Levy ratchets in the spatially tempered fractional Fokker-Planck equation

Abstract

L\'evy ratchets are minimal models of fluctuation-driven transport in the presence of L\'evy noise and periodic external potentials with broken spatial symmetry. In these systems, a net ratchet current can appear even in the absence of time dependent perturbations, external tilting forces, or a bias in the noise. The majority of studies on the interaction of L\'evy noise with external potentials have assumed $\alpha$-stable L\'evy statistics in the Langevin description, which in the continuum limit corresponds to the fractional Fokker-Planck equation. However, the divergence of the low order moments is a potential drawback of $\alpha$-stable distributions because, in applications, the moments represent physical quantities. For example, for $\alpha <1$, the current $J$, in $\alpha$-stable L\'evy ratchets is unbounded. To overcome this limitation, we study ratchet transport using truncated L\'evy distributions which in the continuum limit correspond to the spatially tempered fractional Fokker-Planck equation. The main object of study is the dependence of the ratchet current on the level of tempering, $\lambda$. For $\lambda \neq 0$, the statistics ultimately converges (although very slowly) to Gaussian diffusion in the absence of a potential. However, it is shown here that in the presence of a ratchet potential a finite current persists asymptotically for any finite value of $\lambda$. The current converges exponentially in time to the steady state value. The steady state current exhibits algebraically decay, $J\sim \lambda^{-\zeta}$, for $\alpha \geq 1.75$. However, for $\alpha \leq 1.5$, the decay is exponential, $J \sim e^{-\xi \lambda}$. In the presence of a bias in the L\'evy noise, it is shown that the tempering can lead to a current reversal. A detailed numerical study is presented on the dependence of the current on $\lambda$ and the physical parameters of the system.