Eigen Micostate Theory of Complex Systems and its Application in Earth System

For a complex system consisting of large collection of interacting objects, the states of all individuals at a certain time are defined as a microstate of the system following the Gibbs’s theory of statistical physics. All microstates during the time period of observation constitute an ensemble of the complex system. The normalized N×M matrix A of the ensemble, whose columns represent microstates and rows characterize time evolution, can be decomposed as A=∑I=1r σI UI⊗VI, where r=min⁡(N,M),  σI2 is the probability of the eigen microstate UI with time evolution VI. The change of σI from zero to finite indicates the condensation of eigen microstate and a phase transition of complex system. This eigen microstate approach (EMA) has applied successfully to different complex systems. We anticipate that the EMA can be used to study phase transitions of subsystems of different scales in Earth System.