Stochastic Equations of Motion for the Linear Response. II. Simple Example —An Extended Kubo-Anderson Model—

Abstract

An extended model of the Kubo-Anderson oscillator with random frequency modulation under the action of an external driving force is considered. The time-convolutionless and time-convolution equations of motion for the averaged linear response of the complex coordinate of this oscillator to the driving force are compared with the exact equation under the assumption that the random process of the frequency is Gaussian. It is shown that the time-convolutionless equation coincides with the exact one up to fourth order in powers of the random modulation, and that the time-convolution equation coincides with the exact one in the narrowing limit. For the Gaussian-Markoffian process, the exact solution is shown to coincide with the solution of the time-convolution equation up to second order in powers of the random modulation in the narrowing limit, but not for small times.