Browsing by Author "Tsallis, Constantino"

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Central Limit Behavior at the Edge of Chaos in the Z-Logistic Map
(American Physical Society, 2025) Saberi, Abbas Ali; Tirnakli, Ugur; Tsallis, Constantino
We focus on the Feigenbaum-Coullet-Tresser point of the dissipative one-dimensional z-logistic map x(t+1) = 1-a|x(t )|(z) (z 1). We show that sums of iterates converge to q-Gaussian distributions P-q(y) = P-q(0) exp(q)(-beta(q)y(2)) = P-q(0 )[1 + (q-1)beta(q) y(2)](1/(1-q))(q >= 1; beta(q) > 0), which optimize the nonadditive entropic functional Sq under simple constraints. We propose and justify heuristically a closed-form prediction for the entropic index, q(z) = 1 + 2/(z + 1), and validate it numerically via data collapse for typical z values. The formula captures how the limiting law depends on the nonlinearity order and implies finite variance for z > 2 and divergent variance for 1 <= z <= 2. These results extend edge-of-chaos central limit behavior beyond the standard (z = 2) case and provide a simple predictive law for unimodal maps with varying maximum order.
Citation - WoS: 3
Citation - Scopus: 3
First-Principle Validation of Fourier's Law in D=1, 2, 3 Classical Systems
(Elsevier, 2023) Tsallis, Constantino; Lima, Henrique Santos; Tırnaklı, Uğur; Eroğlu, Deniz
We numerically study the thermal transport in the classical inertial nearest-neighbor XY ferromagnet in d = 1, 2, 3, the total number of sites being given by N = Ld, where L is the linear size of the system. For the thermal conductance sigma, we obtain sigma(T, L)L delta(d)= A(d) e-B(d) [L gamma (d)T ]eta(d) (with ez q(d) q equivalent to [1+(1-q)z]1/(1-q); ez1 = ez; A(d) > 0; B(d) > 0; q(d) > 1; eta(d) > 2; delta >= 0; gamma(d) > 0), for all values of L gamma(d)T for d = 1, 2, 3. In the L -> infinity limit, we have sigma proportional to 1/L rho sigma(d) with rho sigma(d) = delta(d)+gamma(d)eta(d)/[q(d)-1]. The material conductivity is given by kappa = sigma Ld proportional to 1/L rho kappa(d) (L -> infinity) with rho kappa(d) = rho sigma(d) - d. Our numerical results are consistent with 'conspiratory' d-dependences of (q, eta, delta, gamma), which comply with normal thermal conductivity (Fourier law) for all dimensions.(c) 2023 Published by Elsevier B.V.
Fourier's Law Breakdown for the Planar-Rotor Chain With Long-Range Interactions
(Elsevier, 2026) Lima, Henrique Santos; Tsallis, Constantino; Eroglu, Deniz; Tirnakli, Ugur
Fourier's law, which linearly relates heat flux to the negative gradient of temperature, is a fundamental principle in thermal physics and widely applied across materials science and engineering. However, its validity in low-dimensional systems with long-range interactions remains only partially understood. We investigate here the thermal transport along a onedimensional chain of classical planar rotators with algebraically decaying interactions 1/ with distance ( >= 0), known as the inertial a-XY model. Using nonequilibrium simulations with thermal reservoirs at the boundaries, we numerically study the thermal conductance as a function of system sizea, temperature , and . We find that the results obey a universal scaling law characterized by a stretched-exponential function with -dependent parameters. Notably, a threshold at approximate to 2 separates two regimes: for >= , Fourier's law holds with size-independent conductivity = , while for < , anomalous transport is observed, corroborating (with higher precision) the results reported in Phys.Rev.E94,042117(2016). These findings provide a quantitative framework for understanding the breakdown of Fourier's law in systems with long-range interactions. The simulation is carried out by assuming the equations of motion, which include Langevin heat baths applied to the first and last particles, and are integrated using the Velocity Verlet algorithm. The conductance is calculated from the connection between Lagrangian heat flux and heat equation for typical values of (, , ). For large , the results can be collapsed into an universal -stretched exponential form, namely proportional to -() , where = [1 + (1-)]1/(1-). The parameters (, , ,) are -dependent, and is the index of the -stretched exponential. This form is achievable due to the ratio /( - 1) being almost constant with respect to the lattice size. These findings provide significant insights into heat conduction mechanisms in systems with long-range interactions.
Citation - WoS: 4
Citation - Scopus: 4
Generalization of the Gauss Map: a Jump Into Chaos With Universal Features
(Amer Physical Soc, 2024) Beck, Christian; Tirnakli, Ugur; Tsallis, Constantino
The Gauss map (or continued fraction map) is an important dissipative one-dimensional discrete-time dynamical system that exhibits chaotic behavior, and it generates a symbolic dynamics consisting of infinitely many different symbols. Here we introduce a generalization of the Gauss map, which is given by xt+1 = 1 where alpha 0 is a parameter and xt is an element of [0, 1] (t = 0, 1, 2, 3, ...). The symbol [... ] denotes the integer part. This map reduces to the ordinary Gauss map for alpha = 1. The system exhibits a sudden "jump into chaos" at the critical parameter value alpha = alpha c equivalent to 0.241485141808811 ... which we analyze in detail in this paper. Several analytical and numerical results are established for this new map as a function of the parameter alpha. In particular, we show that, at the critical point, the invariant density approaches a q-Gaussian with q = 2 (i.e., the Cauchy distribution), which becomes infinitely narrow as alpha -* alpha c+. Moreover, in the chaotic region for large values of the parameter alpha we analytically derive approximate formulas for the invariant density, by solving the corresponding Perron-Frobenius equation. For alpha -* infinity the uniform density is approached. We provide arguments that some features of this transition scenario are universal and are relevant for other, more general systems as well.
Citation - WoS: 6
Citation - Scopus: 8
Nonextensive Footprints in Dissipative and Conservative Dynamical Systems
(Mdpi, 2023) Rodriguez, Antonio; Pluchino, Alessandro; Tirnakli, Ugur; Rapisarda, Andrea; Tsallis, Constantino
Despite its centennial successes in describing physical systems at thermal equilibrium, Boltzmann-Gibbs (BG) statistical mechanics have exhibited, in the last several decades, several flaws in addressing out-of-equilibrium dynamics of many nonlinear complex systems. In such circumstances, it has been shown that an appropriate generalization of the BG theory, known as nonextensive statistical mechanics and based on nonadditive entropies, is able to satisfactorily handle wide classes of anomalous emerging features and violations of standard equilibrium prescriptions, such as ergodicity, mixing, breakdown of the symmetry of homogeneous occupancy of phase space, and related features. In the present study, we review various important results of nonextensive statistical mechanics for dissipative and conservative dynamical systems. In particular, we discuss applications to both discrete-time systems with a few degrees of freedom and continuous-time ones with many degrees of freedom, as well as to asymptotically scale-free networks and systems with diverse dimensionalities and ranges of interactions, of either classical or quantum nature.