Occurrence and robustness of current reversals in overdamped deterministic ratchets under symmetric forcing.

We analyze the conditions for the existence of current reversals in overdamped deterministic tilting ratchets under symmetric forcing. To this end, we use and extend the formalism recently introduced in R. Salgado-García, M. Aldana, and G. Martínez-Mekler [Phys. Rev. Lett. 96, 134101 (2006)] to transform the equations of motion of the ratchet into discrete circle maps. For a periodic dichotomous forcing we show that the phenomenon of current reversal is not uncommon and exists for a nonzero measure set of the parameter space. Additionally, we show numerically that, for a wide class of ratchet potentials, current reversals also occur when the discontinuous dichotomous forcing is replaced by symmetric continuous driving forces. The likelihood of the occurrence of current reversals is a consequence of the structural stability under small perturbations of the associated circle map with rational rotation number.

Analytical approach to the time-dependent probability density function in tilted periodic potentials.

In this work we introduce a scheme for the calculation of an approximate closed expression for the time-dependent probability density function for overdamped particles in tilted periodic potentials. Our derivation is based on an ansatz for the solution of the corresponding Fokker-Planck equation and on a self-consistent cumulant calculation. The high accuracy of our expression for the time-dependent probability density function is exhibited by comparisons with Langevin dynamics simulations and exact analytic results for the drift and diffusion coefficients. Good agreement is found both, for large and intermediate times.

Synchronization phenomena for a pair of locally coupled chaotic electrochemical oscillators: a survey.

Chaotic synchronization of two locally coupled electrochemical oscillators is studied numerically. Both bidirectional and unidirectional couplings are considered. For both these coupling scenarios, varying the characteristics of the coupling terms (functional form and/or strength) reveals a wide variety of synchronization phenomena. Standard diagnostic tests are performed to verify and classify the different types of synchronizations observed.

Deterministic ratchets, circle maps, and current reversals.

In this work we transform the deterministic dynamics of an overdamped tilting ratchet into a discrete dynamical map by looking stroboscopically at the continuous motion originally ruled by differential equations. We show that, for the simple and widely used case of periodic dichotomous driving forces, the resulting discrete map belongs to the class of circle homeomorphisms. This approach allows us to apply the well-known properties of such maps to derive the necessary and sufficient conditions that the ratchet potential must satisfy in order to have a vanishing current. Furthermore, as a consequence of the above, we show (i) that there is a class of periodic potentials which do not exhibit the rectification phenomenon in spite of their asymmetry and (ii) that current reversals occur in the deterministic case for a large class of ratchet potentials.

Parameter space reduction criteria to search for synchronization domains in coupled discrete systems.

Based on analytical considerations, we introduce criteria that enable us to encapsulate the parameter domains for which chaotic synchronization in linearly coupled map systems may be attained. Our aim is to provide means to readily determine parameter regions which preclude synchronization. This results in a significant reduction of parameter space that one needs to explore. Our findings hold for both identical and quasi-identical (small parameter mismatch) maps subjected to unidirectional and bidirectional coupling. As a testing ground we present numerical calculations for the logistic and cubic maps which validate the predictive capability of our approach. Our main contribution relies on the applicability of one of our criteria to experimental situations. Since in real life it is almost impossible to construct two truly identical systems, the results for quasi-identical maps are of particular relevance.

Translocation properties of primitive molecular machines and their relevance to the structure of the genetic code.

We address the question, related with the origin of the genetic code, of why are there three bases per codon in the translation to protein process. As a follow-up to our previous work (Aldana et al., 1998, Martínez-Mekler et al., 1999a,b), we approach this problem by considering the translocation properties of primitive molecular machines, which capture basic features of ribosomal/messenger RNA interactions, while operating under prebiotic conditions. Our model consists of a short one-dimensional chain of charged particles (rRNA antecedent) interacting with a polymer (mRNA antecedent) via electrostatic forces. The chain is subject to external forcing that causes it to move along the polymer which is fixed in a quasi-one-dimensional geometry. Our numerical and analytic studies of statistical properties of random chain/polymer potentials suggest that, under very general conditions, a dynamics is attained in which the chain moves along the polymer in steps of three monomers. By adjusting the model in order to consider present-day genetic sequences, we show that the above property is enhanced for coding regions. Intergenic sequences display a behavior closer to the random situation. We argue that this dynamical property could be one of the underlying causes for the three-base codon structure of the genetic code

Transport properties of the diluted Lorentz slab.

We study the behavior of a point particle incident on a slab of a randomly diluted triangular array of circular scatterers. Various scattering properties, such as the reflection and transmission probabilities and the scattering time are studied as a function of thickness and dilution. We show that a diffusion model satisfactorily describes the mentioned scattering properties. We also show how some of these quantities can be evaluated exactly and their agreement with numerical experiments. Our results exhibit the dependence of these scattering data on the mean free path. This dependence again shows excellent agreement with the predictions of a Brownian motion model.

Statistical characterization of random electrostatic potentials

In this work we study statistical properties of random electrostatic potentials generated by one dimensional lattices with random charges. We show that the resulting random potentials are correlated Gaussian processes, satisfying the Lindeberg version of the central limit theorem, if certain restrictions are imposed on the individual potentials generated by the particles on the lattice. Since most of the point-particle electrostatic potentials occurring in nature satisfy the Lindeberg condition, the correlation properties of the random potentials are not arbitrary and must comply with the central limit theorem. Based on this theorem we can obtain explicit expressions for these correlations. We thus are able to give a characterization of a broad class of potentials yielding feasible physical scenarios. We illustrate some consequences of our findings by considering dynamical properties of a test particle interacting with the lattice. We show how the long range correlations generate statistical features in these properties, which are best exhibited when considering different length scales.

Primitive molecular machine scenario for the origin of the three base codon composition.

We set up a scenario for the operation of primordial synthesis machines operating in outer space quasi one dimensional channels, where polymers interact with fixed particles. The scheme allows for polymerization, translocation and translation. We will show that under very general conditions the particle/polymer interaction potential has spatial regularities with an average distance of three between neighboring minima. We present a model that exhibits how primitive molecular machines may convert the structural properties of the potential into locomotion regularities. On average, polymer movement takes place by shifts with long time intervals every three displacements. We argue that this feature is generic and lies at the origin of the three base codon composition.

Coupled trivial maps.

The first nontrivial example of coupled map lattices that admits a rigorous analysis in the whole range of the strength of space interactions is considered. This class is generated by one-dimensional maps with a globally attracting superstable periodic trajectory that are coupled by a diffusive nearest-neighbor interaction.