Fractal Series — Paper 1: Fractal Entropy — The E³ Model of State Transition and Emergent Steady State

Juan F. Culajay

Abstract:

Title:
Fractal Series — Paper 1: Fractal Entropy — The E³ Model of State Transition and Emergent Steady State

Abstract:
This first paper in the Fractal Series introduces the E³ Model — a universal framework describing how all systems evolve through the interplay of Energy, Environment, and Entropy. It reframes entropy not as disorder but as a law of flow that drives open systems toward adaptive balance. Using the dimensionless entropic potential (Ψ*), the model quantifies how energy throughput, entropy export, and environmental gradients co-determine the stability of physical, biological, and planetary systems. A variational derivation (Appendix 4) establishes Ψ* from non-equilibrium thermodynamics, while a stochastic toy model (Appendix 3) demonstrates dΨ/dt ≥ 0* during recovery from perturbation — confirming entropy’s constructive role in adaptation. The framework bridges physics, biology, and information theory into a single thermodynamic grammar of flow, offering testable predictions for molecular evolution, ecosystem resilience, and cosmic structure formation.

Keywords:
fractal thermodynamics, entropy, equilibrium, E³ model, complex systems, systems physics

Authors:
Juan F. Culajay — Principal Researcher, Fractalism Framework Research Institute

Affiliation:
Fractalism Framework Research Institute (fractalismframework.com)