Diffusion in discrete ratchets

Effect of particle size oscillations on drift and diffusion along a periodically corrugated channel

We study diffusive transport of a particle in a channel with periodically varying cross-section, occurring when the size of the particle periodically switches between two values. In such a situation, the entropy potential, which accounts for the area accessible for diffusion particle, varies both spatially (along the channel axis) and temporally. This underlies the complex interplay between different timescales of the system and leads to novel dynamic regimes. The most notable observations are: emergence of directed motion (in case of asymmetric channel) and resonant diffusion, both controlled by the switching frequency. Resonantlike behaviors of the drift velocity and the effective diffusion coefficient are shown and discussed. Based on heuristic arguments, an approximate analytical treatment of the transport process is proposed. As a comparison with the results obtained from Brownian dynamics simulations indicates, this approach provides a satisfactory way to handle the problem analytically.

Landauer's blowtorch effect as a thermodynamic cross process: Brownian cooling

The local heating of a selected region in a double-well potential alters the relative stability of the two wells and gives rise to an enhancement of population transfer to the cold well. We show that this Landauer's blowtorch effect may be considered in the spirit of a thermodynamic cross process linearly connecting the flux of particles and the thermodynamic force associated with the temperature difference and consequently ensuring the existence of a reverse cross effect. This reverse effect is realized by directing the thermalized particles in a double-well potential by application of an external bias from one well to the other, which suffers cooling.

Thermally driven Casimir ratchet-oscillator system

We study a fluctuation-driven ratchet-oscillator system that consists of two ratchet pinions in contact with thermal baths at different temperatures. Coupling between noncontact parts of the system is mediated by the Casimir force through a thin corrugated plate. Due to mutual rectification, the two ratchets achieve directed average motion in opposite directions. We numerically probe the average velocity, the Peclet number, and the thermal efficiency of the system as functions of the effective temperatures of the thermal baths and other important dimensionless control parameters. It is found that optimal parameter values exist which maximize the directed transport rate but with very low efficiency. We expect that the obtained results will help in the design of noncontact mechanical devices in the future.

SYNCHRONIZATION AND TRANSPORT IN AN OSCILLATING PERIODIC POTENTIAL

We study the motion of overdamped Brownian particles in a periodic potential with a temporally oscillating amplitude. First we investigate diffusive motion in the untitled potential. Furthermore, if a constant force is applied, the oscillating potential induces a synchronized motion. The deterministic dynamics becomes in resonance with the potential oscillations. This dynamics gives rise to a transport with extremely low dispersion. We distinguish slow and fast oscillatory driving and give analytical expressions for the mean velocity and effective diffusion.

Synchronization of weakly perturbed Markov chain oscillators

Rate processes are simple and analytically tractable models for many dynamical systems that switch stochastically between a discrete set of quasistationary states; however, they may also approximate continuous processes by coarse-grained, symbolic dynamics. In contrast to limit-cycle oscillators that are weakly perturbed by noise, in such systems, stochasticity may be strong, and topologies more complicated than a circle can be considered. Here we apply a second-order time-dependent perturbation theory to derive expressions for the mean frequency and phase diffusion constant of discrete-state oscillators coupled or driven through weakly time-dependent transition rates. We also describe a method of global control to optimize the response of the mean frequency in complex transition networks.

Quasideterministic transport of Brownian particles in an oscillating periodic potential

We consider overdamped Brownian dynamics in a periodic potential with temporally oscillating amplitude. We analyze the transport which shows effective diffusion enhanced by the oscillations and derive approximate expressions for the diffusion coefficient. Furthermore we analyze the effect of the oscillating potential on the transport if additionally a constant force is applied. We show the existence of synchronization regimes at which the deterministic dynamics is in resonance with the potential oscillations, giving rise to transport with extremely low dispersion. We distinguish slow and fast oscillatory driving and give analytical expressions for the mean velocity and effective diffusion.

Optimizing rotary processes in synthetic molecular motors

We deal with the issue of quantifying and optimizing the rotation dynamics of synthetic molecular motors. For this purpose, the continuous four-stage rotation behavior of a typical light-activated molecular motor was measured in detail. All reaction constants were determined empirically. Next, we developed a Markov model that describes the full motor dynamics mathematically. We derived expressions for a set of characteristic quantities, i.e., the average rate of quarter rotations or "velocity," V, the spread in the average number of quarter rotations, D, and the dimensionless Péclet number, Pe = V/D. Furthermore, we determined the rate of full, four-step rotations (Omega(eff)), from which we derived another dimensionless quantity, the "rotational excess," r.e. This quantity, defined as the relative difference between total forward (Omega(+)) and backward (Omega(-)) full rotations, is a good measure of the unidirectionality of the rotation process. Our model provides a pragmatic tool to optimize motor performance. We demonstrate this by calculating V, D, Pe, Omega(eff), and r.e. for different rates of thermal versus photochemical energy input. We find that for a given light intensity, an optimal temperature range exists in which the motor exhibits excellent efficiency and unidirectional behavior, above or below which motor performance decreases.

Polymer dynamics in time-dependent periodic potentials

The dynamics of a discrete polymer in time-dependent external potentials is studied with the master equation approach. We consider both stochastic and deterministic switching mechanisms for the potential states and give the essential equations for computing the stationary-state properties of molecules with internal structure in time-dependent periodic potentials on a lattice. As an example, we consider standard and modified Rubinstein-Duke polymers and calculate their mean drift and effective diffusion coefficient in the two-state nonsymmetric flashing potential and symmetric traveling potential. Rich nonlinear behavior of these observables is found. By varying the polymer length, we find current inversions caused by the rebound effect that is only present for molecules with internal structure. These results depend strongly on the polymer type. We also notice increased transport coherence for longer polymers.

Flux reversal in a simple random-walk model on a fluctuating symmetric lattice

A rather simple random-walk model on a one-dimensional lattice is put forward. The lattice as a whole switches randomly between two possible states which are spatially symmetric. Both lattice states are identical, but translated by one site with respect to each other, and consist of infinite arrays of absorbing sites separated by two nonabsorbing sites. Exact explicit expressions for the long-time velocity and the effective diffusion coefficient are obtained and discussed. In particular, it is shown that the direction of the steady motion can be reversed by conveniently varying the values of either the mean residence times in the lattice states or the transition rates to the absorbing and nonabsorbing sites.

Single-polymer Brownian motor: a simulation study

Numerical simulation is used to study a single polymer chain in a flashing ratchet potential to determine how the mechanism of this Brownian motor system is affected by the presence of internal degrees of freedom. The polymer is modeled by a freely jointed chain with N monomers in which the monomers interact via a repulsive Lennard-Jones potential and neighboring monomers on the chain are connected by finite extensible nonlinear elastic bonds. Each monomer is acted upon by a 1D asymmetric, piecewise linear potential of spatial period L comparable to the radius of gyration of the polymer. This potential is also characterized by a localization time, t(on), and by a free diffusion time, t(off). We characterize the average motor velocity as a function of L, t(off), and N to determine optimal parameter ranges, and we evaluate motor performance in terms of finite dispersion, Peclet number, rectification efficiency, stall force, and transportation of a load against a viscous drag. We find that the polymer motor performs qualitatively better than a single particle in a flashing ratchet: with increasing N, the polymer loses velocity much more slowly than expected in the absence of internal degrees of freedom, and the motor stall force increases linearly with N. To understand these cooperative aspects of motor operation, we analyze relevant Rouse modes. The experimental feasibility is analyzed and the parameters of the model are scaled to those of lambda-DNA. Finally, in the context of experimental realization, we present initial modeling results for a 2D flashing ratchet constructed using an electrode array, and find good agreement with the results of 1D simulations although the polymer in the 2D potential sometimes briefly "detaches" from the electrode surface.

Periodic driving controls random motion of Brownian steppers

We study the motion of Brownian steppers, which are objects moving unidirectionally by discrete steps. A single step is composed of two processes. An activation process describing the random attachment of a fuel molecule is followed by a conformational change of the stepper, leading to the forward motion. Whereas activation is given by a Markovian rate process, the forward motion is defined by a gamma distribution. In this paper we propose a periodic modulation of the fuel concentration in order to control the random motion of the stepper. We show that the driving may reduce the fluctuations of the stepper. Corresponding minima of the diffusion coefficient and maxima of the Péclet number prove the regularity of the motion.

Performance characteristics of Brownian motors

Brownian motors are nonequilibrium systems that rectify thermal fluctuations to achieve directed motion, using spatial or temporal asymmetry. We provide a tutorial introduction to this basic concept using the well-known example of a flashing ratchet, discussing the micro- to nanoscopic scale on which such motors can operate. Because of the crucial role of thermal noise, the characterization of the performance of Brownian motors must include their fluctuations, and we review suitable performance measures for motor coherency and efficiency. Specifically, we highlight that it is possible to determine the energy efficiency of Brownian motors by measuring their velocity fluctuations, without detailed knowledge of the motor function and its energy input. Finally, we exemplify these concepts using a model for an artificial single-molecule motor with internal degrees of freedom.

Phase velocity and phase diffusion in periodically driven discrete-state systems

We develop a theory to calculate the effective phase diffusion coefficient and the mean phase velocity in periodically driven stochastic models with two discrete states. This theory is applied to a dichotomically driven Markovian two-state system. Explicit expressions for the mean phase velocity, the effective phase diffusion coefficient, and the Pe clet number are analytically calculated. The latter indicates as a measure of phase-coherence forced synchronization of the stochastic system with respect to the periodic driving and exhibits a "bona fide" resonance. In a second step, the theory is applied to a non-Markovian two-state system modeling excitable systems. The results prove again stochastic synchronization to the periodic driving and are in good agreement with simulations of a stochastic FitzHugh-Nagumo system.

Brownian motors: current fluctuations and rectification efficiency

With this work, we investigate an often neglected aspect of Brownian motor transport, namely, the role of fluctuations of the noise-induced current and its consequences for the efficiency of rectifying noise. In doing so, we consider a Brownian inertial motor that is driven by an unbiased monochromatic, time-periodic force and thermal noise. Typically, we find that the asymptotic, time-, and noise-averaged transport velocities are small, possessing rather broad velocity fluctuations. This implies a corresponding poor performance for the rectification power. However, for tailored profiles of the ratchet potential and appropriate drive parameters, we can identify a drastic enhancement of the rectification efficiency. This regime is marked by persistent, unidirectional motion of the Brownian motor with few back-turns only. The corresponding asymmetric velocity distribution is then rather narrow, with a support that predominantly favors only one sign for the velocity.

Diffusion and coherence in tilted piecewise linear double-periodic potentials

In the present paper we study the overdamped motion of Brownian particles in tilted periodic piecewise linear potentials of two maxima per period. This system allows one to observe additional phenomena with respect to the case of single-periodic potentials. We show that for certain types of potentials the effective diffusion coefficient D (F) can be enhanced or suppressed compared to the simple sawtooth case. As the most unexpected result it is found that the curve of diffusion coefficient vs tilt can have two maxima. In the region of Poissonian hopping processes we demonstrate the possibility to have a resonantlike peak in the Péclet number Pe(F) . At the tilting force corresponding to the maximum of Pe(F) , the curve of Péclet number vs temperature possesses two maxima.

Noise-induced transport with low randomness

We study the transport of overdamped Brownian particles in periodic potentials subject to a spatially modulated Gaussian white noise. We derive an analytical expression for the diffusion coefficient of particles. By means of velocity, diffusion coefficient, and their ratio (Péclet number) we discuss (a) symmetric potential and modulation of noise intensity and (b) a ratchet profile with strong noise modulation. It is shown that state dependent fluctuations may not only induce directed transport, but also a pronounced coherence of transport if the potential exhibits a strong asymmetry.

Giant diffusion and coherent transport in tilted periodic inhomogeneous systems

We investigate the dynamics of an overdamped Brownian particle moving in a washboard potential with space dependent friction coefficient. Analytical expressions have been obtained for current and diffusion coefficient. We show that the effective diffusion coefficient can be enhanced or suppressed compared to that of the uniform friction case. The diffusion coefficient is maximum near the critical threshold (F(c)), which is sensitive to temperature and the frictional profile. In some parameter regime, we observe that increase in noise (temperature) decreases the diffusion, which is counterintuitive. This leads to coherent transport with large mean velocity accompanied by small diffusion. This is shown explicitly by analysis of Péclet number, which has been introduced to study coherent or optimal transport. This phenomenon is complementary to giant diffusion.

Hopping motion of interacting particles: from time-dependent interaction to directed transport

The hopping motion of three or more pairwise interacting particles on a linear equipotential chain is investigated. It is shown that a directed transport of the particle array can be obtained by introducing time-dependent interactions such as switching the interaction strength periodically between two values. The direction and mean velocity of the motion are determined by the nature of the imposed time dependence and can, hence, be chosen dynamically. The possibility of transporting a passive cargo and some other applications are discussed.