Noise-induced enhancement of stability in a metastable system with damping

The mean first passage time of a Brownian particle from an initial unstable state in a metastable system with damping is investigated. The system is analyzed in the low to high damping regime, and the role played by the damping parameter is studied. We observe the noise enhanced stability effect for all the initial unstable states under study and for all values of the damping parameter $\gamma$ investigated. The curves show a behavior of the mean first passage time vs $\gamma$ very close to that observed for an overdamped particle in the presence of colored noise as a function of the correlation time.

Figure 1

(Color online) The cubic potential $U\left(x\right)=a{x}^2-b{x}^3$ with some of the initial positions investigated (dots), namely, $x_0=1.2,1.3,1.4,1.5,1.6$. The position $x_c=1.5$ represents, for the overdamped motion in the presence of white noise [see Eq. (2)], the critical initial position which separates the set of the initial unstable states producing divergency for $D\to 0$ from those which give only a nonmonotonic behavior of the average escape time 6, 8, 20, 21.

Figure 2

(Color online) Log-log plots of the mean first passage time $\tau$ as a function of $D$ for different values of the damping, namely, $\gamma =0.1,1,20$, and for the initial positions $x_0=1.3,1.4,1.5,1.6,1.7$. The panels (a)–(c) show the values rescaled in time to allow the comparison with the overdamped case [panel (d)] obtained by numerical simulation in Eq. (2).

Figure 3

(Color online) Log-log plot of the standard deviation $\sigma$ as a function of $D$ for $\gamma =0.1,1,20$ [panels (a)–(c), respectively] and the same initial positions in Fig. 1. Panel (d): semilogarithmic plot of the fraction of particle $N_i/N$ entering the potential well, within the maximum simulation time $t_{\text{max}}$, as a function of the scaled noise intensity $D$.

Figure 4

(Color online) Log-log plot of the MFPT $\tau$ as a function of $D$ for fixed initial positions $\left(x_0=1.2,1.3,1.4,1.5\right)$ and for different values of the damping parameter, namely, $\gamma =0.18,0.32,0.56,1$. We see how the divergent trend depends on $\gamma$ for a fixed $x_0$ value. This means that the critical position $x_c^{\ast }$ depends on the damping parameter $x_c^{\ast }=x_c^{\ast }\left(\gamma \right)$.

Figure 5

(Color online) Critical initial position as a function of the damping parameter $\gamma$. By increasing the damping, $x_c^{\ast }$ tends asymptotically to the overdamped value $x_c=1.5$.