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 ; for a wide class of initial conditions, it follows a stretched exponential law (t)exp[-(t/0)], <1, up to times t<R, and (t)=const for tR. By taking advantage of this fact, a good definition of a relaxation time becomes possible. Below a critical value Ec (model dependent) of the energy density E, the relaxation time R is found to follow a scaling with E, which is compatible with a Nekhoroshev-like law, i.e., R=0exp(E0/E), for both the Fermi-Pasta-Ulam (FPU) model and the classical lattice 4 model; a remarkable difference with respect to Nekhoroshevs theorem (where the exponent scales as 1/N2) 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 (E>Ec) R is almost independent of E. The maximum Lyapunov characteristic exponent 1 is measured in both models as a function of the energy density. The striking result is a change in the scaling 1(E) occurring at the same E (=Ec) at which R has its crossover. At E>Ec, the scaling 1E2/3 is found for both the FPU and the 4 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 Ec, 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. © 1990 The American Physical Society.