Signatures of noise-enhanced stability in metastable states

The lifetime of a metastable state in the transient dynamics of an overdamped Brownian particle is analyzed, both in terms of the mean first passage time and by means of the mean growth rate coefficient. Both quantities feature nonmonotonic behaviors as a function of the noise intensity, and are independent signatures of the noise enhanced stability effect. They can therefore be alternatively used to evaluate and estimate the presence of this phenomenon, which characterizes metastability in nonlinear physical systems.

Figure 1

(Color online) MFPT $\tau (x_0,x_F)$ evaluated from expression (3) vs the noise intensity $D$, for $x_F=2.2$ and for different initial positions $x_0$ (see the legend). The NES effect is observed for $x_0>x_c$, while the divergence regime corresponds to $x_u<x_0<x_c$. Inset: the cubic potential $U\left(x\right)$ with the metastable state at $x=0$. The arrow indicates the intersection point $x_c=1.5$ between the potential curve and the horizontal axis.

Figure 2

Critical stability positions for noiseless and noisy statistics. In the first case $(D=0)$ a particle starting from the position $x_0=x_u-\epsilon$ has an escape time $\left({\tau }_{x_u^{-}}\right)$ equal to infinity, and a particle starting form the position $x_0=x_u+\epsilon$ has an escape time equal to the deterministic one ${\tau }_{x_u^{+}}={\tau }_{\mathit{det}}$. The same behavior is recognized in the second case $(D\ne 0)$ in the limit $D\to 0$ for the critical position $x_c$. This means that the “unstable equilibrium” position in the stochastic case (from the point of view of the stability of the metastable state) is not $x_0=x_u$, but $x_0=x_c$.

Figure 3

(Color online) Numerical estimation of the MFPT (symbols) vs $D$ for $x_0=1.46$ (filled circles), $x_0=1.47$ (empty circles), and $x_0=1.48$ (filled triangles). The solid curves are the corresponding estimations of the MFPT from Eq. (3). Inset: standard deviation $\sigma$ of MFPT’s corresponding to different noise realizations vs $D$ for the same initial positions $x_0$.

Figure 4

${\lambda }_i(x,t)$ (see text for definition) vs time, for $x_0=1.3$, $x_F=3$, and $D=1$.

Figure 5

(Color online) (a) Numerical estimation of the MFPT for various initial positions (as in the legend) and (b) the corresponding MGRC vs the noise intensity $D$. The average has been performed over 20 000 realizations. The absorbing boundary is $x_F=20$, and the maximum waiting time used in the simulations is $T_{\mathit{max}}=20000$.

Figure 6

Number of trajectories reaching the position $x=0.5$ inside the potential well. In this position the instantaneous growth rate ${\lambda }_i(x,t)$ is equal to zero.