Noise driven current reversal and stabilisation in the tilted ratchet potential subject to tempered stable L\'{e}vy noise

Abstract

We consider motion of a particle in a one-dimensional tilted ratchet potential subject to two-sided tempered stable L\'{e}vy noise characterised by strength $\Omega$, fractional index $\alpha$, skew $\theta$ and tempering $\lambda$. We derive analytic solutions to the corresponding Fokker-Planck L\'{e}vy equations for the probability density. Due to the periodicity of the potential, we carry out reduction to a compact domain and solve for the analogue there 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 threshold for metastability of the system, namely when the particle stabilises 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 stabilised in a well in circumstances where deterministically it should flow with the tilt.