First-order integrability-breaking phase transitions in dynamical systems
Abstract
We investigate a discontinuous route from integrability to chaos using a confined stochastic random walk and a deterministic stadium-like billiard. In both systems, the stationary diffusive observable exhibits a finite jump at the transition: it vanishes at the unperturbed limit but approaches a finite, geometry-controlled value for arbitrarily small nonzero perturbations, providing the characteristic order-parameter signature of a first-order transition. Despite this discontinuity, the relaxation timescale diverges as the transition is approached, revealing critical slowing down. Both models exhibit normal diffusion with β=1/2, a perturbation-independent stationary state with α=0, and a crossover iteration scaling as nx∝λ−2, yielding z=−2, where λ denotes the corresponding perturbation parameter. The common exponent set (α,β,z)=(0,1/2,−2) originates from the same coarse-grained mechanism: normal diffusion within a finite accessible domain with a diffusion coefficient that vanishes quadratically at the transition. The agreement between stochastic transport and deterministic chaotic scattering provides strong evidence for a common class of discontinuous dynamical transitions and extends the statistical-mechanics description of phase transitions to integrability-breaking dynamics.
Related
- Integrability-breaking phase transitions in stadium-like billiards
- Digital Twins of Networked Dynamical Systems: Machine Learning Prediction of Synchronization Transitions in Coupled Chaotic Oscillators
- Intermittency-induced transitions in fast-slow dynamical systems
- Out-of-equilibrium finite-time percolation transitions and spinodal-like behaviors after quenches across magnetic first-order transitions of Ising systems
- Breaking of integrability in elliptic deformations of AdS3 strings with mixed flux
BibTeX
@misc{fonseca2026firstorder,
title = {{First-order integrability-breaking phase transitions in dynamical systems}},
author = {Anne Ketri P. da Fonseca and Marcelo de Almeida Presotto and Diego F. M. Oliveira and Edson D. Leonel},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.13616},
eprint = {2609.13616},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.13616},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - First-order integrability-breaking phase transitions in dynamical systems
AU - Anne Ketri P. da Fonseca
AU - Marcelo de Almeida Presotto
AU - Diego F. M. Oliveira
AU - Edson D. Leonel
PY - 2026
DA - 2026/09/12
PB - arXiv
DO - 10.48550/arxiv.2609.13616
UR - https://arxiv.org/abs/2609.13616
AB - We investigate a discontinuous route from integrability to chaos using a confined stochastic random walk and a deterministic stadium-like billiard. In both systems, the stationary diffusive observable exhibits a finite jump at the transition: it vanishes at the unperturbed limit but approaches a finite, geometry-controlled value for arbitrarily small nonzero perturbations, providing the characteristic order-parameter signature of a first-order transition. Despite this discontinuity, the relaxation timescale diverges as the transition is approached, revealing critical slowing down. Both models exhibit normal diffusion with β=1/2, a perturbation-independent stationary state with α=0, and a crossover iteration scaling as n_x∝λ^−2, yielding z=−2, where λ denotes the corresponding perturbation parameter. The common exponent set (α,β,z)=(0,1/2,−2) originates from the same coarse-grained mechanism: normal diffusion within a finite accessible domain with a diffusion coefficient that vanishes quadratically at the transition. The agreement between stochastic transport and deterministic chaotic scattering provides strong evidence for a common class of discontinuous dynamical transitions and extends the statistical-mechanics description of phase transitions to integrability-breaking dynamics.
ER -