Stochastic Resonance: A Comparative Study of Two-State Models

Abstract

We consider a dynamical system describing the motion of a particle in a double well potential with a periodic perturbation of very small frequency, and an additive stochastic perturbation of amplitude ε. It is in stochastic resonance if the solution trajectories amplify the small periodic perturbation in a “best possible way”. Systems of this type first appeared in simple energy balance models designed for a qualitative explanation of global glacial cycles. Large deviations theory provides a lower bound for the proportion of the amplitude and the logarithm of the period above which quasi-deterministic periodic behavior can be observed. However, to obtain optimality, one has to measure periodicity with a measure of quality of tuning such as spectral power amplification favored in the physical literature. In a situation where the potential switches discontinuously between two spatially antisymmetric double well states we encounter a surprise. Contrary to physical intuition, the stochastic resonance pattern is not correctly given by the reduced dynamics described by a two state Markov chain with periodic hopping rates between the potential minima which mimic the large (spatial) scale motion of the diffusion. Only if small scale fluctuations inside the potential wells where the diffusion spends most of its time are carefully eliminated, the Markov chain gives the correct picture.