Levy ratchets in the spatially tempered fractional Fokker-Planck equation

TL;DR

This paper investigates how truncated Le9vy distributions affect ratchet transport, showing persistent currents under certain conditions and revealing how tempering influences current decay and reversal in Le9vy ratchets.

Contribution

It introduces the use of spatially tempered fractional Fokker-Planck equations to model Le9vy ratchets, addressing limitations of e1-stable distributions and analyzing current behavior with respect to tempering parameter bb.

Findings

Finite steady-state current persists for any finite bb.

Current decays algebraically for b e 1.75 and exponentially for b e 1.5.

Tempering can cause current reversal in biased Le9vy noise.

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.