Ruiz Alejandro
Independent Researcher, Sacramento, CA 95814, USA.
Entropy (Basel). 2025 Jul 11;27(7):745. doi: 10.3390/e27070745.
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 PGL(2,Q(5)). 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 62%:38% split between entropy production and useful power, an RG-invariant diffusion coefficient linking relaxation time and correlation length Dα=ξz/τ, and a ϑ=45∘ 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.
我们发展了一种基于对称性的变分理论，该理论表明在一个驱动耗散系统中，当弛豫到黄金比例时，功流入与热流出之间的粗粒化平衡。两个二阶莫比乌斯变换——一个自对偶翻转和一个自相似移位——生成了PGL(2,Q(√5))的一个离散非阿贝尔子群。要求任何光滑、严格凸的李雅普诺夫泛函在这两个映射下都不变，会强制产生一个单一的非平衡不动点：黄金均值。我们通过以下方式证实了这一结果：(i) 一个梯度流偏微分方程；(ii) 一个生死马尔可夫链，其连续极限是福克 - 普朗克方程；(iii) 一个马丁 - 西格西亚 - 罗斯场论；以及(iv) 保护不动点免受噪声影响的精确沃德恒等式。微观动力学仅仅设定了趋近速率；出现了三个无参数不变量：熵产生与有用功率之间62%:38%的比例、一个将弛豫时间和关联长度联系起来的重整化群不变扩散系数Dα = ξ^z / τ，以及一个映射到黄金对数螺旋的θ = 45°特征角。相同的对偶对称性是旋转湍流、植物叶序、皮层雪崩、量子临界金属甚至德西特宇宙学中尺度定律的基础，为远离平衡态的模式形成提供了一个可证伪的统一原理。
