Mirror symmetry breaking through an internal degree of freedom leading to directional motion

Chaos in kicked ratchets

We present a minimal one-dimensional deterministic continuous dynamical system that exhibits chaotic behavior and complex transport properties. Our model is an overdamped rocking ratchet with finite dissipation, that is periodically kicked with a δ function driving force, without finite inertia terms or temporal or spatial stochastic forces. To our knowledge this is the simplest model reported in the literature for a ratchet, with this complex behavior. We develop an analytical approach that predicts many key features of the system, such as current reversals, as well as the presence of chaotic behavior and bifurcation. Our analytical approach allows us to study the transition from regular to chaotic motion as well as a tangent bifurcation associated with this transition. We show that our approach can be easily extended to other types of periodic driving forces. The square wave is shown as an example.

Collective dynamics of a network of ratchets coupled via a stochastic dynamical environment

We investigate the collective dynamics of a network of inertia particles diffusing in a ratchet potential and interacting indirectly through their stochastic dynamical environment. We obtain analytically the condition for the existence of a stable collective state, and we show that the number N of particles in the network, and the strength k of their interaction with the environment, play key roles in synchronization and transport processes. Synchronization is preceded by symmetry-breaking associated with double-resonance oscillations and is shown to be strongly dependent on the network size: convergence to the synchronization manifold occurs much faster with a large network. For small networks, increasing the noise level enhances synchronization in the weakly coupled regime, while particles in a large network are weakly synchronized. Similarly, in the strongly coupled regime, particles in a small network are weakly synchronized; whereas the synchronization is strong and robust against noise when the network-size is large. Small and moderate networks maximize and stabilize efficient transport. Although the dynamics for larger networks is highly correlated, the transport current is erratic.

Directional motion of forced polymer chains with hydrodynamic interaction

We study the propulsion of a one-dimensional (1D) polymer chain under sinusoidal external forces in the overdamped (low Reynolds number) regime. We show that, when hydrodynamical interactions are included, the polymer presents directional motion which depends on the phase differences of the external force applied along the chain. Moreover, the velocity shows a maximum as a function of the frequency. We discuss the relevance of all these results in light of recent nanotechnology experiments.

Dimer motion on a periodic substrate: spontaneous symmetry breaking and absolute negative mobility

We consider two coupled particles moving along a periodic substrate potential with negligible inertia effects (overdamped limit). Even when the particles are identical and the substrate spatially symmetric, a sinusoidal external driving of appropriate amplitude and frequency may lead to spontaneous symmetry breaking in the form of a permanent directed motion of the dimer. Thermal noise restores ergodicity and thus zero net velocity, but entails arbitrarily fast diffusion of the dimer for sufficiently weak noise. Moreover, upon application of a static bias force, the dimer exhibits a motion opposite to that force (absolute negative mobility). The key requirement for all these effects is a nonconvex interaction potential of the two particles.

Active particles with broken symmetry

We discuss and analyze the driving a polar active particle with a head-tail asymmetry based on the dynamics of an internal motor variable driven by an energy depot and a broken symmetry of friction with respect to the internal degree of freedom. We show that such a driving may be advantageous for driving large masses with small energy uptake from the environment and exhibits interesting properties such as resonance-driven optimal propulsion.

Current reversals and synchronization in coupled ratchets

Current reversal is an intriguing phenomenon that has been central to recent experimental and theoretical investigations of transport based on ratchet mechanism. By considering a system of two interacting ratchets, we demonstrate how the coupling can be used to control the reversals. In particular, we find that current reversal that exists in a single driven ratchet system can ultimately be eliminated with the presence of a second ratchet. For specific coupling strengths a current-reversal free regime has been detected. Furthermore, in the fully synchronized state characterized by the coupling threshold k(th), a specific driving amplitude a(opt) is found for which the transport is optimum.

Ratcheting of neutral elastic dimers on a charged filament

We consider an elastic neutral dimer formed by two bound equal masses carrying opposite charges and moving along an electrically active filament in one dimension. An ac electrical field drives the two dimer heads, when set free or bound together to form a rigid rod, to opposite directions, thus yielding a zero net dimer current for zero and infinite elastic constants. Under the same driving conditions, an elastically deformable dimer can get rectified and the ensuing net current maximized for an optimal value of dimer elastic constant. The dependence of the dimer current on the periodic charge distribution along the filament is analyzed in terms of global symmetries of the dimer dynamics.

Driven and damped double sine-Gordon equation: the influence of internal modes on the soliton ratchet mobility

This work studies the damped double sine-Gordon equation driven by a biharmonic force, where a parameter lambda controls the existence and the frequency of an internal mode. The role of internal oscillations of the kink width in ratchet dynamics of kink is investigated within the framework of collective coordinate theories. It is found that the ratchet velocity of the kink, when an internal mode appears in this system, decreases contrary to what was expected. It is also shown that the kink exhibits a higher mobility in the double sine-Gordon without internal mode, but with a quasilocalized first phonon mode.

Ratchet effect of a dimer with broken friction symmetry in a symmetric potential

The one-dimensional overdamped Brownian motion of a dimer consisting of two harmonically interacting components is considered. Both components are coupled to the same heat bath and feel the same spatially periodic symmetric potential, whose amplitude is modulated periodically in time. The friction coefficients may differ between dimer components, thus breaking the dynamical symmetry of the system. In the absence of any external bias, a ratchet effect (directed transport) arises generically. Two accurate approximations for the dimer's velocity and diffusion coefficient are obtained for weak and strong couplings. The velocity of the system can be maximized for each direction by adding an optimal amount of noise and by tuning the driving frequency to an optimal value. Furthermore, there exist two optimal coupling strengths at which the velocity is the largest.

Dynamics of a dimer in a symmetric potential: ratchet effect generated by an internal degree of freedom

The one-dimensional dynamics of a dimer consisting of two harmonically coupled components is considered. The mutual distance between the dimer components plays the role of an internal degree of freedom. Both components are in contact with the same heat bath and are coupled to a spatially periodic, symmetric potential, whose amplitude is modulated periodically in time and whose coupling strength is different for the two components. In the absence of any external bias, a ratchet effect (directed transport) arises generically unless the mutual coupling of the dimer components tends to zero or infinity. In other words, the ratchet effect is generated by the internal degree of freedom. An accurate analytical approximation for the dimer's velocity and diffusion coefficient is obtained. The velocity of the system is maximized by adding an optimal amount of noise and by tuning the driving frequency to an optimal value. Furthermore, there exists an optimal coupling strength at which the velocity is the largest.

Model for hand-over-hand motion of molecular motors

A simple flashing ratchet model in two dimensions is proposed to simulate the hand-over-hand motion of two head molecular motors such as kinesin. Extensive Langevin simulations of the model are performed. Good qualitative agreement with the expected behavior is observed. We discuss different regimes of motion and efficiency depending on model parameters.

Active elastic dimers: self-propulsion and current reversal on a featureless track

We present a Brownian inchworm model of a self-propelled elastic dimer in the absence of an external potential. Nonequilibrium noise together with a stretch-dependent damping form the propulsion mechanism. Our model connects three key nonequilibrium features -- position-velocity correlations, a nonzero mean internal force, and a drift velocity. Our analytical results, including striking current reversals, compare very well with numerical simulations. The model unifies the propulsion mechanisms of DNA helicases, polar rods on a vibrated surface, crawling keratocytes and Myosin VI. We suggest experimental realizations and tests of the model.

Controlling the ratchet effect through the symmetries of the systems: Application to molecular motors

We discuss a novel generic mechanism for controlling the ratchet effect through the breaking of relevant symmetries. We review previous works on ratchets where directed transport is induced by the breaking of standard temporal symmetries f(t)=-f(t+T/2) and f(t)=f(-t) (or f(t)=-f(-t)). We find that in seemingly unrelated systems the average velocity (or the current) of particles (or solitons) exhibits common features. We show that, as a consequence of Curie's symmetry principle, the average velocity (or the current) is related to the breaking of the symmetries of the system. This relationship allows us to control the transport in a systematic way. The qualitative agreement between the present analytical predictions and previous experimental, numerical, and theoretical results leads us to suggest that for the given breaking of the temporal symmetries there is an optimal wave form for a given time-periodic force. Also, we comment on how this mechanism can be applied to the case where a ratchet effect is induced by breaking of spatial symmetries. Finally, we conjecture that the ratchet potential underlying biological motor proteins might be optimized according to the breaking of the relevant symmetries.

Analytical approach to soliton ratchets in asymmetric potentials

We use soliton perturbation theory and collective coordinate ansatz to investigate the mechanism of soliton ratchets in a driven and damped asymmetric double sine-Gordon equation. We show that, at the second order of the perturbation scheme, the soliton internal vibrations can couple effectively, in presence of damping, to the motion of the center of mass, giving rise to transport. An analytical expression for the mean velocity of the soliton is derived. The results of our analysis confirm the internal mode mechanism of soliton ratchets proposed in [Phys. Rev. E 65, 025602(R) (2002)].

Deterministic directed transport of inertial particles in a flashing ratchet potential

Deterministic directed transport of inertia particles in a periodically on-off ratchet potential is investigated. We find that the directed transport can be induced by a finite inertia; i.e., in the overdamped case, no directed motion of the system can be observed. It is shown that a critical threshold of the ratchet asymmetry is required for the system to achieve a net current. Directed transport can be greatly enhanced when the coupling strength of particles is increased. An appropriate match of the coupling, the flashing period, and the damping can give rise to the best efficiency of transport. The commensurate effect on the directed transport, which originates from the spatial competition between the period of the potential and the static length of the coupling, is analyzed.

Thermal ratchet effects in ferrofluids

Rotational Brownian motion of colloidal magnetic particles in ferrofluids under the influence of an oscillating external magnetic field is investigated. It is shown that for a suitable time dependence of the magnetic field, a noise-induced rotation of the ferromagnetic particles due to rectification of thermal fluctuations takes place. Via viscous coupling, the associated angular momentum is transferred from the magnetic nanoparticles to the carrier liquid and can then be measured as macroscopic torque on the fluid sample. A thorough theoretical analysis of the effect in terms of symmetry considerations, analytical approximations, and numerical solutions is given which is in accordance with recent experimental findings.

Stimulated diffusion of an adsorbed dimer

The mobility and the diffusivity of a dimer (two atoms coupled by an elastic spring) in a periodic substrate potential under the action of the dc and ac external forces are studied. It is shown that the dimer diffusivity may be strongly enhanced due to driving.

Collective directed transport of symmetrically coupled lattices in symmetric periodic potentials

A mechanism responsible for the directed transport of symmetrically coupled lattices in a symmetric periodic potential is proposed. Under the action of an external wave that breaks the spatiotemporal symmetry and introduces inhomogeneities of the lattice, a net unidirectional current can be observed, originating from a collaboration of the lattice (periodic potential and mutual interactions) and the wave (amplitude, frequency, and phase shifts). Mode-locking induced resonant steps are observed and predicted. The directed transport can be optimized and furthermore controlled by suitably adjusting the parameters of the system.

Broken space-time symmetries and mechanisms of rectification of ac fields by nonlinear (non)adiabatic response

We consider low-dimensional dynamical systems exposed to a heat bath and to additional ac fields. The presence of these ac fields may lead to a breaking of certain spatial or temporal symmetries, which in turn cause nonzero averages of relevant observables. Nonlinear (non)adiabatic response is employed to explain the effect. We consider a case of a particle in a periodic potential as an example and discuss the relevant symmetry breakings and the mechanisms of rectification of the current in such a system.

Broken symmetries and directed collective energy transport in spatially extended systems

We study the appearance of directed energy current in homogeneous spatially extended systems coupled to a heat bath in the presence of an external ac field E(t). The systems are described by nonlinear field equations. By making use of a symmetry analysis, we predict the right choice of E(t) and obtain directed energy transport for systems with a nonzero topological charge Q. We demonstrate that the symmetry properties of motion of topological solitons (kinks and antikinks) are equivalent to the ones for the energy current. Numerical simulations confirm the predictions of the symmetry analysis and, moreover, show that the directed energy current drastically increases as the dissipation parameter alpha reduces.

Ratchet due to broken friction symmetry

A ratchet mechanism that occurs due to asymmetric dependence of the friction of a moving system on its velocity or a driving force is reported. For this kind of ratchet, instead of a particle moving in a periodic potential, the dynamics of which have broken space-time symmetry, the system must be provided with some internal structure realizing such a velocity- or force-friction dependence. For demonstration of a ratchet mechanism of this type, an experimental setup (gadget) that converts longitudinal oscillating or fluctuating motion into a unidirectional rotation has been built and experiments with it have been carried out. In this device, an asymmetry of friction dependence on an applied force appears, resulting in rectification of rotary motion. In experiments, our setup is observed to rotate only in one direction, which is in accordance with given theoretical arguments. Despite the setup being three dimensional, the ratchet rotary motion is proved to be described by one dynamical equation. This kind of motion is a result of the interplay of friction and inertia. We also consider a case with viscous friction, which is irrelevant to this gadget, but it can be a possible mechanism of rotary unidirectional motion of some swimming organisms in a liquid.