Subdiffusive Brownian ratchets rocked by a periodic force

Abstract

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

Transport properties of elastically coupled fractional Brownian motors

“…. , N. And we proceed to express the proposed model (17) in terms of dimensionless one [54,55]. The distance is scaled in units of L, the energy is scaled in units of E 0 = mL 2 , coupling force and external harmonic force in units of…”

“…Thus, generalized Langevin equation (GLE) [48][49][50][51] and fractional Langevin equation (FLE) [38,52,53] were proposed to describe the motion behaviors of particles in various memorable mediums. Igor et al respectively studied the rocking Markovian ratchets [54] and fluctuating tilt ratchets [55] based on fractional subdiffusion in sticky viscoelastic medium characterized by a power law memory kernel, and found that subdiffusive current could flow in the counterintuitive direction upon a change of temperature or driving frequency. However, all of these literatures were mainly restricted to the study on dynamics of single particle.…”

Flashing subdiffusive ratchets in viscoelastic media

“…Such viscoelastic subdiffusion has recently been found relevant also for transport in polymer networks [27,28], cytosol of biological cells [29] and akin crowded fluids [30]. The corresponding fluctuating tilt or rocking ratchets in viscoelastic media have recently been proposed and studied in [31,32] with a number of quite unexpected and surprising properties revealed. There is growing interest in the ratchet effect in glass-like environments [33].…”

Trichotomous Noise Induced Resonance Behavior for a Fractional Oscillator with Random Mass

“…When ω 2 0 = 0, the Eq. (2) describes the normal diffusion with the mean-square displacement (MSD) which satisfies x 2 (t) ∼ t. However, a situation that should not be neglected is that the classical theory of Brownian diffusion cannot explain the subdiffusion phenomenon with MSD which satisfies x 2 (t) ∼ t α , 0 < α < 1 [2,3]. In mathematical descriptions of the subdiffusion phenomenon, the fractional dynamical systems play very important roles.…”