Precursors to rare events in stochastic resonance

Abstract

In stochastic resonance, a periodically forced Brownian particle in a double-well potential jumps between minima at rare increments, the prediction of which pose a major theoretical challenge. Here, we use a path-integral method to predict these transitions by determining the most probable (or "optimal") space-time path of a particle. We characterize the optimal path using a direct comparison principle between the Langevin and Hamiltonian dynamical descriptions, allowing us to express the jump condition in terms of the accumulation of noise around the stable periodic path. In consequence, as a system approaches a rare event these fluctuations approach one of the deterministic minimizers, thereby providing a precursor for predicting the stochastic transition. We demonstrate the method numerically, which allows us to determine whether a state is following a stable periodic path or will experience an incipient jump. The vast range of systems that exhibit stochastic resonance behavior insures broad relevance of our framework, which allows one to extract precursor fluctuations from data.

The Thermodynamic Cost of Erasing Information in Finite-time

“…The comparison is splendid, with only minor fluctuations around this linear behavior, the origin of which is the relative magnitude of the two summands in Eq. (33) as discussed above. Namely, the fluctuations associated with the noise-driven transitions between the two states 0 and 1, which are captured in the distribution P(τ 0 ) of sojourn times τ 0 , dominates the thermal fluctuations of the trajectories within the potential wells.…”

“…Therefore, each state is dealt with individually as a tilted harmonic potential with the minima at ±a + F erase (t), with the plus (minus) sign referring to the right (left) well. Each instanton X inst (t) satisfies the following coupled system of equations (see [33] and references therein)…”

“…We find that the principal sources of randomness are the transitions over the potential barrier between the two memory states, whereas the fluctuations within the potential wells have negligible effects on the work distribution. Thirdly, by using recently developed methods from stochastic resonance and early warning quantifiers [32][33][34][35][36], we provide a concrete recipe for calculating the jump-time distributions (19) using instantons and survival probabilities. By combining these results with (17), (30) and (31), the averages in (11) and (16) reduce to simple numerical quadratures.…”

“…Here we derive an analytical approximation of the integrals in Eqs. (32) and (33) for rapid erasure protocol, which yield corresponding approximations for the mean and the variance of the work respectively. The brevity of the protocol insures that the contribution to the mean and variance of the work from trajectories with more than one inter-well transition is negligible.…”

Extending Transition Path Theory: Periodically Driven and Finite-Time Dynamics

“…What are bottlenecks in the transient dynamics toward the equilibrium state (Schonmann 1992;Hollander et al 2000)? How does tipping occur under the joint effect of noise and parameter changes (Ashwin et al 2012;Giorgini et al 2019) or periodic forcing (Herrmann et al 2005)?…”