Energy-based stochastic resetting can avoid noise-enhanced stability

Abstract

The theory of stochastic resetting asserts that restarting a stochastic process can expedite its completion. In this paper, we study the escape process of a Brownian particle in an open Hamiltonian system that suffers noise-enhanced stability. This phenomenon implies that under specific noise amplitudes the escape process is delayed. Here, we propose a new protocol for stochastic resetting that can avoid the noise-enhanced stability effect. In our approach, instead of resetting the trajectories at certain time intervals, a trajectory is reset when a predefined energy threshold is reached. The trajectories that delay the escape process are the ones that lower their energy due to the stochastic fluctuations. Our resetting approach leverages this fact and avoids long transients by resetting trajectories before they reach low energy levels. Finally, we show that the chaotic dynamics (i.e., the sensitive dependence on initial conditions) catalyzes the effectiveness of the resetting strategy.

Figures (13)

• Figure 1: Isopotential curves of the Hénon-Heiles system for various values of the potential $V(x,y)=\frac{1}{2}(x^2+y^2)+x^2y-\frac{1}{3}y^3$. The color of the curves indicates the different values of the potential, which are marked in the color bar. For energy values below the escape energy $E_e=1/6$, the isopotential curves are closed, while the potential exhibits three symmetrical exits for energy values above $E_e$. Particles can escape through these exits towards $\pm\infty$.
• Figure 2: Average escape time for $200$ different noise amplitudes. The escape time of $10^6$ initial conditions is averaged for each value of $\xi$. The blue shaded region represents a $99.7\%$ confidence interval and the peak, emphasized by a red dashed line, corresponds to the noise-enhanced stability phenomenon.
• Figure 3: (a) Average escape times and (b) average minimum energy during the escape path for $100$ different values of $\theta$. In both panels, $10^6$ initial conditions are computed for each angle.
• Figure 4: Coefficient of variation, $CV=\sigma(T)/\bar{T}$, for $100$ values of $\theta$. For each value of $\theta$, the average escape time and the corresponding standard deviation have been estimated from $10^6$ realizations of the process.
• Figure 5: Escape time distribution based on initial conditions with (a) low $CV$ and (b) high $CV$. The escape time is averaged over $10^7$ initial conditions in the intervals: $\theta\in[0,0.24]\cup [1.85,2\pi/3]$ for low $CV$ and $\theta\in[0.37,1.71]$ for high $CV$. The relative frequency $f$ is normalized to unity. The distribution for high $CV$ shows a long tail of escape times, which we aim to avoid by resetting.