R. Salgado-García

Exact Scaling in the Expansion-Modification System

This work is devoted to the study of the scaling, and the consequent power-law behavior, of the correlation function in a mutation-replication model known as the expansion-modification system. The latter is a biology inspired random substitution model for the genome evolution, which is defined on a binary alphabet and depends on a parameter interpreted as a mutation probability. We prove that the time-evolution of this system is such that any initial measure converges towards a unique stationary one exhibiting decay of correlations not slower than a power-law. We then prove, for a significant range of mutation probabilities, that the decay of correlations indeed follows a power-law with scaling exponent smoothly depending on the mutation probability. Finally we put forward an argument which allows us to give a closed expression for the corresponding scaling exponent for all the values of the mutation probability. Such a scaling exponent turns out to be a piecewise smooth function of the parameter.

Normal-to-anomalous diffusion transition in disordered correlated potentials: from the central limit theorem to stable laws.

We study the diffusion of an ensemble of overdamped particles sliding over a tilted random potential (produced by the interaction of a particle with a random polymer) with long-range correlations. We found that the diffusion properties of such a system are closely related to the correlation function of the corresponding potential. We model the substrate as a symbolic trajectory of a shift space which enables us to obtain a general formula for the diffusion coefficient when normal diffusion occurs. The total time that the particle takes to travel through n monomers can be seen as an ergodic sum to which we can apply the central limit theorem. The latter can be implemented if the correlations decay fast enough in order for the central limit theorem to be valid. On the other hand, we presume that when the central limit theorem breaks down the system give rise to anomalous diffusion. We give two examples exhibiting a transition from normal to anomalous diffusion due to this mechanism. We also give analytical expressions for the diffusion exponents in both cases by assuming convergence to a stable law. Finally we test our predictions by means of numerical simulations.

Markov approximations of Gibbs measures for long-range interactions on 1D lattices

We study one-dimensional lattice systems with pair-wise interactions of infinite range. We show projective convergence of Markov measures to the unique equilibrium state. For this purpose we impose a slightly stronger condition than summability of variations on the regularity of the interaction. With our condition we are able to explicitly obtain stretched exponential bounds for the rate of mixing of the equilibrium state. Finally we show convergence for the entropy of the Markov measures to that of the equilibrium state via the convergence of their topological pressure (free energy).

Unbiased diffusion of Brownian particles on disordered correlated potentials

In this work we study the diffusion of non-interacting overdamped particles, moving on unbiased disordered correlated potentials, subjected to Gaussian white noise. We obtain an exact expression for the diffusion coefficient which allows us to prove that the unbiased diffusion of overdamped particles on a random polymer does not depend on the correlations of the disordered potentials. This universal behavior of the unbiased diffusivity is a direct consequence of the validity of the Einstein relation and the decay of correlations of the random polymer. We test the independence on correlations of the diffusion coefficient for correlated polymers produced by two different stochastic processes, a one-step Markov chain and the expansion-modification system. Within the accuracy of our simulations, we found that the numerically obtained diffusion coefficient for these systems agree with the analytically calculated ones, confirming our predictions.

Interface Magnetopolaron in III–V Nitride Single Heterostructures

Polaronic corrections due to the electron-interface-phonon interaction in AIN/GaN single heterostructures are calculated within the improved Wigner-Brillouin perturbation theory using a dielectric continuum approach for the optical phonons, and a simplified triangular quantum well potential model for the conduction band electrons. The renormalization of the GaN electron effective mass is particularly investigated in order to be compared with cyclotron resonance measurements.

Normal and anomalous diffusion of Brownian particles on disordered potentials

In this work we study the transition from normal to anomalous diffusion of Brownian particles on disordered potentials. The potential model consists of a series of “potential hills” (defined on a unit cell of constant length) whose heights are chosen randomly from a given distribution. We calculate the exact expression for the diffusion coefficient in the case of uncorrelated potentials for arbitrary distributions. We show that when the potential heights have a Gaussian distribution (with zero mean and a finite variance) the diffusion of the particles is always normal. In contrast, when the distribution of the potential heights is exponentially distributed the diffusion coefficient vanishes when the system is placed below a critical temperature. We calculate analytically the diffusion exponent for the anomalous (subdiffusive) phase by using the so-called “random trap model”. Our predictions are tested by means of Langevin simulations obtaining good agreement within the accuracy of our numerical calculations.

Degenerated ground-states in a spin chain with pair interactions: a characterization by symbolic dynamics

In this paper we study a class of one-dimensional spin chain having a highly degenerated set of ground-state configurations. The model consists of spin chain having infinite-range pair interactions with a given structure. We show that the set of ground-state configurations of such a model can be fully characterized by means of symbolic dynamics. Particularly we found that the set ground-state configurations define what in symbolic dynamics is called sofic shift space. Finally we prove that this system has a non-vanishing residual entropy (the topological entropy of the shift space), which can be exactly calculated.

Symbolic complexity for nucleotide sequences: a sign of the genome structure

We introduce a method for estimating the complexity function (which counts the number of observable words of a given length) of a finite symbolic sequence, which we use to estimate the complexity function of coding DNA sequences for several species of the Hominidae family. In all cases, the obtained symbolic complexities show the same characteristic behavior: exponential growth for small word lengths, followed by linear growth for larger word lengths. The symbolic complexities of the species we consider exhibit a systematic trend in correspondence with the phylogenetic tree. Using our method, we estimate the complexity function of sequences obtained by some known evolution models, and in some cases we observe the characteristic exponential-linear growth of the Hominidae coding DNA complexity. Analysis of the symbolic complexity of sequences obtained from a specific evolution model points to the following conclusion: linear growth arises from the random duplication of large segments during the evolution of the genome, while the decrease in the overall complexity from one species to another is due to a difference in the speed of accumulation of point mutations.

Replication ratchets: polymer transport enhanced by complementarity

We show that a rudimentary model of two complementary polymer chains confined to a quasi-one-dimensional geometry, each one interacting separately with a fixed particle via a ratchet potential, can be reformulated in terms of one polymer subject to a ratchet potential produced by its interaction with a dumbbell (or effective dimer). In the over-dampened regime, we look at transport properties when the system is under zero mean periodic external forcing. For a piece-wise linear asymmetric potential we exhibit the general mechanism by which the average mean velocity of a single polymer is enhanced by complementarity. We show that same behavior holds for a continuous ratchet potential. We relate this modelling to primitive molecular replication machines operating in outer space origin of life scenarios and argue that complementarity is not only essential to transcription but may also contribute to the efficiency of the dynamics involved in the replication process.