Universal exponent for transport in mixed Hamiltonian dynamics.

Area of Science:

• Dynamical systems theory
• Statistical mechanics
• Chaos theory

Background:

• Transport in mixed phase space is complex.
• Understanding chaotic transport is crucial in various scientific fields.
• Previous models struggled to capture the nuances of transport near hierarchical structures.

Purpose of the Study:

• To compute universal distributions for transition probabilities in a Markov model for transport.
• To verify the applicability of the Meiss-Ott Markov tree model to mixed systems.
• To analyze the survival probability distribution of trajectories in complex phase spaces.

Main Methods:

• Developed a Markov model for transport in area-preserving maps.
• Analyzed transition probabilities within the mixed phase space.
• Computed survival probability distributions for trajectories near island-around-island hierarchies.
• Compared results with simulations of the Hénon and Chirikov-Taylor maps.

Main Results:

• Universal distributions for transition probabilities were computed.
• A power-law decay with exponent γ=1.57 was observed for survival probabilities.
• This exponent aligns with simulation results from the Hénon and Chirikov-Taylor maps.
• The observed power-law decay provides strong evidence for the Meiss-Ott Markov tree model.

Conclusions:

• The Meiss-Ott Markov tree model effectively describes transport in mixed dynamical systems.
• The universal power-law decay is a key characteristic of chaotic transport in these systems.
• This study validates a theoretical model using computational and simulation approaches.