm theory : We studied the properties of a functional determinant D(E), which appears in BEM for 2d quantum billiards, and its semiclassical limit with the aid of the Fredholm theory. 4)Quantum nonequilibrium stationary states : We studied the behaviors of a 1d noninteracting electron system placed between two perfect conductors with the aid of CィイD1*ィエD1-algebraic method. Two different stationary states are obtained in the limit of t →±∞. By comparing their properties, the dynamical reversibility is shown to be consistent with irreversible evolution of states. 5)Unstable quantum states : Unstable quantum states can be represented in terms of generalized eigenfunctions of the Hamiltonian such as Gamow vectors. We investigated the properties of two interacting unstable states and necessary mathematical backgrounds.Also we investigated other related topics such as 6)Optical properties of carbon nanotubes, 7)Spectral properties of evolution operators of distribution functions for Hopf bifurcation and intermittent maps, 8)Transport properties of periodic intermittent maps, and 9)Stationary states for a reaction-diffusion type MultiBaker map.

我们用解析的方法精确地研究了包括混沌系统在内的动力系统的分布函数的行为。1)耗散Baker映射的SRB测度：借助于De Rham函数方程构造了不变分布和物理测度，并研究了它们的性质。2)能量多重贝克映射耗散的微观机制：我们引入了一个动能坐标对应的多重贝克映射，研究了非平衡定态、弛豫模式、标度极限行为、向过去的时间演化。我们发现，动态可逆性与分布函数的不可逆演化是一致的。3)二维量子台球的边界元方法(BEM)和Fredholm理论：利用Fredholm理论研究了二维量子台球边界元中出现的泛函行列式D(E)的性质及其半经典极限。4)量子非平衡定态：用CィイD_1*ィエD_1-代数方法研究了置于两个理想导体之间的一维非相互作用电子系统的行为。在t→±∞的极限下，得到了两种不同的定态。通过比较它们的性质，表明动态可逆性与状态的不可逆演化是一致的。5)不稳定量子态：不稳定量子态可以用哈密顿量的广义本征函数来表示，如Gamow矢量。我们研究了两个相互作用的不稳定态的性质和必要的数学背景，还研究了其他相关问题，如6)碳纳米管的光学性质，7)Hopf分叉和间歇映射分布函数的演化算符的光谱性质，8)周期间歇映射的输运性质，9)反应扩散型多重Baker映射的定态。