Kramers’ law for a bistable system with time-delayed noise

We demonstrate that the classical Kramers’ escape problem can be extended to describe a bistable system under the influence of noise consisting of the superposition of a white Gaussian noise with the same noise delayed by time $\tau$. The distribution of times between two consecutive switches decays piecewise exponentially, and the switching rates for $0<t<\tau$ and $\tau <t<2\tau$ are calculated analytically using the Langevin equation. These rates are different since, for the particles remaining in one well for longer than $\tau$, the delayed noise acquires a nonzero mean value and becomes negatively autocorrelated. To account for these effects we define an effective potential and an effective diffusion coefficient of the delayed noise.

Figure 1

(a) Experimental RTD with (circles) and without (dots) the delayed noise. (b) Numerical RTD calculated using Eq. (4) with $\epsilon =0$ and $D=0.15$ (dashed curve) and $\epsilon =-0.34$ and $D=0.167$ (solid curve). The thin gray lines are included to show the difference in the escape rates before and after $\tau$ in the presence of delayed noise.

Figure 2

(a) The ACF approaches the white noise limit as $\epsilon$ increases. The parameters are $D=0.15$, $\tau =100$, and $x_{\mathit{th}}=0.5$. The ACF does not depend on $\tau$. (b) Values of $A$ found from simulation together with the expression $A=A_0∕1+{\epsilon }^2$ with $A_0\approx 0.1788$ and $\gamma \approx 1$ for parameters $D=0.15$, $\tau =100$, and $x_{\mathit{th}}=0.5$.

Figure 3

Comparison of theoretical (curves) and numerically calculated (symbols) escape rates $r_1$ (triangles) and $r_2$ (circles). The numerical data are for $D=0.15$, $\tau =100$, and $x_{\mathit{th}}=0.5$. For the numerical simulations the value of $x_{\mathit{th}}=0.5$ is used to match $r_1$ from simulation with the theoretical $r_1$ for $D=0.15$.