复动力系统的淹没点与玻璃态相变  

Buried points in complex dynamical systems and glassy transitions

作　　者：乔建永 Jianyong Qiao(School of Science,Beijing University of Posts and Telecommunications,Beijing 100876,China)

机构地区：[1]北京邮电大学理学院,北京100876

出　　处：《科学通报》2022年第1期11-17,共7页Chinese Science Bulletin

基　　金：国家重点研发计划(2019YFB1406500);国家自然科学基金(12071047)。

摘　　要：重整化群理论是在粒子物理研究中为克服微扰发散困难而进行标度变换,从而得到群不变性的一种理论.该理论于1971年由Wilson[1]提出,被广泛用于研究凝聚态物理的相变问题,其影响巨大,于1982年获得诺贝尔物理学奖.近年来,重整化群理论在信息传播、系统科学、材料科学和工程技术等许多学术领域都有十分深刻的应用,展示出极强的普遍应用价值.本文推介复动力系统与统计物理的交叉研究方向,希望以重整化群理论为桥梁,进一步探讨复动力系统混沌集合的拓扑复杂性与玻璃态相变的深层次联系.A new interdisciplinary research direction is addressed across the fields of statistical mechanics and complex dynamical systems together with specific research topics,such as the connection between buried points and glassy transitions.First,pertinent results on complex dynamical systems are delivered.Let f be a complex analytical mapping from the Riemann sphere(or the complex plane)to itself,i.e.,f is rational or entire.The stable set F(f)of this dynamical system is called the Fatou set of f,and the unstable set J(f)the Julia set.Every component of F(f)is called a Fatou component.Obviously,the boundary of each Fatou component belongs to J(f).If a point in J(f)does not belong to the boundary of any Fatou component,then it is a buried point of f,which is often used to describe the topological complexity of the Julia set.In this quest,Makienko’s conjecture is undoubtedly one of the most important problems which is stated as follows:Let R be a rational mapping and F(R)contain infinitely many components.If every Fatou component is not completely invariant,then R has buried points.The known result is the following:If J(R)is disconnected,or connected and locally connected,then Makienko’s conjecture is true;if J(R)is disconnected and R has a buried point,then it must have buried components.Concerning entire functions,we have:If F(f)is not empty,then f has a buried point if and only if F(f)is disconnected.Following the above results,interesting topological properties on the sets of buried points are deduced.However,many open problems remain unsolved in regard to these special points.In the second part,we explained the topological complexity of the set of Yang-Lee zeros.In 1952,Yang C.N.and Lee T.D.established an analytic theory in statistical mechanics,summarized as the circle theorem:For statistical models like 2-dimensional lattice gas,the zeros of the partition function condense to the unit circle.Henceforth,the importance of the distribution of Yang-Lee zeros in the complex plane is emphasized for general models.

关 键 词：重整化群理论 诺贝尔物理学奖 凝聚态物理 玻璃态 信息传播 标度变换 统计物理 工程技术 

分 类 号：O414.2[理学—理论物理] O19[理学—物理]