New statistical models of nonergodic cognitive systems and their pathologies

Area of Science:

• Cognitive Science
• Information Theory
• Statistical Physics

Background:

• Cognition fundamentally involves reducing uncertainty, implying an external information source.
• Traditional information theory (Shannon entropy) is insufficient for path-dependent, non-ergodic systems where time and ensemble averages diverge.

Purpose of the Study:

• To extend information theory to non-ergodic systems by conceptualizing information as free energy.
• To develop new analytical tools for understanding complex cognitive dynamics and failure modes.

Main Methods:

• Utilizing directed homotopy groupoids instead of traditional groups for algebraic expression of complex dynamic topologies.
• Extending the Data Rate Theorem by linking control and information theories through spontaneous symmetry breaking analogies.
• Constructing dynamic empirical Onsager models based on the Legendre transform of path-dependent information uncertainties.

Main Results:

• Information is identified as a form of free energy, applicable to non-ergodic cognitive systems.
• A novel mathematical framework using directed homotopy groupoids is established for complex system dynamics.
• The methodology allows for the construction of dynamic Onsager models from information uncertainties.

Conclusions:

• The proposed framework offers new analytic tools for studying cognitive phenomena across various scales.
• Understanding information as free energy provides insights into failure modes in physiological, systemic, and economic contexts.