Current reversals in a rocking ratchet: the frequency domain

A multiple-timing analysis of temporal ratcheting

We develop a two-timing perturbation analysis to provide quantitative insights on the existence of temporal ratchets in an exemplary system of a particle moving in a tank of fluid in response to an external vibration of the tank. We consider two-mode vibrations with angular frequencies ω and α ω , where α is a rational number. If α is a ratio of odd and even integers (e.g.,2 1 , 3 2 , 4 3 ), the system yields a net response: here, a nonzero time-average particle velocity. Our first-order perturbation solution predicts the existence of temporal ratchets for α = 2 . Furthermore, we demonstrate, for a reduced model, that the temporal ratcheting effect for α = 3 2 and4 3 appears at the third-order perturbation solution. More importantly, we find closed-form formulas for the magnitude and direction of the induced net velocities for these α values. On a broader scale, our methodology offers a new mathematical approach to study the complicated nature of temporal ratchets in physical systems.

Net motion induced by nonantiperiodic vibratory or electrophoretic excitations with zero time average

It is well established that application of an oscillatory excitation with zero time-average but temporal asymmetry can yield net drift. To date this temporal symmetry breaking and net drift has been explored primarily in the context of point particles, nonlinear optics, and quantum systems. Here, we present two new experimental systems where the impact of temporally asymmetric force excitations can be readily observed with mechanical motion of macroscopic objects: (1) solid centimeter-scale objects placed on a uniform flat surface made to vibrate laterally, and (2) charged colloidal particles in water placed between parallel electrodes with an applied oscillatory electric potential. In both cases, net motion is observed both experimentally and numerically with nonantiperiodic, two-mode, sinusoids where the frequency modes are the ratio of odd and even numbers (e.g., 2Hz and 3Hz). The observed direction of motion is always the same for the same applied waveform, and is readily reversed by changing the sign of the applied waveform, for example, by swapping which electrode is powered and grounded. We extend these results to other nonlinear mechanical systems, and we discuss the implications for facile control of object motion using tunable periodic driving forces.

Current inversion in a periodically driven two-dimensional Brownian ratchet

It is well known that Brownian ratchets can exhibit current reversals, wherein the sign of the current switches as a function of the driving frequency. We introduce a spatial discretization of such a two-dimensional Brownian ratchet to enable spectral methods that efficiently compute those currents. These discrete-space models provide a convenient way to study the Markovian dynamics conditioned upon generating particular values of the currents. By studying such conditioned processes, we demonstrate that low-frequency negative values of current arise from typical events and high-frequency positive values of current arises from rare events. We demonstrate how these observations can inform the sculpting of time-dependent potential landscapes with a specific frequency response.

Multiple Current Reversals Using Superimposed Driven Lattices

We demonstrate that directed transport of particles in a two dimensional driven lattice can be dynamically reversed multiple times by superimposing additional spatially localized lattices on top of a background lattice. The timescales of such current reversals can be flexibly controlled by adjusting the spatial locations of the superimposed lattices. The key principle behind the current reversals is the conversion of the particle dynamics from chaotic to ballistic, which allow the particles to explore regions of the underlying phase space which are inaccessible otherwise. Our results can be experimentally realized using cold atoms in driven optical lattices and allow for the control of transport of atomic ensembles in such setups.

Directed motion of spheres induced by unbiased driving forces in viscous fluids beyond the Stokes' law regime

The emergence of directed motion is investigated in a system consisting of a sphere immersed in a viscous fluid and subjected to time-periodic forces of zero average. The directed motion arises from the combined action of a nonlinear drag force and the applied driving forces, in the absence of any periodic substrate potential. Necessary conditions for the existence of such directed motion are obtained and an analytical expression for the average terminal velocity is derived within the adiabatic approximation. Special attention is paid to the case of two mutually perpendicular forces with sinusoidal time dependence, one with twice the period of the other. It is shown that, although neither of these two forces induces directed motion when acting separately, when added together, the resultant force generates directed motion along the direction of the force with the shortest period. The dependence of the average terminal velocity on the system parameters is analyzed numerically and compared with that obtained using the adiabatic approximation. Among other results, it is found that, for appropriate parameter values, the direction of the average terminal velocity can be reversed by varying the forcing strength. Furthermore, certain aspects of the observed phenomenology are explained by means of symmetry arguments.

Deterministic particle transport in a ratchet flow

This study is motivated by the issue of the pumping of particle through a periodic modulated channel. We focus on a simplified deterministic model of small inertia particles within the Stokes flow framework that we call "ratchet flow." A path-following method is employed in the parameter space in order to retrace the scenario which from bounded periodic solutions leads to particle transport. Depending on whether the magnitude of the particle drag is moderate or large, two main transport mechanisms are identified in which the role of the parity symmetry of the flow differs. For large drag, transport is induced by flow asymmetry, while for moderate drag, since the full transport solution bifurcation structure already exists for symmetric settings, flow asymmetry only makes the transport effective. We analyzed the scenarios of current reversals for each mechanism as well as the role of synchronization. In particular we show that, for large drag, the particle drift is similar to phase slip in a synchronization problem.

Control of transport in two-dimensional systems via dynamical decoupling of degrees of freedom with quasiperiodic driving fields

We consider the problem of the control of transport in higher-dimensional periodic structures by applied ac fields. In a generic crystal, transverse degrees of freedom are coupled, and this makes the control of motion difficult to implement. We show, both with simulations and with an analytical functional expansion on the driving amplitudes, that the use of quasiperiodic driving significantly suppresses the coupling between transverse degrees of freedom. This allows a precise control of the transport, and does not require a detailed knowledge of the crystal geometry.