Escape of coupled Brownian particles across a fluctuating barrier

The escape of two harmonically coupled Brownian particles across the fluctuating barrier of a bistable potential is investigated with correlated additive and multiplicative fluctuations. Positive correlations enhance the rate of escape across the barrier when the coupling is effective, whereas for weakly coupled particles, escape becomes difficult. It is found that the system exhibits the phenomenon of resonant activation when the rate of barrier fluctuations is comparable to the relaxation time in the bistable potential. Using a decoupling ansatz, we derive the Markovian limit of the problem in the steady state, under the constraint that the barriers fluctuate on a time scale faster than the relative oscillation of the two particles. Adiabatic elimination of the fast variable of the dynamical system is discussed in appropriate limits.

Figure 1

Variation of the MFPT with correlation time $\tau$ of the barrier fluctuations. The nonmonotonic dependence of the MFPT on $\tau$ is evident from the figure and occurs for $\tau \approx 1$, which is the relaxation time in the bistable potential $U$. It should be noted that both axes are represented in logarithmic scale. The intensity of the barrier fluctuations ${\sigma }^2=D^m/\tau$. The MFPT is calculated over 10 000 ensembles.

Figure 2

Variation of the MFPT with noise correlation ${\rho }^m$ for a fixed correlation time $\tau =1$ for (a) $\sigma =0.5$ and (b) $\sigma =2$. It is shown in (b) that for a higher intensity of barrier fluctuations, the MFPT is lower when the coupling between the particles is large, e.g., $k=1$ as compared to nearly independent particles, $k=0.01$, for strongly correlated barrier fluctuations. The $y$ axis is shown in logarithmic scale and the MFPT is calculated over 10 000 ensembles.

Figure 3

Distribution of FPTs of the center of mass starting at $x_c=-1$ and reaching $x_c=1$, for fixed temperature $D^a=0.1$, correlation time $\tau =1$, and independent barrier fluctuations ${\rho }^m=0$ for spring constant $k=1$ for (a) $\sigma =0.5$ and (b) $\sigma =2$. The FPT distributions are calculated using 10 000 data points.