Relaxation properties and ergodicity breaking in nonlinear Hamiltonian dynamics

Abstract

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/${\tau }_0$ ${\left)\right]}^{\xi }$, ξ<1, up to times t<${\tau }_R$, and η(t)=const for t≥${\tau }_R$. By taking advantage of this fact, a good definition of a relaxation time becomes possible. Below a critical value ${\varepsilon }_c$ (model dependent) of the energy density ɛ, the relaxation time ${\tau }_R$ is found to follow a scaling with ɛ, which is compatible with a ‘‘Nekhoroshev-like’’ law, i.e., ${\tau }_R$=${\tau }_0$exp(${\varepsilon }_0$/ɛ${)}^{\gamma }$, for both the Fermi-Pasta-Ulam (FPU) β model and the classical lattice ${\phi }^4$ model; 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.