A study of periodically driven stochastic systems using the method of moments

A white-noise driven non-linear stochastic system subject to harmonic external forcing is considered. A general technique to compute the system's first- and higher-harmonic response to such a driving of arbitrary amplitude is elaborated, and its linear response version is derived. In addition, a novel method of computing the rate of escape of a stochastic system from a given region of its state space is proposed.

These developments are applied to study the response of a stochastic oscillator in a bistable quartic potential V(x) = −(x²/2) + (x ⁴/4) to harmonic driving, as well as the rate of escape from one of its potential minima. Two cases are separately considered, namely, the overdamped (Smoluchowski) limit, and a more general case of arbitrary damping constant.

Calculation of the probability of escape from a potential minimum per unit time as a function of noise intensity, T, and damping constant is performed. The results are shown to be in good quantitative agreement with the Kramers-Arrhenius formula in the region of its validity, i.e., at low noise intensities. A subsequent increase of the escape rate with noise intensity in proportion to [special characters omitted] is predicted.

A detailed study of the susceptibility of the average coordinate to external forcing is carried out, and the effect of driving frequency, noise intensity, and damping on the system's driven probability distribution and response is investigated in the context of stochastic resonance (SR). In the overdamped limit, two types of processes contribute to the system's susceptibility, namely, intrawell and interwell relaxation. It is shown that even in the overdamped limit, both of these processes result in stochastic resonant enhancement of the system's sensitivity to driving with respect to noise strength.

Other types of SR appear in the case of small damping, when noise can be used to tune the system to the interwell and/or intrawell vibrational state characterized by the frequency matching that of external driving.

A study of the dependence of the system's sensitivity on damping constant, γ, is carried out, and it is shown that at low driving frequency this dependence is nonmonotonic and exhibits a peak at some optimal value of γ. The relation of such a dissipative response enhancement to the interwell mechanism of SR is pointed out. It is demonstrated that the traditionally accepted interpretation of SR in terms of the synchronisation of external forcing with the thermally activated interwell hopping is not quite accurate, and an alternative explanation of SR is proposed.