On the dynamics of large-N O(N)-symmetric quantum systems at finite temperature

Time evolution of a perturbed thermal state is studied in a quantum-mechanical system with O(N) symmetry. In the limit of large N, time dependence of O(N)-singlet expectation values can be described by classical equations of motion in a one-dimensional potential well. Time dependence of the perturbation is then described by a linear differential equation with time-dependent periodic coefficient. This equation, depending on the parameters, admits either exponentially growing/decaying or periodically oscillating solutions. It is demonstrated that only the latter possibility is actually realized, thus in such a system there is no redistribution of initial perturbation over all N degrees of freedom.