Eigen microstates and their evolutions in complex systems  

作　　者：Yu Sun Gaoke Hu Yongwen Zhang Bo Lu Zhenghui Lu Jingfang Fan Xiaoteng Li Qimin Deng Xiaosong Chen 

机构地区：[1]Institute of Theoretical Physics,Key Laboratory of Theoretical Physics,Chinese Academy of Sciences,PO Box 2735,Beijing 100190,China [2]School of Physical Sciences,University of Chinese Academy of Science,No.19A Yuquan Road,Beijing 100049,China [3]Beijing Computational Science Research Center,Beijing 100193,China [4]Data Science Research Center,Kunming University of Science and Technology,Kunming 650500,China [5]Laboratory for Climate Studies,National Climate Center,China Meteorological Administration,Beijing 100081,China [6]CAS Key Laboratory of Regional Climate Environment for Temperate East Asia,Institute of Atmospheric Physics,Chinese Academy of Sciences,Beijing 100029,China [7]School of Systems Science,Beijing Normal University,Beijing 100875,China [8]Harvest Fund Management,Beijing 100021,China [9]Lab for Climate and Ocean-Atmosphere Studies,Dept,of Atmospheric and Oceanic Sciences,School of Physics,Peking University,Beijing,100871,China

出　　处：《Communications in Theoretical Physics》2021年第6期140-157,共18页理论物理通讯（英文版）

基　　金：supported by the Key Research Program of Frontier Sciences,Chinese Academy of Sciences(Grant No.QYZD-SSW-SYS019)。

摘　　要：Emergence refers to the existence or formation of collective behaviors in complex systems.Here,we develop a theoretical framework based on the eigen microstate theory to analyze the emerging phenomena and dynamic evolution of complex system.In this framework,the statistical ensemble composed of M microstates of a complex system with N agents is defined by the normalized N×M matrix A,whose columns represent microstates and order of row is consist with the time.The ensemble matrix A can be decomposed as■,where r=min(N,M),eigenvalueσIbehaves as the probability amplitude of the eigen microstate U_I so that■and U_I evolves following V_I.In a disorder complex system,there is no dominant eigenvalue and eigen microstate.When a probability amplitudeσIbecomes finite in the thermodynamic limit,there is a condensation of the eigen microstate UIin analogy to the Bose–Einstein condensation of Bose gases.This indicates the emergence of U_I and a phase transition in complex system.Our framework has been applied successfully to equilibrium threedimensional Ising model,climate system and stock markets.We anticipate that our eigen microstate method can be used to study non-equilibrium complex systems with unknown orderparameters,such as phase transitions of collective motion and tipping points in climate systems and ecosystems.

关 键 词：complex system phase transition critical phenomena Earth system statistical ensemble eigen microstate dynamic evolution ECONOPHYSICS 

分 类 号：O414.2[理学—理论物理]