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Scaling Invariance: A Gateway to Phase Transitions
1Departamento de Física, Universidade Estadual Paulista (UNESP), Av. 24A, 1515, São Paulo 13506-900, SP, Brazil.
Entropy (Basel, Switzerland)
|August 28, 2025
Summary
This study reveals scaling invariance in dynamical systems transitioning from regularity to chaos. This behavior mirrors a continuous phase transition, evidenced by diverging susceptibility near the critical point.
Area of Science:
- Nonlinear dynamics
- Statistical mechanics
- Chaos theory
Background:
- Dynamical systems can transition between ordered (regular) and disordered (chaotic) states.
- Hamiltonian systems, characterized by action and angle variables, are often used to model such transitions.
- Phase transitions in physical systems involve critical phenomena and scaling laws.
Purpose of the Study:
- To investigate scaling invariance in a specific class of 2D nonlinear area-preserving mappings.
- To characterize the transition from regular to chaotic behavior in these systems.
- To determine if this transition exhibits properties of a continuous phase transition.
Main Methods:
- Utilized a two-dimensional nonlinear mapping preserving area in phase space.
- Analyzed the behavior of action and angle variables under variation of a control parameter.
- Calculated the average squared action and susceptibility within the chaotic regime.
Main Results:
- Observed scaling invariance in the average squared action in the chaotic region.
- Identified an order parameter that approaches zero at the transition point.
- Demonstrated that susceptibility diverges as the order parameter approaches zero.
Conclusions:
- The transition from regularity to chaos in these systems exhibits characteristics of a second-order or continuous phase transition.
- Scaling invariance provides evidence for universality in the transition to chaos.
- The diverging susceptibility further supports the analogy to phase transitions in statistical mechanics.
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