Noise-driven current reversal and stabilization in the tilted ratchet potential subject to tempered stable Lévy noise

We consider motion of a particle in a one-dimensional tilted ratchet potential subject to two-sided tempered stable Lévy noise characterized by strength $\mathrm{\Omega }$, fractional index $\alpha$, skew $\theta$, and tempering $\lambda$. We derive analytic solutions to the corresponding Fokker-Planck Lévy equations for the probability density. Due to the periodicity of the potential, we carry out reduction to a compact domain and solve for the analog of steady-state solutions which we represent as wrapped probability density functions. By solving for the expected value of the current associated with the particle motion, we are able to determine thresholds for metastability of the system, namely when the particle stabilizes in a well of the potential and when the particle is in motion, for example as a consequence of the tilt of the potential. Because the noise may be asymmetric, we examine the relationship between skew of the noise and the tilt of the potential. With tempering, we find two remarkable regimes where the current may be reversed in a direction opposite to the tilt or where the particle may be stabilized in a well in circumstances where deterministically it should flow with the tilt.

Figure 1

Examples of Eq. (17) (insets) and Eq. (18) (main) for $\rho =-1.5$ and $\gamma =1$ and various values of the diffusivity $\mathrm{\Omega }$ which governs noise strength. The leftmost inset details a case with stable fixed points in the potential ($\mu =0.99$), the rightmost inset details a case with no fixed points in the potential ($\mu =1.01$). Both insets are given as a parametric plot governed by Eq. (19).

Figure 2

Examples of Eq. (23) (insets) and Eq. (27) (main) for $\rho =-1.5$, $\gamma =1$, $\alpha =0.5$, $\lambda =0.5$, $\mathrm{\Omega }=0.1$ and various values of the asymmetry $\theta$. The left-most inset details a case with stable fixed points in the potential ($\mu =0.99$), the right-most inset details a case with no fixed points in the potential ($\mu =1.01$). Both insets are given as a parametric plot governed by Eq. (19).

Figure 3

Examples of Eq. (23) (insets) and Eq. (27) (main) for $\rho =-1.5$, $\gamma =1$, $\alpha =1.25$, $\lambda =0.001$, $\mathrm{\Omega }=0.1$ and various values of the asymmetry $\theta$. The left-most inset details a case with stable fixed points in the potential ($\mu =0.5$), the middle inset is $\mu =0.99$, and the right-most inset details a case with no fixed points in the potential ($\mu =1.01$). The insets are given as a parametric plot governed by Eq. (19).

Figure 4

Examples of Eq. (27) for $\rho =-1.5$, $\gamma =1$, $\mathrm{\Omega }=0.1$. The tempering $\lambda$ is being varied on the horizontal axis, with various values of the asymmetry $\theta$. Down the rows (A–F) the tilt $\mu$ is varied, with the values $\mu =\{0.1,0.995,1.005,1.01,1.03,1.05\}$ being applied from top to bottom row. Across the columns (I–IV) the fractional power $\alpha$ is varied, with the values $\alpha =\{0.25,0.55,0.85,1.25\}$ applied from left-most to right-most column.