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Time evolution of correlation functions and thermalization

We investigate the time evolution of a classical ensemble of isolated periodic chains of O(N)-symmetric anharmonic oscillators. Our method is based on an exact evolution equation for the time dependence of correlation functions. We discuss its solutions in an approximation which retains all contributions in next-to-leading order in a 1/N expansion and preserves time reflection symmetry. We observe effective irreversibility and approximate thermalization. At large time the system approaches stationary solutions in the vicinity of, but not identical to, thermal equilibrium. The ensemble therefore retains some memory of the initial condition beyond the conserved total energy. Such a behavior with incomplete thermalization is referred to as "mesoscopic dynamics". It is expected for systems in a small volume. Surprisingly, we find that the nonthermal asymptotic stationary solutions do not change for large volume. This raises questions on Boltzmann's conjecture that macroscopic isolated systems thermalize.