Scaling Invariance: A Gateway to Phase Transitions

Edson D. Leonel

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arXiv · 2504.06187

Scaling Invariance: A Gateway to Phase Transitions

Edson D. Leonel

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Abstract

We explore the concept of scaling invariance in a type of dynamical systems that undergo a transition from order (regularity) to disorder (chaos). The systems are described by a two-dimensional, nonlinear mapping that preserves the area in the phase space. The key variables are the action and the angle, as usual from Hamiltonian systems. The transition is influenced by a control parameter giving the form of the order parameter. We observe a scaling invariance in the average squared action within the chaotic region, providing evidence that this change from regularity (integrability) to chaos (non-integrability) is akin to a second-order or continuous phase transition. As the order parameter approaches zero, its response against the variation of the control parameter (susceptibility) becomes increasingly pronounced (indeed diverging), resembling a phase transition. These findings could not be obtained without a seminal paper on Phys. Rev. Lett. {\bf 2004}, {\em 93}, 014101.

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Edson D. Leonel. 2025-04-08. Scaling Invariance: A Gateway to Phase Transitions. https://arxiv.org/abs/2504.06187

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