Stability measures in metastable states with Gaussian colored noise

We present a study of the escape time from a metastable state of an overdamped Brownian particle in the presence of colored noise generated by Ornstein-Uhlenbeck process. We analyze the role of the correlation time on the enhancement of the mean first passage time through a potential barrier and on the behavior of the mean growth rate coefficient as a function of the noise intensity. We observe the noise-enhanced stability effect for all the initial unstable states used and for all values of the correlation time ${\tau }_c$ investigated. We can distinguish two dynamical regimes characterized by weak and strong correlated noises, depending on the value of ${\tau }_c$ with respect to the relaxation time of the system.

Figure 1

(Color online) The cubic potential $U\left(x\right)=a{x}^2-b{x}^3$ with the various initial positions investigated (dots), namely, $x_o=1.2,1.3,1.4,1.5,1.6$. The parameters of the potential are $a=0.3,b=0.2$. For the white-noise case, $x_c=1.5$ is the critical initial position which separates the set of the initial unstable states producing divergency for $D$ tending to zero from those which give only a nonmonotonic behavior of the average escape time [6, 19].

Figure 2

(Color online) Panel a: log-log plot of the mean first passage time $\tau$ as a function of noise intensity $D$ in the case of correlated noise with ${\tau }_c=0.01$ for the four initial positions investigated (see Fig. 1). Inset: the related standard deviation as a function of the noise intensity $D$. The dotted straight line at $D=D_s$ separates the simulation data representing the Brownian particles escaped within the maximum simulation time $T_{\text{max}}$ for $D>D_s$ from those representing the particles partially trapped within the well for a time greater or equal to $T_{\text{max}}$ for $D<D_s$. Panel b: mean growth rate coefficient $\Lambda$ as a function of the noise intensity $D$, with the same initial positions of Fig. 1.

Figure 3

(Color online) Semilogarithmic plot of the fraction of particles $N_i/N$ reaching the threshold position $x_t=0.5$ into the potential well, within the $T_{\text{max}}$, as a function of noise intensity $D$. This threshold position $x_t$ corresponds to the flex point of the potential, where the instantaneous growth rate ${\lambda }_i\left(x,t\right)$ is equal to zero. The correlation time of the noise is ${\tau }_c=0.01$ with the same initial conditions of Fig. 1.

Figure 4

(Color online) Log-log plot of the MFPT $\tau$ as a function of noise intensity $D$ for the same initial positions of Fig. 1 and for different values of the correlation times ${\tau }_c$, namely, ${\tau }_c=0.1,0.6,1$, corresponding, respectively, to the weak, intermediate, and strong colored noise dynamical regimes. The dotted straight line at $D=D_s$ separates the noise values for which all the Brownian particles escape $\left(D>D_s\right)$ from those for which the particles are partially trapped into the well within the $T_{\text{max}}$ $\left(D<D_s\right)$.

Figure 5

(Color online) Log-log plot of the standard deviation $\sigma$ as a function of the noise intensity $D$ for ${\tau }_c=0.6$ and the same initial positions of Fig. 1. The dotted straight line indicates the value of the noise intensity $D_s$ which separates the simulation data representing the trapped Brownian particles from those escaped within $T_{\text{max}}$. Inset: log-log plot of the ratio $\sigma /\tau$ as a function of noise intensity $D$.

Figure 6

(Color online) Semilogarithmic plot of mean growth rate coefficient $\Lambda$ as a function of noise intensity $D$ for the same initial positions $x_o$ and the same values of the correlation times ${\tau }_c$ of Fig. 4, namely, ${\tau }_c=0.1,0.6,1$, corresponding respectively to the weak, intermediate, and strong colored noise dynamical regimes.

Figure 7

(Color online) Semilogarithmic plot of the fraction of particles $N_i/N$ reaching the threshold position $x_t=0.5$ into the potential well, within the $T_{\text{max}}$, as a function of noise intensity $D$. This threshold position $x_t$ corresponds to the flex point of the potential, where the instantaneous growth rate ${\lambda }_i\left(x,t\right)$ is equal to zero. The correlation time of the noise is ${\tau }_c=0.6$, with the same initial conditions of Fig. 1.