Lévy-noise-induced transport in a rough triple-well potential

Rough energy landscape and noisy environment are two common features in many subjects, such as protein folding. Due to the wide findings of bursting or spiking phenomenon in biology science, small diffusions mixing large jumps are adopted to model the noisy environment that can be properly described by Lévy noise. We combine the Lévy noise with the rough energy landscape, modeled by a potential function superimposed by a fast oscillating function, and study the transport of a particle in a rough triple-well potential excited by Lévy noise, rather than only small perturbations. The probabilities of a particle staying in the middle well are considered under different amplitudes of roughness to find out how roughness affects the steady-state probability density function. Variations in the mean first passage time from the middle well to the right well have been investigated with respect to Lévy parameters and amplitudes of the roughness. In addition, we have examined the influences of roughness on the splitting probabilities of the first escape from the middle well. We uncover that the roughness can enhance significantly the first escape of a particle from the middle well, especially for different skewness parameters, but weak differences are found for stability index and noise intensity on the probabilities a particle staying in the middle well and splitting probability to the right.

Figure 1

Diagrams illustrating of the rough potential Eq. (2) for different $\varepsilon$.

Figure 2

The probability density functions (PDFs) of Lévy noise. (a) $\beta =0,D=0.5$, (b) $\alpha =1.2,D=0.5$.

Figure 3

The PDFs for different Lévy parameters. (a)$\beta =0,D=0.3$, the larger is $\alpha$ the fatter is the PDF. (b) $\alpha =1.5,\beta =0$, the changes are similar with $\alpha$, but PDFs for different $D$ are always slim. (c) $\alpha =1.5,D=0.3$, for $\beta >0$ the PDFs are right-skewed, for $\beta <0$ the PDFs are left-skewed.

Figure 4

The PDFs for different amplitudes of roughness in the potential under the parameters $\alpha =1.5,D=0.3$. (a) Plots of PDFs for $\varepsilon$ varying from zero to 0.15. (b) Compasions of the PDFs in the region (1.5, 2.3) for $\varepsilon =0$ and $\varepsilon =0.05$ corresponding to their potentials.

Figure 5

$P_{\mathrm{mid}}$ for different amplitudes of roughness with respect to $\alpha , D$, and $\beta$, (a) $\beta =0,D=0.3, P_{\mathrm{mid}}$ increases with $\alpha$, but the differences between different roughness are small. (b) $\alpha =1.5,\beta =0, P_{\mathrm{mid}}$ decreases along with the noise intensity $D$ in the absence of roughness, when $\varepsilon =0.15P_{\mathrm{mid}}$ tends to relatively stationarily for large $D$. (c) $\alpha =1.5,D=0.3, P_{\mathrm{mid}}$ is almost symmetric around $\beta =0$, and the roughness decreases $P_{\mathrm{mid}}$ significantly.

Figure 6

MFPTs in the absence of roughness ($\varepsilon =0$) with respect to $\alpha ,D$, and $\beta$. (a) $D=0.3$, with the increase of $\alpha$ particles need more and more time to escape. (b) $\beta =0$, large $D$ will decrease the MFPT of particles. (c) $\alpha =1.5$, the increase of $\beta$ can promote the escape of particles.

Figure 7

MFPTs with respect to different amplitudes of the roughness in the potential. (a) $\beta =0,D=0.3$, when $\alpha =1.2$ the MFPT is almost unchanged for varying $\varepsilon$, but when $\alpha$ is large the roughness can decrease the MFPT significantly. (b) $\alpha =1.5,\beta =0$, contrary to the case of $\alpha$, the MFPTs of small $D$ acquires more influences from the roughness and decrease obviously. (c) $\alpha =1.5,D=0.3$, for $\beta >0$, roughness will decrease the MFPTs, but for $\beta <0$, it is the opposite.

Figure 8

The schematic illustration of the transitions for particles from rough potential well.

Figure 9

The splitting probabilities from the middle well to the right well with respect to different amplitudes of roughness and $\alpha ,\beta ,D$. (a)$\beta =0,D=0.3, \mathrm{s}{\mathrm{p}}_r$ for different $\alpha$ change only a bit. (b) $\alpha =1.5,\beta =0$, it is similar with $\alpha$, little change happens on different $D$. (c) $\alpha =1.5,D=0.3$, the influences of roughness with respect $\beta$ is significant, for $\beta >0$, roughness will accelerate particles to jump to the right well, while for $\beta <0$, roughness will weaken the process.