Statistical properties of complex systems with subexponetnatial instability and phase transition

具有次指数不稳定和相变的复杂系统的统计特性

基本信息

批准号：
15540135
负责人：
YURI Michiko
金额：
$ 1.98万
依托单位：
Sapporo University
依托单位国家：
日本
项目类别：
Grant-in-Aid for Scientific Research (C)
财政年份：
2003
资助国家：
日本
起止时间：
2003 至 2004
项目状态：
已结题

来源：
https://kaken.nii.ac.jp/en/grant/KAKENHI-PROJECT-15540135/
关键词：
Nonhyperbolicity Intermittency Phase transition Variational principle Multifractal formalism Equilibrium state Weak Gibbs measure Indifferent periodic point Variational Principle Large deviation indifferent periodic point iterated fu

项目摘要

One of the purpose of this project is to clarify how statistical properties of complex systems is influenced by subexponential instability of the dynamics. In particular, we direct our attention to non-hyperbolic phenomena exhibiting phase transitions. For this purpose, piecewise invertible systems with generalized indifferent periodic orbits associated to a given potential function are considered. For such systems, it was shown in [1] that the presence of such orbits causes non-uniqueness of equilibrium states (phase transitions) and non-Gibbsianness of equilibrium measures. More specifically, we established in [1] that non-Gibbsian behaviour of equilibrium states in the sense of Bowen, non-differentiability of the pressure function (phase transiton), powerlike tails of the distribution of the stopping times over hyperbolic regions, and the Hausdorff dimension of level sets associated to pointwise dimension. In particular, non-differentiability of pressure functions is related to multifractal problem. We also established in [2] that the natural extensions of invariant ergodic weak Gibbs measures absolutely continuous with respect to weak Gibbs conformal measures possess a version of u -Gibbs property. In particular, if dynamical potentials admit generalized indifferent periodic points, then the natural extensions exhibit non-Gibbsian character in statistical mechanics. Another purpose of this project is to associate non-Gibbsian weak Gibbs measures for intermittent maps to non-Gibbsian weakly Gibbssian states in statistical mechanics in the sense of Dobrushin. This purpose was achiedved in [ 31 and we showed a higher dimensional intermittent map of which Sinai-Bowen-Ruelle measure is a weak Gibbs equilibrium state and a weakly Gibbsian state in the sense of Dobrushin admitting essential discontinuities in its conditional probabilities.

该项目的目的之一是阐明复杂系统的统计属性如何受动力学的次指定不稳定的影响。特别是，我们将注意力转移到表现出相变的非纤维化现象上。为此，考虑了与给定潜在功能相关的具有广义无关的周期性轨道的分段可逆系统。对于此类系统，在[1]中表明，这种轨道的存在会导致平衡状态（相变）和平衡度量的非吉布斯主义性。更具体地说，我们在[1]中确定了平衡状态在鲍恩（Bowen）意义上，压力函数的非差异性（相通顿）的非差异性，止血区域上停止时间分布的尾巴的非差异性，以及与点循环减小相关的Hausdorff尺寸。特别是，压力函数的非差异性与多重分子问题有关。我们还在[2]中确定，不变的千古弱吉布斯的自然扩展与弱Gibbs共形度量绝对连续，具有U -Gibbs属性的版本。特别是，如果动态电势允许通用的漠不关心的周期点，则自然扩展在统计力学中表现出非吉布斯的特征。该项目的另一个目的是将间歇地图的非Gibbsian弱Gibbs措施与统计力学的非Gibbsian弱Gibbssian状态相关联。在[31中对此目的进行了痛苦，我们显示了一个较高维度的间歇性图，西奈 - 鲍恩 - 荷兰措施是弱的gibbs平衡状态，而在多布鲁什（Dobrushin）的意义上是弱小的吉布斯（Gibbsian）状态。