C.M. Arizmendi

Quenched disorder effects on deterministic inertia ratchets.

The effect of quenched disorder on the underdamped motion of a periodically driven particle on a ratchet potential is studied. As a consequence of disorder, current reversal and chaotic diffusion may take place on regular trajectories. On the other hand, on some chaotic trajectories disorder induces regular motion. A localization effect similar to the Golosov phenomenon sets in whenever a disorder threshold that depends on the mass of the particle is reached. Possible applications of the localization phenomenon are discussed.

Current basins of attraction in inertia ratchets

Inertia ratchets are characterized by a complex dynamics with multiple current reversals. We recently studied the problem of control of current on inertia ratchets and demonstrated the feasibility of a control method associated with a process of locking to different mean velocity attractors. Here we present the results of a study of the fractal characteristics of basins of attraction for different mean velocities as well as the influence of quenched disorder on the attractors. We find that our previous conjecture that the domains of attraction of the two velocities are intermixed fractals is verified. We also find that the attractor for the negative velocity dissolves as the strength of disorder is increased. These results have important implications for the design of a reliable technique for controlling the particle mean velocities.

Control of current reversal in single and multiparticle inertia ratchets

We have studied the deterministic dynamics of underdamped single and multiparticle ratchets associated with current reversal, as a function of the amplitude of the external driving force. Two experimentally inspired methods are used. In the first method, the same initial condition is used for each new value of the amplitude. In the second method, the last position and velocity is used as the new initial condition when the amplitude is changed. The two methods are found to be complementary for control of current reversal, because the first one elucidates the existence of different attractors and gives information about their basins of attraction, while the second method, although history dependent, shows the locking process. We show that control of current reversals in deterministic inertia ratchets is possible as a consequence of a locking process associated with different mean velocity attractors. An unlocking effect is produced when a chaos to order transition limits the control range.

Chaotic dynamics and control of deterministic ratchets

Deterministic ratchets, in the inertial and also in the overdamped limit, have a very complex dynamics, including chaotic motion. This deterministically induced chaos mimics, to some extent, the role of noise, changing, on the other hand, some of the basic properties of thermal ratchets; for example, inertial ratchets can exhibit multiple reversals in the current direction. The direction depends on the amount of friction and inertia, which makes it especially interesting for technological applications such as biological particle separation. We overview in this work different strategies to control the current of inertial ratchets. The control parameters analysed are the strength and frequency of the periodic external force, the strength of the quenched noise that models a non-perfectly-periodic potential, and the mass of the particles. Control mechanisms are associated with the fractal nature of the basins of attraction of the mean velocity attractors. The control of the overdamped motion of noninteracting particles in a rocking periodic asymmetric potential is also reviewed. The analysis is focused on synchronization of the motion of the particles with the external sinusoidal driving force. Two cases are considered: a perfect lattice without disorder and a lattice with noncorrelated quenched noise. The amplitude of the driving force and the strength of the quenched noise are used as control parameters.

Memory correlation effect on thermal ratchets

We introduce correlation between steps in a random walk model with asymmetric piecewise linear potential where either a force or the barrier height fluctuates between two states. We call the correlation between steps that we introduce in the model memory correlation to distinguish it from the usual time correlation of the fluctuations on ratchets. The model gives a qualitative idea of the effect of finite inertia on thermal ratchets. The statistical measures that we have used to characterize the particle movement are the ratchet current and the temporal autocorrelation function. We find steady-state current reversals upon varying the memory effect parameter. We find also a transition in the autocorrelation function with increasing memory correlation that indicates a change from long memory correlated to randomly correlated motion.

Approach to steady-state current in ratchets

We have studied the approach to the steady state in thermal ratchets looking for signs of the future development of a non-zero current. We have concentrated on the evolution of the statistical root mean square fluctuations of the position of the walker as well as its temporal correlations. We find oscillations and abrupt slope variations in the fluctuations in the position of the ratchet particle and the temporal correlation functions. These effects are due to the fact that the flipping time of the potential is the same order of magnitude as the adiabatic adjustment time for the barrier. We find no oscillations when the flipping time is too low or too high compared with the adiabatic adjustment time.

Trapping mechanism in overdamped ratchets with quenched noise.

A trapping mechanism is observed and proposed as the origin of the anomalous behavior recently discovered in transport properties of overdamped ratchets subject to an external oscillatory drive in the presence of quenched noise. In particular, this mechanism is shown to appear whenever the quenched disorder strength is greater than a threshold value. The minimum disorder strength required for the existence of traps is determined by studying the trap structure in a disorder configuration space. An approximation to the trapping probability density function in a disordered region of finite length included in an otherwise perfect ratchet lattice is obtained. The mean velocity of the particles and the diffusion coefficient are found to have a nonmonotonic dependence on the quenched noise strength due to the presence of the traps.

Algorithmic complexity and efficiency of a ratchet

Molecular motors are characterized by a high degree of efficiency of energy transformation in the presence of thermal fluctuations. A fundamental question is how the efficiency of thermal ratchets depend on temperature and the flow of physical information (or negentropy). In order to address this question, in this work we have calculated the algorithmic complexity (or Kolmogorov information entropy) of a smoothly varying potential ratchet. The complexity is measured in terms of the average number of bits per time unit necessary to specify the sequence generated by the system. For a wide range of values of the flipping rate, the algorithmic complexity is found to be proportional to the efficiency in a flashing thermal ratchet. In addition, we find that at low temperatures, the algorithmic complexity (or efficiency) of a thermal ratchet increases with temperature. This is a highly counterintuitive result that may be important in the operation of molecular motors.

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.

Control of Current Reversal in Single and Multiparticle Inertia Ratchets

We have studied the deterministic dynamics of underdamped single and multiparticle ratchets associated with current reversal, as a function of the amplitude of the external driving force. Two experimentally inspired methods are used. In the first method the same initial condition is used for each new value of the amplitude. In the second method the last position and velocity is used as the new initial condition when the amplitude is changed. The two methods are found to be complementary for control of current reversal, because the first one elucidates the existence of different attractors and gives information about their basins of attraction, while the second method, although history dependent shows the locking process. We show that control of current reversals in deterministic inertia ratchets is possible as a consequence of a locking process associated with different mean velocity attractors. An unlocking effect is produced when a chaos to order transition limits the control range.