Subdiffusive Brownian ratchets rocked by a periodic force

This work puts forward a generalization of the well-known rocking Markovian Brownian ratchets to the realm of antipersistent non-Markovian subdiffusion in viscoelastic media. A periodically forced subdiffusion in a parity-broken ratchet potential is considered within the non-Markovian generalized Langevin equation (GLE) description with a power-law memory kernel $\eta(t)\propto t^{-\alpha}$ ($0<\alpha<1$). It is shown that subdiffusive rectification currents, defined through the mean displacement and subvelocity $v_{\alpha}$, $<\delta x(t)>\sim v_{\alpha} t^{\alpha}/ \Gamma(1+\alpha)$, emerge asymptotically due to the breaking of the detailed balance symmetry by driving. The asymptotic exponent is $\alpha$, the same as for free subdiffusion, $<\delta x^2(t)>\propto t^\alpha$. However, a transient to this regime with some time-dependent $\alpha_{\rm eff}(t)$ gradually decaying in time, $\alpha\leq \alpha_{\rm eff}(t)\leq 1$, can be very slow depending on the barrier height and the driving field strength. In striking contrast to its normal diffusion counterpart, the anomalous rectification current is absent asymptotically in the limit of adiabatic driving with frequency $\Omega\to 0$, displaying a resonance like dependence on the driving frequency. However, an anomalous current inversion occurs for a sufficiently fast driving, like in the normal diffusion case. In the lowest order of the driving field, such a rectification current presents a quadratic response effect. Beyond perturbation regime it exhibits a broad maximum versus the driving field strength. Moreover, anomalous current exhibits a maximum versus the potential amplitude.