Risken's matrix continued fraction method is used to solve the Fokker-Planck equation to calculate particle current in an inertial symmetric (sinusoidal) periodic potential under the action of a constant force. The particle moves in a medium with friction coefficient also varying periodically in space as the potential but with a finite phase difference $ϕ(\ne nπ, n=0,1,2, ...)$. The algebraic sum of current with applied forces $\pm|F|$ gives the ratchet current in the adiabatic limit. Even though, the effect of frictional inhomogeneity is weak, the ratchet current shows very rich qualitative characteristics. The effects of variation of $F$, the temperature $T$, and the average friction coefficient $γ_0$ on the performance characteristics of the ratchet are presented.
We investigate the performance of an inertial frictional ratchet in a sinusoidal potential driven by a sinusoidal external field. The dependence of the performance on the parameters of the sinusoidally varying friction, such as the mean friction coefficient and its phase difference with the potential, is studied in detail. Interestingly, under certain circumstances, the thermodynamic efficiency of the ratchet against an applied load shows a non-monotonic behavior as a function of the mean friction coefficient. Also, in the large friction ranges, the efficiency is shown to increase with increas…
We obtain ratchet effect in inertial structureless systems in symmetric periodic potentials where the asymmetry comes from the nonuniform friction offered by the medium and driven by symmetric periodic forces. In the adiabatic limit the calculations are done by extending the matrix continued fraction method and also by numerically solving the appropriate Langevin equation. For finite frequency field drive the ratchet effect is obtained only numerically. In the transient time scales the system shows dispersionless behaviour as reported earlier when a constant force is applied. In the periodic d…
We study the nature of currents, input energy and entropy production in different types of adiabatically rocked ratchets using the method of stochastic energetics. The currents exhibit a peak as a function of noise strength. We show that there is no underlying resonance or synchronisation phenomena in the dynamics of the particle with these current peaks. This follows from the analysis of energy loss in the medium. We also show that the maxima seen in current as well as the total entropy production are not directly correlated.
We discuss a problem of optimization of the energetic efficiency of a simple rocked ratchet. We concentrate on a low-temperature case in which the particle's motion in a ratchet potential is deterministic. We show that the energetic efficiency of a ratchet working adiabatically is bounded from above by a value depending on the form of ratchet potential. The ratchets with strongly asymmetric potentials can achieve ideal efficiency of unity without approaching reversibility. On the other hand we show that for any form of the ratchet potential a set of time-protocols of the outer force exist unde…
We study the efficiency of frictional thermal ratchets driven by finite frequency driving force and in contact with a heat bath. The efficiency exhibits varied behavior with driving frequency. Both nonmonotonic and monotonic behavior have been observed. In particular the magnitude of efficiency in finite frequency regime may be more than the efficiency in the adiabatic regime. This is our central result for rocked ratchets. We also show that for the simple potential we have chosen, the presence of only spatial asymmetry (homogeneous system) or only frictional ratchet (symmetric potential profi…
We study the transport of an overdamped particle adiabatically driven by an asymmetric potential which is periodic in both space and time. We develop an adiabatic perturbation theory after transforming the Fokker-Planck equation into a time-dependent hermitian problem, and reveal an analogy with quantum adiabatic particle transport. An analytical expression is obtained for the ensemble average of the particle velocity in terms of the Berry phase of the Bloch states. Its time average is shown to be quantized as a Chern number in the deterministic or tight-binding limit, with exponentially small…
The transport of a walker in rocking feedback-controlled ratchets are investigated. The walker consists of two coupled "feet" that allow the interchange of the order of the particles while the walker moves. In the underdamped case, the deterministic dynamics of the walker in a tilted asymmetric ratchet with an external periodic force is considered. It is found that the delayed feedback ratchets with a switching-on-and-off dependence of the states of the system can lead to the absolute negative mobility (ANM). In such a novel phenomenon the particles move against the bias. Moreover, the walker …
The underdamped Langevin equation of motion of a particle, in a symmetric periodic potential and subjected to a symmetric periodic forcing with mean zero over a period, with nonuniform friction, is solved numerically. The particle is shown to acquire a steady state mean velocity at asymptotically large time scales. This net particle velocity or the ratchet current is obtained in a range of forcing amplitudes $F_0$ and peaks at some value of $F_0$ within the range depending on the value of the average friction coefficient and temperature of the medium. At these large time scales the position di…
The energetic efficiency of an overdamped Brownian particle in a saw tooth potential in the presence of time asymmetric forcing is studied in the adiabatic limit. An error made in the earlier work on the same problem in literature is corrected. We find that asymmetry in potential together with temporal asymmetry in forcing leads to much enhanced efficiency without fine tuning of parameters. The origin of this is traced to the suppression of backward current. We also present a comparative study between the role of continuous and discontinuous ratchet forces on these measurable quantities. We fi…
We investigate the dynamics of quantum particles in a ratchet potential subject to an ac force field. We develop a perturbative approach for weak ratchet potentials and force fields. Within this approach, we obtain an analytic description of dc current rectification and current reversals. Transport characteristics for various limiting cases -- such as the classical limit, limit of high or low frequencies, and/or high temperatures -- are derived explicitly. To gain insight into the intricate dependence of the rectified current on the relevant parameters, we identify characteristic scales and ob…
We investigate a dissipative, deterministic ratchet model in the overdamped regime driven by a {\it rectangular force}. Extensive numerical calculations are presented in a diagram depicting the drift velocity as a function of a wide range of the driving parameter values. We also present some theoretical considerations which explain some features of the mentioned diagram. In particular, we proof the existence of regions in the driving parameter space with bounded particle motion possessing zero current. Moreover, we present an explicit analytical expression for the drift velocity in the adiabat…
We study the generalized efficiency of an adiabatically rocked ratchet with both spatial and temporal asymmetry. We obtain an analytical expression for the generalized efficiency in the deterministic case. Generalized efficiency of the order of 50% is obtained by fine tuning of the parameter range. This is unlike the case of thermodynamic efficiency where we could readily get an enhanced efficiency of upto 90%. The observed higher values of generalized efficiency is attributed to be due to the suppression of backward current. We have also discussed briefly the differences between thermodynamic…
We explore the dependence of the thermally activated barrier crossing rate on various model parameters for a dimer that undergoes a Brownian motion on a piecewise linear bistable potential employing the method of adiabatic elimination of fast variable. By introducing a different model system and approaches than the previous works \cite{c4,c5}, not only we recapture the previous results but we further show that systematic elimination of the fast changing variable leads to an effective Kramers type potential. It is shown that for rigid dimer, the escape rate $R$ monotonously decreases with $k$. …
A duality relation between the long-time dynamics of a quantum Brownian particle in a tilted ratchet potential and a driven dissipative tight-binding model is reported. It relates a situation of weak dissipation in one model to strong dissipation in the other one, and vice versa. We apply this duality relation to investigate transport and rectification in ratchet potentials: From the linear mobility we infer ground-state delocalization for weak dissipation. We report reversals induced by adiabatic driving and temperature in the ratchet current and its dependence on the potential shape.
We study the noise-induced currents and reliability or coherence of transport in two different classes of rocking ratchets. For this, we consider the motion of Brownian particles in the over damped limit in both adiabatic and non-adiabatic regimes subjected to unbiased temporally symmetric and asymmetric periodic driving force. In the case of a time symmetric driving, we find that even in the presence of a spatially symmetric simple sinusoidal potential, highly coherent transport occurs. These ratchet systems exhibit giant coherence of transport in the regime of parameter space where unidirect…
When a Brownian particle in contact with a heat bath at a constant temperature is controlled by a time-dependent harmonic potential, its distribution function can be rigorously derived from the Kramers equation with the consideration of the inertial effect of the Brownian particle. Based on this rigorous solution and the concept of shortcuts to adiabaticity, we construct a stochastic heat engine by employing the time-dependent harmonic potential to manipulate the Brownian particle to complete a thermodynamic cycle. We find that the efficiency at maximum power of this stochastic heat engine is …
We construct a unifying theory of geometric effects in mesoscopic stochastic kinetics. We demonstrate that the adiabatic pump and the reversible ratchet effects, as well as similar new phenomena in other domains, such as in epidemiology, all follow from geometric phase contributions to the effective action in the stochastic path integral representation of the moment generating function. The theory provides the universal technique for identification, prediction and calculation of pump-like phenomena in an arbitrary mesoscopic stochastic framework.
Efficiency of generation of net unidirectional current in an adiabatically driven symmetric periodic potential system is studied. The efficiency shows a maximum, in the case of an inhomogeneous system with spatially varying periodic friction coefficient, as a function of temperature. The ratchet is not most efficient when it gives maximum current. The direction of current may also be reversed as a function of noise strength when, instead, an asymmetric periodic potential is considered.
We investigate directed motion in non-adiabatically rocked ratchet systems sustaining few bands below the barrier. Upon restricting the dynamics to the lowest M bands, the total system-plus-bath Hamiltonian is mapped onto a discrete tight-binding model containing all the information both on the intra- and inter-well tunneling motion. A closed form for the current in the incoherent tunneling regime is obtained. In effective single-band ratchets, no current rectification occurs. We apply our theory to describe rectification effects in vortex quantum ratchets devices. Current reversals upon varia…
An asymmetric Brownian particle subjected to an external time-dependent force may acquire a net drift velocity, and thus operate as a motor or ratchet, even if the external force is represented by an unbiased time-periodic function or by a zero-centered noise. For an adequate description of such ratchets, a conventional Langevin equation linear in the particle's velocity is insufficient, and one needs to take into account the first nonlinear correction to the dissipation force which emerges beyond the weak coupling limit. We derived microscopically the relevant nonlinear Langevin equation by e…
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 $η(t)\propto t^{-α}$ ($0<α<1$). It is shown that subdiffusive rectification currents, defined through the mean displacement and subvelocity $v_α$, $<δx(t)>\sim v_α t^α/ Γ(1+α)$, emerge asymptotically due to the breaking of the detailed…
We describe how adiabatically rocked quantum electron ratchets can act as heat pumps. In general, ratchets may be described as non-equilibrium systems in which directed particle motion is generated using spatial or temporal asymmetry. In a rocked ratchet, which may also be described as a non-linear rectifier, an asymmetric potential is tilted symmetrically and periodically. The potential deforms differently during each half-cycle, producing a net current of particles when averaged over a full period of rocking. Recently it was found that in the quantum regime, where tunnelling contributes to t…
We explore heat transfer in molecular junctions between two leads in the absence of a finite net thermal bias. The application of an unbiased, time-periodic temperature modulation of the leads entails a dynamical breaking of reflection symmetry, such that a directed heat current may emerge (ratchet effect). In particular, we consider two cases of adiabatically slow driving, namely (i) periodic temperature modulation of only one lead and (ii) temperature modulation of both leads with an ac driving that contains a second harmonic, thus generating harmonic mixing. Both scenarios yield sizeable di…
We consider the motion of a overdamped Brownian particle in periodic asymmetric potential with space dependent friction coefficient. In the presence of external time periodic forcing, the system shows multiple current reversals on varying the amplitude of the external forcing and the temperature of the thermal bath. In the adiabatic regime we find a single reversal of current as a function of noise strength which can only be accounted due to the presence of space dependent friction coefficient. For very large forcing term, the current does not go to zero, instead it asymptotically tends to a l…
We present a detailed study of the transport and energetics of a Brownian particle moving in a periodic potential in the presence of an adiabatic external periodic drive. The particle is considered to move in a medium with periodic space dependent friction with the same periodicity as that of the potential but with a phase lag. We obtain several results, most of them arising due to the medium being inhomogeneous and are sensitive to the phase lag. When the potential is symmetric we show that efficiency of energy transduction can be maximised as a function of noise strength or temperature. Howe…
The efficiency of energy transduction in a temporally asymmetric rocked ratchet is studied. Time asymmetry favours current in one direction and suppresses it in the opposite direction due to which large efficiency ~ 50% is readily obtained. The spatial asymmetry in the potential together with system inhomogeneity may help in further enhancing the efficiency. Fine tuning of system parameters considered leads to multiple current reversals even in the adiabatic regime.
The motion of self-propelled particles can be rectified by asymmetric or ratchet-like periodic patterns in space. Here we show that a non-zero average drift can already be induced in a periodic potential with symmetric barriers when the self-propulsion velocity is also symmetric and periodically modulated but phase-shifted against the potential. In the adiabatic limit of slow rotational diffusion we determine the mean drift analytically and discuss the influence of temperature. In the presence of asymmetric barriers modulating the self-propulsion can largely enhance the mean drift or even reve…
We study the Hamiltonian dynamics of a one-dimensional chain of linearly coupled particles in a spatially periodic potential which is subjected to a time-periodic mono-frequency external field. The average over time and space of the related force vanishes and hence, the system is effectively without bias which excludes any ratchet effect. We pay special attention to the escape of the entire chain when initially all of its units are distributed in a potential well. Moreover for an escaping chain we explore the possibility of the successive generation of a directed flow based on large accelerati…
We present an analytical study of a nonlinear oscillator subject to an additive Ornstein-Uhlenbeck noise. Known results are mainly perturbative and are restricted to the large dissipation limit (obtained by neglecting the inertial term) or to a quasi-white noise (i.e., a noise with vanishingly small correlation time). Here, in contrast, we study the small dissipation case (we retain the inertial term) and consider a noise with finite correlation time. Our analysis is non perturbative and based on a recursive adiabatic elimination scheme: a reduced effective Langevin dynamics for the slow actio…
We report a theoretical study of an overdamped Brownian particle dynamics in the presence of both a spatially modulated one-dimensional periodic potential and a periodic alternating force (AF). As the periodic potential has a low symmetry (a ratchet potential) the Brownian particle displays a broken symmetry motion with a nonzero time average velocity. By making use of the Green function method and a mapping to the theory of Brillouin bands the probability distribution of the particle coordinate is derived and the nonlinear dependence of the macroscopic velocity on the frequency and the amplit…
We consider the motion of overdamped particles on random potentials subjected to a Gaussian white noise and a time-dependent periodic external forcing. The random potential is modeled as the potential resulting from the interaction of a point particle with a random polymer. The random polymer is made up, by means of some stochastic process, from a finite set of possible monomer types. The process is assumed to reach a non-equilibrium stationary state, which means that every realization of a random polymer can be considered as an out-of-equilibrium structure. We show that the net flux of partic…
We study far from equilibrium transport of a periodically driven inertial Brownian particle moving in a periodic potential. As detected recently for a SQUID ratchet dynamics (Spiechowicz J. & Luczka J. Phys. Rev. E 91, 062104 (2015)), the mean square deviation of the particle position from its average may involve three distinct intermediate, although extended diffusive regimes: initially as superdiffusion, followed by subdiffusion and finally, normal diffusion in the asymptotic long time limit. Even though these anomalies are transient effects, their lifetime can be many, many orders of magnit…
