We consider the dynamics of a complex quantum system subjected to a time-dependent perturbation, using a random matrix approach. The dynamics are described by a diffusion constant characterizing the spread of the probability distribution for the energy of a particle which was initially in an eigenstate.
We discuss a system of stochastic differential equations which are a model for the Schrodinger equation written in an adiabatic basis. We examine the dependence of the diffusion constant D on the rate of change of the perturbation parameter, X. Our analysis indicates that D alpha X(2), in agreement with the Kubo formula, up to a critical velocity X*; for faster perturbations, the rate of diffusion is lower than that predicted from the Kubo formula. These predictions are confirmed in numerical experiments on a banded random matrix model. The implications of this result are discussed.