Time evolution of correlation functions for classical and quantum anharmonic oscillators

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The time evolution of the correlation functions of an ensemble of anharmonic N-component oscillators with O(N) symmetry is described by a flow equation, exact up to corrections of order 1/N². We find effective irreversibility. Nevertheless, analytical and numerical investigation reveals that the system does not reach thermal equilibrium for large times, even when N→ ∞. Depending on the initial distribution, the dynamics is asymptotically stable or it exhibits growing modes which break the conditions for the validity of the 1/N expansion for large time. We investigate both classical and quantum systems, the latter being the limit of an O(N) symmetric scalar quantum field theory in zero spatial dimensions.