Dynamic Balance: A Thermodynamic Principle for the Emergence of the Golden Ratio in Open Non-Equilibrium Steady States

Abstract

We propose that a single irrational constant, ϕ ≈ 1.618, serves as a universal attractor in far-from-equilibrium systems in a steady-state. In our Dynamic Balance framework, the dimensionless ratio α = Ė/T Ṡ-comparing the system's energy throughput Ė to its entropic heat loss T Ṡ-naturally converges to ϕ in a wide range of open, dissipative processes. This ratio enforces a balance condition that partitions energy into useful work (order) and dissipative loss (disorder) in a way that maximizes both stability and adaptability. We demonstrate the principle using local cost functions, gradient-flow PDE expansions, Wilsonian renormalization group methods, and Markov master-equation analyses, showing in particular how it unifies nearcritical behavior in the brain: the resulting neurodynamic PDE reproduces avalanche scaling, fractal wave expansions, short-term plasticity responses, and morphological growth patterns without ad hoc saturations. More broadly, the Dynamic Balance principle complements maximum entropy production and self-organized criticality in physics, parallels metabolic efficiency and fractal organization in biology, and implies optimal resource division in geology. We review empirical evidence-from stellar pulsations and fluid vortices to branching structures, quantum critical phenomena, and neural avalanche data-indicating that ϕ emerges as a robust signature of dynamic organization in non-equilibrium state-states.

Keywords: golden ratio, Fibonacci sequence, TBG magic angle, variable star, rotating turbulence, quantum critical chains, neural activity exponents, criticality, Phyllotaxis, Neural avalanches, Fibonacci anyons, Penrose Tiling, Galactic spirals, Non-Equilibrium Steady States, far-from-equilibrium dynamics