On the transient stochastic dynamics driven by Gaussian colored noise of systems with time-dependent control parameters. The effect of initial conditions

On a recent work we studied the transient stochastic dynamics driven by Gaussian white noise (GWN), of linear systems with time-dependent control parameters, by means of the connection between the nonlinear relaxation times (NLRT) and the quasideterministic (QD) approach. In the present study we make an extension of the analysis to the Gaussian colored noise (GCN) problem. Here, we first calculate the characteristic time associated with the decay of the unstable state of such linear systems, when the control parameter is a linear function of time of the form a(t) = bt − a0, with b> a0 > 0, which is continuously swept from below to above a threshold . This type of linear modulation is known as the ramp model. Then, going further, we consider a general case where the control parameter is modulated by a family of functions a(t) = btδ a0, with δ > 0. The effects of the coupling between the initial state, at time t = 0, of the system with the noise are specially emphasized. The results of NLRT for the ramp model and the general case are compared.