Dynamic Balance: A Thermodynamic Principle for the Emergence of the Golden Ratio in Open Non-Equilibrium Steady States
We develop a symmetry-based variational theory that shows the coarse-grained balance of work inflow to heat outflow in a driven, dissipative system relaxed to the golden ratio. Two order-2 Möbius transformations—a self-dual flip and a self-similar shift—generate a discrete non-abelian subgroup of &l...
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| description | We develop a symmetry-based variational theory that shows the coarse-grained balance of work inflow to heat outflow in a driven, dissipative system relaxed to the golden ratio. Two order-2 Möbius transformations—a self-dual flip and a self-similar shift—generate a discrete non-abelian subgroup of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>PGL</mi><mo>(</mo><mn>2</mn><mo>,</mo><mi mathvariant="double-struck">Q</mi><mrow><mo>(</mo><msqrt><mn>5</mn></msqrt><mo>)</mo></mrow><mo>)</mo></mrow></semantics></math></inline-formula>. Requiring any smooth, strictly convex Lyapunov functional to be invariant under both maps enforces a single non-equilibrium fixed point: the golden mean. We confirm this result by (i) a gradient-flow partial-differential equation, (ii) a birth–death Markov chain whose continuum limit is Fokker–Planck, (iii) a Martin–Siggia–Rose field theory, and (iv) exact Ward identities that protect the fixed point against noise. Microscopic kinetics merely set the approach rate; three parameter-free invariants emerge: a <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mn>62</mn><mo>%</mo><mo>:</mo><mn>38</mn><mo>%</mo></mrow></semantics></math></inline-formula> split between entropy production and useful power, an RG-invariant diffusion coefficient linking relaxation time and correlation length <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi mathvariant="script">D</mi><mi>α</mi></msub><mo>=</mo><msup><mi>ξ</mi><mi>z</mi></msup><mo>/</mo><mi>τ</mi></mrow></semantics></math></inline-formula>, and a <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>ϑ</mi><mo>=</mo><msup><mn>45</mn><mo>∘</mo></msup></mrow></semantics></math></inline-formula> eigen-angle that maps to the golden logarithmic spiral. The same dual symmetry underlies scaling laws in rotating turbulence, plant phyllotaxis, cortical avalanches, quantum critical metals, and even de-Sitter cosmology, providing a falsifiable, unifying principle for pattern formation far from equilibrium. |
| format | Article |
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| spelling | doaj-art-94074e7fb4d44afc81af51e0907c2d4a2025-08-20T02:45:45ZengMDPI AGEntropy1099-43002025-07-0127774510.3390/e27070745Dynamic Balance: A Thermodynamic Principle for the Emergence of the Golden Ratio in Open Non-Equilibrium Steady StatesAlejandro Ruiz0Independent Researcher, Sacramento, CA 95814, USAWe develop a symmetry-based variational theory that shows the coarse-grained balance of work inflow to heat outflow in a driven, dissipative system relaxed to the golden ratio. Two order-2 Möbius transformations—a self-dual flip and a self-similar shift—generate a discrete non-abelian subgroup of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>PGL</mi><mo>(</mo><mn>2</mn><mo>,</mo><mi mathvariant="double-struck">Q</mi><mrow><mo>(</mo><msqrt><mn>5</mn></msqrt><mo>)</mo></mrow><mo>)</mo></mrow></semantics></math></inline-formula>. Requiring any smooth, strictly convex Lyapunov functional to be invariant under both maps enforces a single non-equilibrium fixed point: the golden mean. We confirm this result by (i) a gradient-flow partial-differential equation, (ii) a birth–death Markov chain whose continuum limit is Fokker–Planck, (iii) a Martin–Siggia–Rose field theory, and (iv) exact Ward identities that protect the fixed point against noise. Microscopic kinetics merely set the approach rate; three parameter-free invariants emerge: a <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mn>62</mn><mo>%</mo><mo>:</mo><mn>38</mn><mo>%</mo></mrow></semantics></math></inline-formula> split between entropy production and useful power, an RG-invariant diffusion coefficient linking relaxation time and correlation length <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi mathvariant="script">D</mi><mi>α</mi></msub><mo>=</mo><msup><mi>ξ</mi><mi>z</mi></msup><mo>/</mo><mi>τ</mi></mrow></semantics></math></inline-formula>, and a <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>ϑ</mi><mo>=</mo><msup><mn>45</mn><mo>∘</mo></msup></mrow></semantics></math></inline-formula> eigen-angle that maps to the golden logarithmic spiral. The same dual symmetry underlies scaling laws in rotating turbulence, plant phyllotaxis, cortical avalanches, quantum critical metals, and even de-Sitter cosmology, providing a falsifiable, unifying principle for pattern formation far from equilibrium.https://www.mdpi.com/1099-4300/27/7/745non-equilibrium thermodynamicsentropycriticalitybranching and phyllotaxisneural avalanchesFibonacci anyons |
| spellingShingle | Alejandro Ruiz Dynamic Balance: A Thermodynamic Principle for the Emergence of the Golden Ratio in Open Non-Equilibrium Steady States Entropy non-equilibrium thermodynamics entropy criticality branching and phyllotaxis neural avalanches Fibonacci anyons |
| title | Dynamic Balance: A Thermodynamic Principle for the Emergence of the Golden Ratio in Open Non-Equilibrium Steady States |
| title_full | Dynamic Balance: A Thermodynamic Principle for the Emergence of the Golden Ratio in Open Non-Equilibrium Steady States |
| title_fullStr | Dynamic Balance: A Thermodynamic Principle for the Emergence of the Golden Ratio in Open Non-Equilibrium Steady States |
| title_full_unstemmed | Dynamic Balance: A Thermodynamic Principle for the Emergence of the Golden Ratio in Open Non-Equilibrium Steady States |
| title_short | Dynamic Balance: A Thermodynamic Principle for the Emergence of the Golden Ratio in Open Non-Equilibrium Steady States |
| title_sort | dynamic balance a thermodynamic principle for the emergence of the golden ratio in open non equilibrium steady states |
| topic | non-equilibrium thermodynamics entropy criticality branching and phyllotaxis neural avalanches Fibonacci anyons |
| url | https://www.mdpi.com/1099-4300/27/7/745 |
| work_keys_str_mv | AT alejandroruiz dynamicbalanceathermodynamicprinciplefortheemergenceofthegoldenratioinopennonequilibriumsteadystates |