Residence-times distribution function of a periodically modulated bistable system subject to noise recycling

Abstract

The research mentioned in the statement focuses on investigating a theoretical method for calculating the residence-times distribution function (RTDF) in a periodically driven, bistable system subject to noise recycling. This situation deviates from a Markovian process due to the recycling lag, making it challenging to determine the RTDF using traditional two-state models. In this paper, the aim is to overcome this issue and provide a systematic analysis of how to calculate the RTDF in such a system. By considering the piecewise escape rate of the system, which relies not only on the current state but also on the previous state, we have successfully derived the recursive expression of RTDF. Then, RTDF for large and small A/D is approximately derived using a piecewise analysis formula, respectively. It is further demonstrated both theoretically and numerically that the RTDF has exhibited a feedback-induced structure as a result of the recycling procedure. The results are shown as follows: for large A/D, the RTDF exhibits a series of sharp peaks located at odd multiples of approximately half the period of the driving signal. This indicates the occurrence of stochastic resonance. Additionally, it is interesting to note that the RTDF displays a sharp dip at t=τ. It is further shown that the process of noise recycling increases the probability of short residence times for t < τ and decreases the probability of long residence times for t>τ. This demonstrates that noise recycling plays a crucial role in facilitating particle hops between the double-well potential. For small A/D, the RTDF displays a phenomenon of piecewise exponential decay and declines sharply at t=τ. Notably, the interval between discontinuities becomes smaller or even disappears with increasing the noise intensity or the relative strength. Furthermore, when driven by an appropriate periodic signal, the RTDF exhibits a sequence of maximum values at odd multiples of approximately half the period of the driving signal. However, these maximum values disappear with increasing the noise intensity or the relative strength. This suggests that moderate noise recycling can induce the occurrence of stochastic resonance. However, excessive noise recycling actually inhibits the generation of stochastic resonance. The theoretical results have been successfully validated via numerical methods, demonstrating the reasonability of the present theoretical approach.