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Chaotic renormalization group flow and entropy gradients over Haros graphs
Jorge Calero-Sanz1,2, Bartolo Luque1, Lucas Lacasa3
1Departamento de Matemática Aplicada a la Ingeniería Aeroespacial, ETSIAE, Universidad Politécnica de Madrid, Madrid, Spain.
Physical Review. E
|May 18, 2023
Summary
This study explores the chaotic dynamics of a graph operator on Haros graphs, revealing a stable fixed point linked to rational numbers and complex orbits for irrationals. Graph entropy decreases nonmonotonically towards this fixed point.
Area of Science:
- Graph Theory
- Dynamical Systems
- Renormalization Group Theory
Background:
- Haros graphs provide a bijective relationship between graphs and real numbers in the unit interval.
- A graph operator R, with a renormalization group (RG) structure, has been defined for characterizing low-dimensional nonlinear dynamics.
Purpose of the Study:
- To investigate the iterated dynamics of the graph operator R on Haros graphs.
- To analyze the chaotic behavior, fixed points, periodic orbits, and entropy changes within this RG flow.
Main Methods:
- Iterated application of the graph operator R to Haros graphs.
- Analysis of the resulting dynamical system, identifying fixed points and orbits.
- Calculation and analysis of graph entropy changes along the RG flow.
Main Results:
- The dynamics of R on Haros graphs exhibit chaotic behavior, including unstable periodic and nonmixing aperiodic orbits.
- A single stable fixed point is identified, with its basin of attraction corresponding to rational numbers.
- Periodic orbits are linked to quadratic irrationals, and aperiodic orbits to algebraic and transcendental numbers; graph entropy decreases nonmonotonically towards the fixed point.
Conclusions:
- The renormalization group flow on Haros graphs is chaotic, with distinct number sets mapping to different dynamical behaviors.
- Graph entropy exhibits a nonmonotonic decrease along the RG flow, with constant entropy observed for metallic ratios.
- The findings offer insights into the physical interpretation of chaotic RG flows and entropy gradients, relating to c-theorems.
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