Nonergodicity and dynamical approach to the anharmonic oscillator

General lower bounds for the time average C^AA(∞)≡lim_T→∞(1/T)∝^T₀ C^AA(t) dt of the correlation C^AA(t)≡A(t)A(0)−A² of an arbitrary variable A are derived, which depend only on the temperature derivatives of the canonical averages of A and the Hamiltonian of the system. The bounds may be used to give good estimations for C^AA(∞) which is different from zero when A is nonergodic. It is important to take care of these terms when dynamical theories made for interacting systems are applied to isolated systems. We show explicitely that our recently developed dynamical approach to phonon systems with quartic anharmonicity yields excellent results for the corresponding isolated system, the anharmonic single well oscillator, when nonvanishing time averages are taken into account.