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Phase space geometry and stochasticity thresholds in hamiltonian dynamics
1Osservatorio Astrofisico di Arcetri, Largo E. Fermi 5, 50125 Firenze, Italy and Istituto Nazionale di Fisica della Materia, UdR di Firenze, Frienze, Italy.
Summary
Numerical computations reveal two thresholds in the Fermi-Pasta-Ulam alpha+beta model. A strong stochasticity threshold (SST) is geometrically characterized by phase space curvature and autocorrelation decay time.
Area of Science:
- Nonlinear Dynamics
- Statistical Mechanics
- Computational Physics
Background:
- The Fermi-Pasta-Ulam (FPU) model is a foundational system in nonlinear dynamics, used to study energy equipartition and thermalization.
- Previous research identified a stochasticity threshold in the FPU alpha model and a strong stochasticity threshold (SST) in the FPU beta model.
Purpose of the Study:
- To investigate the behavior of the largest Lyapunov exponent in the combined FPU alpha+beta model.
- To characterize the strong stochasticity threshold (SST) geometrically and through dynamical properties.
- To understand the interplay between different nonlinear mechanisms in a complex system.
Main Methods:
- Numerical computation of the largest Lyapunov exponent, lambda(1)(varepsilon,N), as a function of energy density (varepsilon) and particle number (N).
- Analysis of the coexistence of stochasticity and strong stochasticity thresholds in the FPU alpha+beta model.
- Geometric characterization of the SST using the mean curvature of constant energy hypersurfaces in phase space.
- Calculation of the characteristic decay time of the time autocorrelation function, tau(c)(varepsilon,N).
Main Results:
- The FPU alpha+beta model exhibits two distinct thresholds at large N: a stochasticity threshold and a strong stochasticity threshold (SST).
- The SST's geometric properties, specifically the mean curvature of energy hypersurfaces, correlate with the decay time of the autocorrelation function.
- The autocorrelation decay time tau(c)(varepsilon,N) shows a strong correlation with the largest Lyapunov exponent lambda(1)(varepsilon,N) for fixed N.
- These findings suggest a deep connection between phase space geometry and dynamical stochasticity in the model.
Conclusions:
- The coexistence of two thresholds in the FPU alpha+beta model is confirmed, demonstrating a superposition of properties from the individual alpha and beta models.
- The study provides a novel geometric interpretation of the strong stochasticity threshold (SST) through phase space curvature.
- The characteristic decay time of the autocorrelation function serves as a valuable indicator of the system's dynamical behavior and correlates with Lyapunov exponents.
- This work offers significant insights into the complex phase space geometry and dynamics of the FPU alpha+beta model.