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.

Eigen microstates and their evolution of global ozone at different geopotential heights

“…The eigen microstate method [33] has shown its great ability to deal with the collective behaviors in complex systems. Here, we introduce how to implement this method into the ozone system.…”

“…To better analyze the mechanism of the emergence of collective behaviors in OMMR, we remove the seasonal effect in the following analysis. For a complex systems without localization of microstate, the weight of an eigen microstate approaches zero in the thermodynamic limit [33]. But for the system of ozone, the variation is affected by both natural and anthropogenic variation.…”

“…For example, it improved the prediction of ENSO [24,25], and revealed the influence of Rossby waves in carbon dioxide (CO 2 ) concentration [40][41][42], air pollution [44] and extreme-rainfall teleconnections [4]. Eigen microstate method [33] is a newly developed method, and can effectively analyze the emerging phenomena and the dynamic evolution of complex systems in non-equilibrium, such as the complex climate systems. It can be also used to identify the phase transition and universality class of complex systems [17].…”

Quantum-Like Criticality in Granular Fluids

“…Recently, criticality phenomena have also been expected to hold for collective motion of other many-body complex systems at different scales, from animal groups 17 to escape panic 18 and individual cells and molecular motors 19,20 . As the weight of an eigen microstate in a complex system has a finite limit, there is condensation of the eigen microstate in the ensemble, which is similar to the Bose-Einstein condensation of a Bose gas [21][22][23] . Granular fluids, which are typical many-body complex systems, are widely observed in both nature and industry [24][25][26] .…”

Renormalization Group Theory of Eigen Microstates

“…[18,19] Nowadays, we need to study urgently the critical phenomena of complex systems without knowledge of their Hamiltonian. [20,21] Recently, we proposed an eigen microstate approach (EMA) [22,23] based on microstates obtained from experiments or simulations. The statistical ensemble composed of the microstates during an observation time period is described by a normalized matrix.…”