反常与非遍历多尺度建模、分析及算法  

Anomalous and nonergodic multiscale modeling,analyses and algorithms

作　　者：邓伟华[1] Weihua Deng

机构地区：[1]兰州大学数学与统计学院,兰州730000

出　　处：《中国科学：数学》2023年第8期1039-1066,共28页Scientia Sinica：Mathematica

基　　金：国家自然科学基金(批准号:12225107);甘肃省基础研究创新群体项目(批准号:22JR5RA391)资助项目。

摘　　要：反常与非遍历现象在自然界中无处不在.随着实验技术的发展,各种新的实验现象仍不断涌现.为解释现象/揭示机理,本文建立微观模型(随机过程),并对微观模型进行随机分析和统计分析.同时,本文推导出一些宏观统计量(如粒子位置、泛函、逃逸概率和首次通过时间等)的概率分布满足的宏观方程,进而研究方程解的正则性及方程的数值计算方法.非静止介质中的反常与非遍历多尺度动力学将是数学和自然科学中有重要理论价值和应用前景的研究课题.流形上的反常与非遍历动力学也是值得研究的课题.所有情形下的跨尺度反常与非遍历动力学是重要的研究课题.Anomalous and nonergodic phenomena are ubiquitous in nature.With the development of experi-mental technology,various new experimental phenomena are still emerging.To explain the phenomena/reveal the mechanism,we establish the microscopic models(i.e.,random processes)and conduct the stochastic analysis and statistical analysis on the microscopic model.At the same time,we derive the macroscopic equation that the probability distribution of some macroscopic statistical observables(such as the particle position,functional of the particle's trajectory,escape probability,and rst passage time)satis es,and then study the regularity of the solution of the equations and the numerical computational method of the equations.Anomalous and nonergodic multiscale dynamics in non-static media will be a research topic with important theoretical values and applica-tion prospects in natural science.Anomalous and nonergodic dynamics on manifolds are also worth studying.Cross-scale anomalous and nonergodic dynamics in all cases are important research topics too.

关 键 词：可计算建模 非平衡态统计物理 非静态介质 分析与算法 

分 类 号：O144.2[理学—数学]