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Phase transition in piecewise linear random maps in the interval
Cesar Maldonado1, Ricardo A Pérez Otero1
1División de Control y Sistemas Dinámicos, IPICYT, Camino a la Presa San José 2055, Lomas 4a sección, C.P. 78216, San Luis Potosí, San Luis Potosí, Mexico.
Chaos (Woodbury, N.Y.)
|October 2, 2021
Summary
This study numerically estimates the critical parameter value for a phase transition in random maps, observing changes in invariant measure existence and correlation decay. Researchers also analyzed random maps lacking a phase transition.
Area of Science:
- Dynamical Systems and Chaos Theory
- Statistical Mechanics
- Numerical Analysis
Background:
- Phase transitions are critical phenomena observed in various systems.
- Random maps provide a simplified model for studying complex dynamical behaviors.
- The existence of absolutely continuous invariant measures is crucial for understanding long-term system dynamics.
Purpose of the Study:
- To numerically estimate the critical parameter value where a phase transition occurs in a one-parameter family of random maps.
- To investigate the behavior of correlation decay rates across the phase transition.
- To analyze a family of random maps without a phase transition for comparison.
Main Methods:
- Numerical computation of invariant densities.
- Calculation of the Lyapunov exponent to identify the critical transition point.
- Analysis of correlation decay rates (power-law vs. exponential).
Main Results:
- The critical parameter value for the phase transition was numerically estimated.
- A transition in correlation decay behavior from power-law-like to exponential-like was observed.
- Invariant densities were computed for random maps with and without a phase transition.
Conclusions:
- The study successfully demonstrated and characterized a phase transition in a simple family of random maps.
- Numerical methods are effective for estimating critical parameters and analyzing dynamical properties.
- The presence or absence of a non-expansive branch influences the occurrence of phase transitions in random maps.
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