Relaxation properties and ergodicity breaking in nonlinear Hamiltonian dynamics

The time evolution toward equipartition of energy is numerically investigated in nonlinear Hamiltonian systems with a large number N of degrees of freedom. The relaxation process is studied by computing the spectral entropy {η}; for a wide class of initial conditions, it follows a stretched exponential law {η}(t){~}exp[-(t/τ₀{)]}^{{ξ}}, ξ1, up to times t{{τ}}R, and η(t)=const for t≥{{τ}}R$. By taking advantage of this fact, a good definition of a relaxation time becomes possible. Below a critical value ${{ε}}c(model dependent) of the energy density ε, the relaxation time{{τ}}Ris found to follow a scaling with ε, which is compatible with a ``Nekhoroshev-like'' law, i.e.,{{τ}}R$=${{τ}}₀$exp(${{ε}}₀/ε{)}^{{γ}}, for both the Fermi-Pasta-Ulam (FPU) β model and the classical lattice{{φ}}⁴model; a remarkable difference with respect to Nekhoroshev's theorem (where the exponent γ scales as 1/{N}²$) is the N independence of numerical experiment results.An important consequence of this fact is the existence of nonequilibrium states of arbitrary lifetimes also at large N values. On the other hand, at high-energy densities ({ε}>εc) τR is almost independent of {ε}. The maximum Lyapunov characteristic exponent λ₁ is measured in both models as a function of the energy density. The striking result is a change in the scaling λ₁({ε}) occurring at the same {ε} (=εc) at which τR has its crossover. At {ε}>εc, the scaling λ₁{~}ε2/3 is found for both the FPU and the φ⁴ models; this is in agreement with a random matrix approximation of the tangent dynamics, which means that the dynamics itself mimics a random process. Below εc, steeper and model-dependent scalings are found. Hence, evidence for the existence of a rather sharp transition of the phase-space structure is provided, and a strong stochasticity threshold (SST) can be defined. A preliminary result, suggesting the stability of the SST in the limit of large N, is also reported.