Noise-induced escape of periodically modulated systems: From weak to strong modulation

D. Ryvkine and M. I. Dykman

Phys. Rev. E 72, 011110 – Published 25 July 2005

Abstract

Noise-induced escape from a metastable state is studied for an overdamped periodically modulated system. We develop an asymptotic technique that gives both the instantaneous and period-average escape rates, including the prefactor, for an arbitrary modulation amplitude $A$ . We find the parameter range where escape is strongly synchronized and the instantaneous escape rate displays sharp peaks. The peaks vary with increasing modulation frequency or amplitude from Gaussian to strongly asymmetric. The prefactor $ν$ in the period-average escape rate depends on $A$ nonmonotonically. Near the bifurcation amplitude $A_{c}$ it scales as $ν \propto {(A_{c} - A)}^{ζ}$ . We identify three scaling regimes, with $ζ = 1 ∕ 4$ , $- 1$ , and $1 ∕ 2$ .

Received 14 April 2005

DOI:https://doi.org/10.1103/PhysRevE.72.011110

Authors & Affiliations

D. Ryvkine and M. I. Dykman

Department of Physics and Astronomy, Michigan State University, East Lansing, Michigan 48824, USA

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Vol. 72, Iss. 1 — July 2005

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Images

Figure 1
(Color online) (a) An oscillating potential barrier. In the limit of slow modulation, the stable and unstable periodic states $q_{a}$ and $q_{b}$ are the instantaneous positions of the potential minimum and barrier top, respectively. The instantaneous escape rate is characterized by the current at an “observation point” located at a sufficiently large distance $Q$ from $q_{b}$ . (b) The dependence of the prefactor $ν$ in the period-average escape rate $\bar{W} = ν \exp (- R ∕ D)$ on the modulation amplitude $A$ (schematically). For $A \to 0$ , $ν$ is given by the Kramers theory [1]. In regions II and III escape is synchronized and $ν \propto D^{1 ∕ 2}$ , where $D$ is the noise intensity. In region III, close to the critical point $A_{c}$ where the metastable state disappears, the prefactor scales as $ν \propto {(A_{c} - A)}^{- 1}$ . In region IV $ν \propto {(A_{c} - A)}^{1 ∕ 2}$ is independent of $D$ .Reuse & Permissions
Figure 2
(Color online) (a) Pulses of escape current in the adiabatic approximation as functions of time scaled by the relaxation time. With increasing parameter $θ$ the pulses change from Gaussian to strongly asymmetric. (b) The same pulses as functions of time scaled by the modulation period $ϕ = ω_{F} t$ .Reuse & Permissions
Figure 3
(Color online) The prefactor $ν$ in the average escape rate $\bar{W}$ [Eq. (44)]. The results refer to the model (52) with $ω_{F} = 0.1$ and describe escape in the regime of strong synchronization, where $ν \propto D^{1 ∕ 2}$ . The solid line for small $A$ shows the scaling $ν \propto A^{- 1 ∕ 2}$ [17]. The solid lines for small $δ A = A - A_{c}$ in the main figure and in the inset show the scaling (51). The dashed line shows the result of the numerical solution of Eq. (21). The squares and crosses show the results of Monte Carlo simulations for $R ∕ D = 5$ and 6, respectively.Reuse & Permissions
Figure 4
(Color online) The prefactor $ν$ in the average escape rate $\bar{W}$ (44) close to the bifurcation point $A = A_{c}$ . The results refer to the model (52) with $ω_{F} = 1$ . The squares and crosses show the results of Monte Carlo simulations for $R ∕ D = 4$ and 5, respectively. The solid line shows the asymptotics $ν = {∣ {\bar{μ}}_{a} {\bar{μ}}_{b} ∣}^{1 ∕ 2} ∕ 2 π \propto {(A_{c} - A)}^{1 ∕ 2}$ .Reuse & Permissions
Figure 5
(Color online) Different regions of escape behavior in modulated overdamped systems depending on the modulation frequency $ω_{F}$ and amplitude $A$ ; $t_{r}^{(0)}$ is the relaxation time in the absence of modulation. The smeared boundaries between the regions are shown by dashed lines. The bold solid line indicates the bifurcational amplitude where the metastable state disappears. The shaded region below it indicates the range where the activation energy of escape $R ≲ D$ . The transition between the regions of exponentially strong and nonexponential synchronization occurs for $ω_{F} t_{r}^{(0)} \sim 1$ .Reuse & Permissions

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