An Exact Theory of Causal Emergence for Linear Stochastic Iteration Systems.

Area of Science:

• Complex Systems Theory
• Statistical Mechanics
• Information Theory

Background:

• Coarse-graining complex systems can reveal emergent causal effects at the macro-state level.
• Causal emergence is quantified by effective information, but lacks frameworks for continuous stochastic systems.
• Existing coarse-graining methods present theoretical challenges.

Purpose of the Study:

• To develop an exact theoretical framework for causal emergence in linear stochastic systems with continuous state spaces.
• To derive an analytical expression for effective information in general dynamics.
• To identify optimal linear coarse-graining strategies for maximizing causal emergence.

Main Methods:

• Developed a theoretical framework for causal emergence in linear stochastic iteration systems.
• Derived analytical expressions for effective information.
• Identified optimal linear coarse-graining strategies based on system parameters.
• Validated models using simplified physical systems and numerical simulations.

Main Results:

• Successfully established a framework for causal emergence in continuous stochastic systems.
• Derived an analytical expression for effective information applicable to general dynamics.
• Determined that maximal causal emergence and optimal coarse-graining depend on principal eigenvalues and eigenvectors.
• Achieved congruent results between analytical models and numerical simulations.

Conclusions:

• The study provides a robust theoretical foundation for causal emergence in continuous stochastic systems.
• Optimal coarse-graining strategies are linked to the spectral properties of the system's parameter matrix.
• The findings offer new insights into understanding and quantifying emergent causality in complex systems.