Eigen microstates and their evolutions in complex systems

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 x 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 = Sigma(r)(I=1) sigma U-I(I) circle times V-I , where r = min(N, M), eigenvalue sigma(I) behaves as the probability amplitude of the eigen microstate U-I so that Sigma(r)(I=1) sigma(2)(I) = 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 sigma(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.