arXiv subjects

A. Robledo

Publications and source records attributed to A. Robledo.

24 records · Page 2Linked to original sources

arXiv2004

Critical fluctuations, intermittent dynamics and Tsallis statistics

It is pointed out that the dynamics of the order parameter at a thermal critical point obeys the precepts of the nonextensive Tsallis statistics. We arrive at this conclusion by putting together two well-defined statistical-mechanical developments. The first is that critical fluctuations are correctly described by the dynamics of an intermittent nonlinear map. The second is that intermittency in the neighborhood of a tangent bifurcation in such map rigorously obeys nonextensive statistics. We comment on the implications of this result. Key words: critical fluctuations, intermittency, nonextensive statistics, anomalous stationary states

cond-mat.stat-mech↗

arXiv2004

Multifractality and nonextensivity at the edge of chaos of unimodal maps

We examine both the dynamical and the multifractal properties at the chaos threshold of logistic maps with general nonlinearity $z>1$. First we determine analytically the sensitivity to initial conditions $ξ_{t}$. Then we consider a renormalization group (RG) operation on the partition function $Z$ of the multifractal attractor that eliminates one half of the multifractal points each time it is applied. Invariance of $Z$ fixes a length-scale transformation factor $2^{-η}$ in terms of the generalized dimensions $D_β$. There exists a gap $Δη$ in the values of $η$ equal to $λ_{q}=1/(1-q)=D_{\infty}^{-1}-D_{-\infty}^{-1}$ where $λ_{q}$ is the $q$-generalized Lyapunov exponent and $q$ is the nonextensive entropic index. We provide an interpretation for this relationship - previously derived by Lyra and Tsallis - between dynamical and geometrical properties. Key Words: Edge of chaos, multifractal attractor, nonextensivity

cond-mat.stat-mech↗

arXiv2003

Criticality in non-linear one-dimensional maps: RG universal map and non-extensive entropy

We consider the period-doubling and intermittency transitions in iterated nonlinear one-dimensional maps to corroborate unambiguously the validity of Tsallis' non-extensive statistics at these critical points. We study the map $x_{n+1}=x_{n}+u| x_{n}| ^{z}$, $z>1$, as it describes generically the neighborhood of all of these transitions. The exact renormalization group (RG) fixed-point map and perturbation static expressions match the corresponding expressions for the dynamics of iterates. The time evolution is universal in the RG sense and the non-extensive entropy $S_{Q}$ associated to the fixed-point map is maximum with respect to that of the other maps in its basin of attraction. The degree of non-extensivity - the index $Q$ in $S_{Q}$ - and the degree of nonlinearity $z$ are equivalent and the generalized Lyapunov exponent $λ_{q}$, $q=2-Q^{-1}$, is the leading map expansion coefficient $u$. The corresponding deterministic diffusion problem is similarly interpreted. We discuss our results.

cond-mat.stat-mech↗

arXiv2003

Universal glassy dynamics at noise-perturbed onset of chaos. A route to ergodicity breakdown

The dynamics of iterates at the transition to chaos in one-dimensional unimodal maps is shown to exhibit the characteristic elements of the glass transition, e.g. two-step relaxation and aging. The properties of the bifurcation gap induced by external noise, including a relationship between relaxation time and entropy, are seen to be comparable to those of a supercooled liquid above a glass transition temperature. Universal time evolution obtained from the Feigenbaum RG transformation is expressed analytically via $q$-exponentials, and interpreted in terms of nonextensive statistics.

cond-mat.stat-mech↗

arXiv2002

Universal renormalization-group dynamics at the onset of chaos in logistic maps and nonextensive statistical mechanics

We uncover the dynamics at the chaos threshold $μ_{\infty}$ of the logistic map and find it consists of trajectories made of intertwined power laws that reproduce the entire period-doubling cascade that occurs for $μ<μ_{\infty}$. We corroborate this structure analytically via the Feigenbaum renormalization group (RG) transformation and find that the sensitivity to initial conditions has precisely the form of a $q$-exponential, of which we determine the $q$-index and the $q$-generalized Lyapunov coefficient $λ_{q}$. Our results are an unequivocal validation of the applicability of the non-extensive generalization of Boltzmann-Gibbs (BG) statistical mechanics to critical points of nonlinear maps.

cond-mat.stat-mech↗

arXiv2002

Sensitivity to initial conditions at bifurcations in one-dimensional nonlinear maps: rigorous nonextensive solutions

Using the Feigenbaum renormalization group (RG) transformation we work out exactly the dynamics and the sensitivity to initial conditions for unimodal maps of nonlinearity $ζ>1$ at both their pitchfork and tangent bifurcations. These functions have the form of $q$-exponentials as proposed in Tsallis' generalization of statistical mechanics. We determine the $q$-indices that characterize these universality classes and perform for the first time the calculation of the $q$-generalized Lyapunov coefficient $λ_{q} $. The pitchfork and the left-hand side of the tangent bifurcations display weak insensitivity to initial conditions, while the right-hand side of the tangent bifurcations presents a `super-strong' (faster than exponential) sensitivity to initial conditions. We corroborate our analytical results with {\em a priori} numerical calculations.

cond-mat.stat-mech↗