Ugur Tirnakli

Closer look at time averages of the logistic map at the edge of chaos

The probability distribution of sums of iterates of the logistic map at the edge of chaos has been recently shown [U. Tirnakli, Phys. Rev. E 75, 040106(R) (2007)] to be numerically consistent with a q-Gaussian, the distribution which-under appropriate constraints-maximizes the nonadditive entropy Sq, which is the basis of nonextensive statistical mechanics. This analysis was based on a study of the tails of the distribution. We now check the entire distribution, in particular, its central part. This is important in view of a recent q generalization of the central limit theorem, which states that for certain classes of strongly correlated random variables the rescaled sum approaches a q-Gaussian limit distribution. We numerically investigate for the logistic map with a parameter in a small vicinity of the critical point under which conditions there is convergence to a q-Gaussian both in the central region and in the tail region and find a scaling law involving the Feigenbaum constant delta. Our results are consistent with a large number of already available analytical and numerical evidences that the edge of chaos is well described in terms of the entropy Sq and its associated concepts.

Central limit behavior of deterministic dynamical systems

We investigate the probability density of rescaled sums of iterates of deterministic dynamical systems, a problem relevant for many complex physical systems consisting of dependent random variables. A central limit theorem (CLT) is valid only if the dynamical system under consideration is sufficiently mixing. For the fully developed logistic map and a cubic map we analytically calculate the leading-order corrections to the CLT if only a finite number of iterates is added and rescaled, and find excellent agreement with numerical experiments. At the critical point of period doubling accumulation, a CLT is not valid anymore due to strong temporal correlations between the iterates. Nevertheless, we provide numerical evidence that in this case the probability density converges to a q -Gaussian, thus leading to a power-law generalization of the CLT. The above behavior is universal and independent of the order of the maximum of the map considered, i.e., relevant for large classes of critical dynamical systems.

Some bounds upon the nonextensivity parameter using the approximate generalized distribution functions

Abstract In this study, the approximate generalized quantal distribution functions and their applications which appeared in the literature so far, have been summarized. Making use of the generalized Planck radiation law, which has been obtained by us [Physica A 240 (1997) 657], some alternative bounds for the nonextensivity parameter q have been estimated. It has been shown that these results are similar to those obtained by Tsallis et al. [Phys. Rev. E 52 (1995) 1447] and by Plastino et al. [Phys. Lett. A 207 (1995) 42].

Generalized distribution functions and an alternative approach to generalized Planck radiation law

In this study, recently introduced generalized distribution functions are summarized and by using one of these distribution functions, namely generalized Planck distribution, an alternative approach to the generalized Planck law for the blackbody radiation has been tackled. The expression obtained is compared with the expression given by C. Tsallis et al. [Phys. Rev. E 52 (1995) 1447], and it is found that this approximate scheme provides bounds to the exact result, depending on the values of q-index.

Nonadditive entropy and nonextensive statistical mechanics – Some central concepts and recent applications

We briefly review central concepts concerning nonextensive statistical mechanics, based on the nonadditive entropy . Among others, we focus on possible realizations of the q-generalized Central Limit Theorem, including at the edge of chaos of the logistic map, and for quasi-stationary states of many-body long-range-interacting Hamiltonian systems.

Circular-like maps: sensitivity to the initial conditions, multifractality and nonextensivity

Dissipative one-dimensional maps may exhibit special points (e.g., chaos threshold) at which the Lyapunov exponent vanishes. Consistently, the sensitivity to the initial conditions has a power-law time dependence, instead of the usual exponential one. The associated exponent can be identified with 1/(1 Ȣ q), where q characterizes the nonextensivity of a generalized entropic form currently used to extend standard, Boltzmann-Gibbs statistical mechanics in order to cover a variety of anomalous situations. It has been recently proposed (Lyra and Tsallis, Phys. Rev. Lett. 80, 53 (1998)) for such maps the scaling law 1/(1 − q) = 1/αmin − 1/αmax, where αmin and αmax are the extreme values appearing in the multifractal f(α) function. We generalize herein the usual circular map by considering inflexions of arbitrary power z, and verify that the scaling law holds for a large range of z. Since, for this family of maps, the Hausdorff dimension df equals unity for all z in contrast with q which does depend on z, it becomes clear that df plays no major role in the sensitivity to the initial conditions.

The standard map: From Boltzmann-Gibbs statistics to Tsallis statistics

As well known, Boltzmann-Gibbs statistics is the correct way of thermostatistically approaching ergodic systems. On the other hand, nontrivial ergodicity breakdown and strong correlations typically drag the system into out-of-equilibrium states where Boltzmann-Gibbs statistics fails. For a wide class of such systems, it has been shown in recent years that the correct approach is to use Tsallis statistics instead. Here we show how the dynamics of the paradigmatic conservative (area-preserving) stan-dard map exhibits, in an exceptionally clear manner, the crossing from one statistics to the other. Our results unambiguously illustrate the domains of validity of both Boltzmann-Gibbs and Tsallis statistical distributions. Since various important physical systems from particle confinement in magnetic traps to autoionization of molecular Rydberg states, through particle dynamics in accelerators and comet dynamics, can be reduced to the standard map, our results are expected to enlighten and enable an improved interpretation of diverse experimental and observational results.

Analysis of return distributions in the coherent noise model.

Return distributions of the coherent noise model are studied for the system-size-independent case. It is shown that, in this case, these distributions are in the shape of q Gaussians, which are the standard distributions obtained in nonextensive statistical mechanics. Moreover, an exact relation connecting the exponent τ of avalanche size distribution and the q value of appropriate q Gaussian has been obtained as q=(τ+2)/τ . Making use of this relation one can easily determine q parameter values of the appropriate q Gaussians a priori from one of the well-known exponents of the system. Since the coherent noise model has the advantage of producing different τ values by varying a model parameter σ , clear numerical evidences on the validity of the proposed relation have been achieved for various cases. Finally, the effect of the system size has also been analyzed and an analytical expression has been proposed, which is corroborated by the numerical results.

Generalization of the Kolmogorov–Sinai entropy: logistic-like and generalized cosine maps at the chaos threshold

Abstract We numerically calculate, at the edge of chaos, the time evolution of the nonextensive entropic form S q ≡[1−∑ i =1 W p i q ]/[ q −1] (with S 1 =−∑ i =1 W p i ln p i ) for two families of one-dimensional dissipative maps, namely a logistic-like and a generalized cosine with arbitrary inflexion z at their maximum. At t =0 we choose N initial conditions inside one of the W small windows in which the accessible phase space is partitioned; to neutralize large fluctuations we conveniently average over a large amount of initial windows. We verify that one and only one value q ∗ exists such that the lim t →∞ lim W →∞ lim N →∞ S q ( t )/ t is finite, thus generalizing the ( ensemble version of the ) Kolmogorov–Sinai entropy (which corresponds to q ∗ =1 in the present formalism). This special, z -dependent, value q ∗ numerically coincides, for both families of maps and all z , with the one previously found through two other independent procedures (sensitivity to the initial conditions and multifractal f ( α ) function).

Two-dimensional maps at the edge of chaos: numerical results for the Henon map.

The mixing properties (or sensitivity to initial conditions) of the two-dimensional Henon map have been explored numerically at the edge of chaos. Three independent methods, which have been developed and used so far for one-dimensional maps, have been used to accomplish this task. These methods are (i) the measure of the divergence of initially nearby orbits, (ii) analysis of the multifractal spectrum, and (iii) computation of nonextensive entropy increase rates. The results obtained closely agree with those of the one-dimensional cases and constitute a verification of this scenario in two-dimensional maps. This obviously makes the idea of weak chaos even more robust.