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Chaos in the Hamiltonian mean-field model
Francesco Ginelli1, Kazumasa A Takeuchi, Hugues Chaté
1Istituto dei Sistemi Complessi, CNR, via dei Taurini 19, I-00185 Roma, Italy.
This study reveals that chaos persists in finite Hamiltonian mean-field (HMF) systems, even with long-range interactions. The largest Lyapunov exponent remains positive, demonstrating chaotic dynamics across different system sizes and phases.
Area of Science:
- Statistical Mechanics
- Dynamical Systems Theory
- Computational Physics
Background:
- The Hamiltonian mean-field (HMF) model describes N particles with global attractive interactions, forming a rotating cluster in its ordered phase.
- Understanding the transition between ordered (clustered) and disordered phases is crucial for systems with long-range interactions.
Purpose of the Study:
- To investigate the dynamical properties and chaos in the canonical ordered phase of the HMF model.
- To analyze the behavior of Lyapunov exponents in the thermodynamic and critical limits.
- To explore the impact of global coupling on system dynamics and chaos.
Main Methods:
- Employed a combination of numerical simulations and analytical techniques.
- Calculated the largest Lyapunov exponent and analyzed the full spectrum of Lyapunov exponents.
- Investigated a 2D extension of the HMF model with increased degrees of freedom.
Main Results:
- The largest Lyapunov exponent remains strictly positive in the infinite-size limit, with 1/lnN corrections.
- Identified scaling laws for the asymptotic Lyapunov exponent in the critical region.
- The Lyapunov spectrum comprises a bulk converging to zero and subextensive bands of finite exponents.
- Chaos exists for finite system sizes, contrasting with the Vlasov equation limit for infinite systems.
Conclusions:
- Global coupling in the HMF model leads to persistent chaos in finite systems.
- The order of taking infinite-time and infinite-size limits is critical for determining the presence of chaos.
- Results highlight the nuanced effects of long-range interactions on dynamical systems.
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