Description: This article discusses a dynamical model for DNA sequences. Abstract: We address the problem of DNA sequences, developing a "dynamical" method based on the assumption that the statistical properties of DNA paths are determined by the joint action of two processes, one deterministic with long-range correlations and the other random and δ-function correlated. The generator of the deterministic evolution is a nonlinear map belonging to a class of maps recently tailored to mimic the processes of weak chaos responsible for the birth of anomalous diffusion. It is assumed that the deterministic process corresponds to unknown biological rules that determine the DNA path, whereas the noise mimics the influence of an infinite-dimensional environment on the biological process under study. We prove that the resulting diffusion process, if the effect of the random process is determined by the joint action of the deterministic and the random process, the correlation effects of the "deterministic dynamics" are canceled on the short-range scale, but show up in the long-range one. We denote their prescription to generate statistical sequences as the copying mistake map (CMM). We carry out their analysis of several DNA sequences and their CMM realizations with a variety of techniques and the authors ...

Description: In this article, the authors study the interplay between a deterministic process of weak chaos, responsible for the anomalous diffusion of a variable x, and a white noise of intensity ≡. The deterministic process of anomalous diffusion results from the correlated fluctuations of a statistical variable ξ between two distinct values +1 and -1, each of them characterized by the same waiting time distribution ψ(t), given by ψ(t)≃ t(-μ) with 2 < μ < 3, in the long-time limit. The authors prove that under the influence of a weak white noise of intensity ≡, the process of anomalous diffusion becomes normal at a time t(c) given by t(c) ~ 1/≡(β)(μ). Here β(μ) is a function of μ which depends on the dynamical generator of the waiting-time distribution ψ(t). The authors derive an explicit expression for β(μ) in the case of two dynamical systems, a one-dimensional superdiffusive map and the standard map in the accelerating state. The theoretical prediction is supported by numerical calculations.

Description: This article discusses the linear response of Hamiltonian chaotic systems as a function of the number of degrees of freedom. Abstract: Using numerical simulations we show that the response to weak perturbations of a variable of Hamiltonian chaotic systems depend on the number of degrees of freedom: When this is small (≈2) the response is not linear, in agreement with the well known objections to the Kubo linear response theory, while, for a larger number of degrees of freedom, the response becomes linear. This is due to the fact that increasing the number of degrees of freedom the shape of the distribution function, projected onto the subspace of the variable of interest, becomes fairly "regular."

Description: This article discusses site correlation, anomalous diffusion, and enhancement of localization length. Herein the authors study the effects on Anderson localizations of correlations in the energy distribution of the sites of a tight-binding Hamiltonian. The lattice correlations are introduced are introduced by means of classical maps generating anomalous diffusion, that have recently been found to account for the correlated disorder of "biological" lattices. The authors show that the enhancement of localization length takes place on a much wider band of energies than in the case of the random-dimer model if the random walk on the site energies of the tight-binding Hamiltonian is determined by the joint action of short- and long-range correlations.

Description: This article discusses fractional Brownian motion as a nonstationary process. Abstract: The long-range correlations in DNA sequences are currently interpreted as an example of stationary fractional Brownian motion (FBM). First the authors show that the dynamics of a dichotomous stationary process with long-range correlations such as that used to model DNA sequences should correspond to Lévy statistics and not to FBM. To explain why, in spite of this, the statistical analysis of the data seems to be compatible with FBM, the authors notice that an initial Gaussian condition, generated by a process foreign to the mechanism establishing the long-range correlations and consequently implying a departure from the stationary condition is maintained approximately unchanged for very long times. This is so because due to the nature itself of the long-range correlation process, it takes virtually an infinite time for the system to reach the genuine stationary state. Then the authors discuss a possible generator of initial Gaussian conditions, based on a folding mechanism of the nucleic acid in the cell nucleus. The model adopted is compatible with the known biological and physical constraints, namely, it is shown to be consistent with the information of current biological literature on folding as well ...