Dynamics of a metastable state nonlinearly coupled to a heat bath driven by external noise

Based on a system-reservoir model, where the system is nonlinearly coupled to a heat bath and the heat bath is modulated by an external stationary Gaussian noise, we derive the generalized Langevin equation with space-dependent friction and multiplicative noise and construct the corresponding Fokker-Planck equation, valid for short correlation time, with space-dependent diffusion coefficient to study the escape rate from a metastable state in the moderate- to large-damping regime. By considering the dynamics in a model cubic potential we analyze the results numerically which are in good agreement with theoretical predictions. It has been shown numerically that enhancement of the rate is possible by properly tuning the correlation time of the external noise.

Figure 1

(Color online) Variation of the escape rate $K$ with dissipation constant $\beta$ with (a) $\mu =0$ and (b) $\mu =0.5$ using the parameter set $b_1=1.0$, $E_b=5.0$, $k_{B}T=0.1$, ${\kappa }_0^2=5.0$, $D_e=0.1$, and ${\tau }_e=0.01$, where the solid line and circle represent analytical and numerical rates, respectively.

Figure 2

(Color online) Variation of the analytical rate constant $K$ with ${\kappa }_0^2$ for different values of the strength of nonlinearity $\mu$ with $b_1=1.0$, $E_b=5.0$, $k_{B}T=0.1$, $\beta =1.0$, $D_e=1.0$, and ${\tau }_e=0.01$.

Figure 3

(Color online) Variation of the rate constant $K$ with strength of nonlinearity $\mu$ (a) for different values of external noise strength $D_e$ (calculated analytically using $k_{B}T=0.1$) and (b) for different values of temperature $k_{B}T$ (calculated numerically using $D_e=1.0$). The values of the other parameters used are $b_1=1.0$, $E_b=5.0$, $\beta =3.0$, ${\kappa }_0^2=5.0$, and ${\tau }_e=0.01$.

Figure 4

(Color online) Turnover of the rate constant $K$ (numerical) with dissipation constant $\beta$ for (a) different values of $\mu$ (for $k_{B}T=0.1$) and (b) different values of $k_{B}T$ (for $\mu =0.5$). The values of the other parameters used are $b_1=1.0$, $E_b=5.0$, $D_e=1.0$, ${\kappa }_0^2=5.0$, and ${\tau }_e=1.0$.

Figure 5

(Color online) Numerical variation of the rate constant $K$ with correlation of external noise ${\tau }_e$ for (a) different values of the strength of external noise $D_e$ (for $k_{B}T=0.1$ and $\mu =0.5$), (b) different values of $k_{B}T$ (for $D_e=1.0$ and $\mu =0.5$), and (c) different values of $\mu$ (for $k_{B}T=1.0$ and $D_e=1.0$). The values of the other parameters used are $b_1=1.0$, $E_b=5.0$, $\beta =5.0$ [for (b) $\beta =3.0$], and ${\kappa }_0^2=5.0$. Note the logarithmic abscissa in (a) and (b).