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Stability of classical chaotic motion under a system's perturbations
Giuliano Benenti1, Giulio Casati, Gregor Veble
1International Center for the Study of Dynamical Systems, Università degli Studi dell'Insubria, Via Valleggio 11, 22100 Como, Italy.
Summary
Classical fidelity decay in chaotic systems is linked to correlations. Decay can be exponential, determined by the Perron-Frobenius operator gap, or algebraic, matching correlation function decay rates.
Area of Science:
- Quantum mechanics
- Statistical mechanics
- Dynamical systems theory
Background:
- Classical fidelity is a measure of how well a system's state is preserved over time.
- Chaotic systems exhibit sensitive dependence on initial conditions, leading to complex dynamics.
- Understanding fidelity decay is crucial for predicting system evolution and information loss.
Purpose of the Study:
- To investigate the detailed time evolution of classical fidelity in chaotic systems.
- To identify the different mechanisms governing fidelity decay.
- To establish the relationship between fidelity decay and system dynamical properties.
Main Methods:
- Analysis of the time behavior of classical fidelity.
- Application of the discretized Perron-Frobenius operator.
- Comparison of fidelity decay rates with correlation function decay powers.
Main Results:
- Demonstrated that fidelity decay can be either exponential or algebraic.
- Identified the gap in the discretized Perron-Frobenius operator as the rate-determining factor for exponential decay.
- Showed that algebraic decay of fidelity shares the same power as correlation function decay.
Conclusions:
- The decay of classical fidelity in chaotic systems is fundamentally connected to correlations.
- System dynamical properties dictate whether fidelity decays exponentially or algebraically.
- This study provides a deeper understanding of information dynamics in chaotic systems.