Resonant activation in piecewise linear asymmetric potentials

This work analyzes numerically the role played by the asymmetry of a piecewise linear potential, in the presence of both a Gaussian white noise and a dichotomous noise, on the resonant activation phenomenon. The features of the asymmetry of the potential barrier arise by investigating the stochastic transitions far behind the potential maximum, from the initial well to the bottom of the adjacent potential well. Because of the asymmetry of the potential profile together with the random external force uniform in space, we find, for the different asymmetries: (1) an inversion of the curves of the mean first passage time in the resonant region of the correlation time $\tau$ of the dichotomous noise, for low thermal noise intensities; (2) a maximum of the mean velocity of the Brownian particle as a function of $\tau$; and (3) an inversion of the curves of the mean velocity and a very weak current reversal in the miniratchet system obtained with the asymmetrical potential profiles investigated. An inversion of the mean first passage time curves is also observed by varying the amplitude of the dichotomous noise, behavior confirmed by recent experiments.

Figure 1

“Up” and “down” configurations of the switching piecewise linear potential $V_{\pm }(x)$ used in our calculations. The position $x_0=0$ represents the starting point of the simulations. A reflecting boundary is assumed at $x=0$ and an absorbing boundary at $x=L$. (a) Symmetric piecewise linear potential the same as that used in Refs. [1, 2] with $V_{\pm }(0)=V_{\pm }(L)$; (b) symmetric piecewise linear potential with the randomly fluctuating external force uniform in space and $V_{\pm }(0)\ne V_{\pm }(L)$, obtained from Eqs. (3) and (4) with $k=0$; (c) asymmetric piecewise linear potential with the randomly fluctuating external force uniform in space, obtained from Eqs. (3) and (4) with $k=-0.25$. In panel (c) we report also the average asymmetric piecewise linear potential. The values of the other parameters are $L=1$, $h=2$ and $x_m=L/2+k$, $x_s=0.5$, and $a=1.2$.

Figure 2

Log-log plot of MFPT as a function of the correlation time of the dichotomous noise. The RA phenomenon is observed for three values of the asymmetry parameter $k$ of the piecewise linear potential [$k=-0.25$, $k=0$ (i.e., symmetric potential), $k=0.25$], with randomly fluctuating external force uniform in the space. The value of the white noise intensity is $D=0.18$. The amplitude of the dichotomous force is $a=1.2$. The left inset shows, in a semilog plot, the mean velocity of the Brownian particle with the same asymmetries as a function of the correlation time $\tau$. The inset on the right shows the three static (average) potential profiles investigated. (The legend is the same for all the plots.) All the other parameter values are as in Fig. 1.

Figure 3

Log-log plot of MFPT as a function of the correlation time. The RA effect is observed for different asymmetrical piecewise linear (PWL) potentials with fixed extremes and equal potential heights in the three cases. Differently from the results in Fig. 2 we do not find any intersection between the three curves, and the resonant correlation time ${\tau }_R$ changes for the different asymmetries. The amplitude of the dichotomous force is $a=0.6$, the same as that corresponding to the symmetric case in Fig. 2. All the other parameter values are as in Fig. 1.

Figure 4

Log-log plot of MFPT as a function of the correlation time for different values of the noise intensity $D$ and asymmetry parameter $k$: $D=0.20,0.25,0.40,0.60$. The values of $k$ and all the other parameters are the same as in Fig. 2. We observe the RA effect for all the noise intensities investigated, the disappearance of the crossings in the MFPT curves, and the shift of the minima by increasing the noise intensity $D$.

Figure 5

Semilog plot of the resonant MFPT values ($T_R$) as a function of the noise intensity $D$ for the three asymmetries investigated. The value $D_T\approx 0.27$ represents the threshold of thermal noise that discriminates if the two crossings in MFPT curves of Fig. 2 are present ($D<D_T$) or absent ($D>D_T$). Inset: mean velocity of the Brownian particle for one value of the asymmetry parameter, $k=0.25$, as a function of the correlation time $\tau$ for the noise intensities $D=0.18,0.20,0.22,0.24,0.26$. All the other parameter values are as in Fig. 1.

Figure 6

Left panel: Semilog plot of the values of MFPT at the two intersections in Fig. 2, ${\tau }_{C_L}$ and ${\tau }_{C_R}$ as a function of the noise intensity for different values of the amplitude of the dichotomous noise: $a=1.2,1.8,2.4,3.0$. Right top panel: Threshold value of the thermal noise intensity $D_T$ as a function of the amplitude $a$. Right bottom panel: MFPT as a function of the correlation time for different values of the amplitude $a$. All the other parameter values are as in Fig. 1

Figure 7

Log-log plot of MFPT as a function of the correlation time for the elementary ratchet composed by two single barriers only. The RA effect is observed. The MFPT is here the mean time taken by the Brownian particle starting at $x=0$ to reach the position $x=1$ or $x=-1$. The mean velocity of the particle is plotted in the upper inset. The mean velocity presents again a maximum at the same resonant value ${\tau }_R$. In the region of very low mean correlation time, the particle shows a very weak negative velocity (left inset), as a consequence of the different behavior of the mean velocities for the various asymmetries and the crossing of the curves shown in the inset of Fig. 2. The value of the noise intensity is $D=0.18$, and the amplitude of the dichotomous force is $a=1.2$. All the other parameters are as in Fig. 1.

Figure 8

Log-log plot of MFPT as a function of the correlation time for different shape of symmetrical potential profiles. The RA is observed in all potential profiles. We note than for values of $\tau$ close to ${\tau }_R$ the logarithmic distances between the MFPT curves are approximatively constant. The parameter values are the same as those in Fig. 2.