Scaling laws for the largest Lyapunov exponent in long-range systems: A random matrix approach

Abstract

We investigate the laws that rule the behavior of the largest Lyapunov exponent (LLE) in many particle systems with long range interactions. We consider as a representative system the so-called Hamiltonian alpha-XY model where the adjustable parameter alpha controls the range of the interactions of N ferromagnetic spins in a lattice of dimension d. In previous work the dependence of the LLE with the system size N, for sufficiently high energies, was established through numerical simulations. In the thermodynamic limit, the LLE becomes constant for alpha greater than d whereas it decays as an inverse power law of N for alpha smaller than d. A recent theoretical calculation based on Pettini's geometrization of the dynamics is consistent with these numerical results (M.-C. Firpo and S. Ruffo, cond-mat/0108158). Here we show that the scaling behavior can also be explained by a random matrix approach, in which the tangent mappings that define the Lyapunov exponents are modeled by random simplectic matrices drawn from a suitable ensemble.

5 pages, no figures

References in corpus (2)

• Non-Gaussian equilibrium in a long-range Hamiltonian system
• Chaos suppression in the large size limit for long-range systems

Cited by in corpus (20)

• Nonextensive statistical mechanics and economics
• On the diffusive anomalies in a long-range Hamiltonian system
• Analytical results for coupled map lattices with long-range interactions
• Universal threshold for the dynamical behavior of lattice systems with long-range interactions
• Metastability, negative specific heat and weak mixing in classical long-range many-rotator system
• Sensitivity to initial conditions of a $d$-dimensional long-range-interacting quartic Fermi-Pasta-Ulam model: Universal scaling
• Chaos in the Hamiltonian mean field model
• Scaling with system size of the Lyapunov exponents for the Hamiltonian Mean Field model
• Controlling the Range of Interactions in the Classical Inertial Ferromagnetic Heisenberg Model: Analysis of Metastable States
• Stability of the entropy for superstatistics
• Quasi-integrable systems are slow to thermalize but may be good scramblers
• Theoretical estimates for the largest Lyapunov exponent of many-particle systems
• Quasi-stationary trajectories of the HMF model: a topological perspective
• Largest Lyapunov exponent of long-range XY systems
• Overdamped dynamics of long-range systems on a one-dimensional lattice: Dominance of the mean-field mode and phase transition
• Lyapunov Exponent and Criticality in the Hamiltonian Mean Field Model
• Lyapunov exponent of many-particle systems: testing the stochastic approach
• Weak chaos and metastability in a symplectic system of many long-range-coupled standard maps
• Critical Exponent for the Lyapunov Exponent and Phase Transitions -- The Generalized Hamiltonian Mean-Field Model
• Non-Universal Finite Size Effects with Universal Infinite-Size Free Energy for the $\alpha$-XY model