Thermally enhanced stability in fluctuating bistable potentials

Overdamped motion of Brownian particles in an asymmetric double-well potential driven by an additive nonequilibrium three-level noise and a thermal noise is considered. In the stationary regime, an exact formula for the mean occupancy of the metastable state is derived, and the phenomenon of enhancement of stability versus temperature is investigated. It is established that in a certain region of the system parameters the mean occupancy can be either multiply enhanced or suppressed by variations of temperature. We show that this effect is due to the involvement of different time scales in the problem. The necessary conditions for several different behaviors of the mean occupancy as a function of temperature are also discussed. The effect is more pronounced when the kurtosis of the three-level noise tends to $-2$, i.e., in the case of dichotomous noise.

Figure 1

Representation of different states of the net potentials $V_n\left(x\right)=U\left(x\right)+z_{n}x$ with $z_1=a$, $z_2=0$, $z_3=-a$. The potential $U\left(x\right)$ is given by Eq. (3) at the parameter values $d=0.24$, $a=2$, and $\epsilon =1$. All quantities are dimensionless.

Figure 2

Ratio $W_0/W_1$ vs the noise correlation time ${\tau }_c$ at various temperatures $D$. The occupancy probabilities $W_0$ and $W_1$ of the left and right potential wells, respectively, are computed by means of Eq. (14). Parameter values: $a=2$, $\epsilon =1$, and $d=0.125$. The different curves correspond to the different values of the parameter $q$ and temperature $D$: (1) $q=0.49$, $D=0.04$; (2) $q=0.35$, $D=0.04$; (3) $q=0.49$, $D=0.07$; (4) $q=0.35$, $D=0.07$.

Figure 3

Occupancy probabilities $W_0$ (solid line) and $W_1$ (dotted line) of the left and right potential wells, respectively, versus the temperature $D$ [Eq. (14)]. The parameter values are $a=2$, $\epsilon =1$, $d=0.24$, and $q=0.45$. At high values of temperature, $D>10$, the probabilities $W_0$ and $W_1$ saturate to the values $d$ and $1-d$, respectively. $\nu =\left(\text{a}\right)1$; (b) ${10}^{-3}$; (c) ${10}^{-9}$.

Figure 4

Ratios $R_1\equiv W_0/W_1$, $R_2\equiv T_0/T_1$, $R_3\equiv {\tilde{T}}_0/{\tilde{T}}_1$ versus the dimensionless temperature $D$ in the case of dichotomous noise, $q=1/2$. Solid line: the function $R_1\left(D\right)$ computed from Eq. (14). Dashed line: the function $R_2\left(D\right)$ computed from Eqs. (A2, A3). Dotted line: the function $R_3\left(D\right)$ computed by means of the kinetic approximation Eq. (18). Parameter values: $a=2,\epsilon =1,d=0.24,\nu ={10}^{-3}$. The inset depicts the region of significant difference between $R_2$ and $R_3$.

Figure 5

Dependence of the probabilities $W_0$ and $W_1$ on the dimensionless temperature $D$ [Eq. (14)] in the case of fast fluctuations, $\nu =40$. Parameter values: $a=2,\epsilon =1,d=0.24,q=0.45$. The dotted line depicts the function $W_0\left(D\right)$ in the case of a nonfluctuating average potential $V_2\left(x\right)$.