Eigen microstates and their evolutions in complex systems

Abstract 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 A = ∑ I = 1 r σ I U I ⨂ V I , where r = min ( N , M ) , eigenvalue σ I behaves as the probability amplitude of the eigen microstate U I so that ∑ I = 1 r σ I 2 = 1 and U I evolves following V I . In a disorder complex system, there is no dominant eigenvalue and eigen microstate. When a probability amplitude σ I becomes finite in the thermodynamic limit, there is a condensation of the eigen microstate U I in 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 three-dimensional 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 order-parameters, such as phase transitions of collective motion and tipping points in climate systems and ecosystems.