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Öğe Aging in coherent noise models and natural time(2004) Tirnakli U.; Abe S.Event correlation between aftershocks in the coherent noise model is studied by making use of natural time, which has recently been introduced in complex time-series analysis. It is found that the aging phenomenon and the associated scaling property discovered in the observed seismic data are well reproduced by the model. It is also found that the scaling function is given by the [Formula presented]-exponential function appearing in nonextensive statistical mechanics, showing power-law decay of event correlation in natural time. © 2004 The American Physical Society.Öğe Analysis of self-organized criticality in Ehrenfest;s dog-flea model(2009) Bakar B.; Tirnakli U.The self-organized criticality in Ehrenfest's historical dog-flea model is analyzed by simulating the underlying stochastic process. The fluctuations around the thermal equilibrium in the model are treated as avalanches. We show that the distributions for the fluctuation length differences at subsequent time steps are in the shape of a q -Gaussian (the distribution which is obtained naturally in the context of nonextensive statistical mechanics) if one avoids the finite-size effects by increasing the system size. We provide clear numerical evidence that the relation between the exponent T, of avalanche size distribution obtained by maximum-likelihood estimation and the q value of appropriate q -Gaussian obeys the analytical result recently introduced by Caruso [Phys. Rev. E 75, 055101(R) (2007)]. This allows us to determine the value of q -parameter a priori from one of the well-known exponents of such dynamical systems. © 2009 The American Physical Society.Öğe Chaos edges of z -logistic maps: Connection between the relaxation and sensitivity entropic indices(2006) Tirnakli U.; Tsallis C.Chaos thresholds of the z -logistic maps xt+1 =1- xt z (z>1; t=0,1,2,...) are numerically analyzed at accumulation points of cycles 2, 3, and 5 (three different cycles 5). We verify that the nonextensive q -generalization of a Pesin-like identity is preserved through averaging over the entire phase space. More precisely, we computationally verify limt› S q sen av (t)t= limt› ln q sen av ?(t)t ? q sen av av, where the entropy Sq (1- i piq) (q-1) (S1 =- i pi ln pi), the sensitivity to the initial conditions ? lim?x(0)›0 ?x(t)?x(0), and lnq x (x1-q -1)(1-q) (ln1 x=ln x). The entropic index q sen av <1, and the coefficient ? q sen av av >0 depend on both z and the cycle. We also study the relaxation that occurs if we start with an ensemble of initial conditions homogeneously occupying the entire phase space. The associated Lebesgue measure asymptotically decreases as 1t1 (qrel -1) (qrel >1). These results (i) illustrate the connection (conjectured by one of us) between sensitivity and relaxation entropic indices, namely, qrel -1 An (1- q sen av) ?n, where the positive numbers (An, ?n) depend on the cycle; (ii) exhibit an unexpected scaling, namely, q sen av (cycle n)= Bn q sen av (cycle 2)+ µ n. © 2006 The American Physical Society.Öğe Convergence dynamics of 2-dimensional isotropic and anisotropic Bak-Sneppen models(2008) Bakar B.; Tirnakli U.The conventional Hamming distance measurement captures only short-time dynamics of the displacement between uncorrelated random configurations. The minimum difference technique introduced by Tirnakli and Lyra [U. Tirnakli, M.L. Lyra. Int. J. Mod. Phys. C 14 (2003) 805] is used to study short-time and long-time dynamics of the two distinct random configurations of isotropic and anisotropic Bak-Sneppen models on a square lattice. Similar to a 1-dimensional case, the time evolution of the displacement is intermittent. The scaling behavior of the jump activity rate and waiting time distribution reveal the absence of typical spatial-temporal scales in the mechanism of displacement jumps used to quantify convergence dynamics. © 2008 Elsevier B.V. All rights reserved.Öğe Critical dynamics of anisotropic Bak-Sneppen model(2004) Tirnakli U.; Lyra M.L.A new damage spreading algorithm, which was introduced very recently in (Int. J. Mod. Phys. C 14 (2003) 85) has been applied to anisotropic Bak-Sneppen model of biological evolution. Since this new algorithm is able to capture both the short-time and long-time dynamics of extended systems which exhibits self-organized criticality, this analysis is expected to shed further light to the recent claim that the dynamics of such systems is similar to the one observed at the usual critical point of continuous phase-transitions and at the chaos threshold of low-dimensional dissipative maps. © 2004 Elsevier B.V. All rights reserved.Öğe Earthquakes, model systems and connections to q-statistics(2012) Celikoglu A.; Tirnakli U.In this work, we make an attempt to review some of the recent studies on earthquakes using either real catalogs or synthetic data coming from some model systems. A common feature of all these works is the use of q-statistics as a tool. © 2012 Versita Warsaw and Springer-Verlag Wien.Öğe Generalized Pesin-Like identity and scaling relations at the chaos threshold of the Rössler system(MDPI AG, 2018) Cetin K.; Afsar O.; Tirnakli U.In this paper, using the Poincaré section of the flow we numerically verify a generalization of a Pesin-like identity at the chaos threshold of the Rössler system, which is one of the most popular three-dimensional continuous systems. As Poincaré section points of the flow show similar behavior to that of the logistic map, for the Rössler system we also investigate the relationships with respect to important properties of nonlinear dynamics, such as correlation length, fractal dimension, and the Lyapunov exponent in the vicinity of the chaos threshold. © 2018 by the authors.Öğe Generalized Tsallis thermostatistics of magnetic systems(1997) Büyükkiliç F.; Tirnakli U.; Demirhan D.Boltzmann-Gibbs statistics fails to study the systems having the conditions (i) the spatial range of the microscopic interactions are long-ranged, (ii) the time range of the microscopic memory is long-ranged and (iii) the system evolves in a (multi)fractal space-time. These kind of systems are said to be nonextensive and a nonextensive formalism of statistics must be needed for them. Recently a generalized thermostatistics is proposed by C. Tsallis to handle the nonextensive systems and up to now, not only the generalization of some of the conventional concepts have been investigated but the formalism has also been properous in some of the physical applications. In this study, our effort is to introduce Tsallis thermostatistics in some details and to give a brief review of the magnetic systems which have been studied in the frame of this formalism.Öğe Mixing and relaxation dynamics of the Hénon map at the edge of chaos(Elsevier, 2004) Borges E.P.; Tirnakli U.The mixing properties (or sensitivity to initial conditions) and relaxation dynamics of the Hénon map, together with the connection between these concepts, have been explored numerically at the edge of chaos. It is found that the results are consistent with those coming from one-dimensional dissipative maps. This constitutes the first verification of the scenario in two-dimensional cases and obviously reinforces the idea of weak mixing. © 2004 Elsevier B.V. All rights reserved.Öğe A new formalism for nonextensive physical systems: Tsallis thermostatistics(1999) Tirnakli U.; Büyükkiliç F.; Demirhan D.Although Boltzmann-Gibbs (BG) statistics provides a suitable tool which enables us to handle a large number of physical systems satisfactorily, it has some basic restrictions : (i) the range of the microscopic interactions must be small compared to the linear size of the macroscopic systems (short-range interactions) , (ii) the time range of the microscopic memory must be small compared to the observation time (Marcovian processes) and (iii) the system must evolve in an Euclidean-like space-time. In the case of a breakdown in one and/or the others of these restrictions, BG statistics fails. More precisely, the situation could be classified in a general manner as follows: (i) For an Euclidean-like space-time, if the forces and/or the memory are long-ranged, as far as we are interested in an equilibrium state, the BG statistics is weakly violated, therefore BG formalism can be used. On the other hand, whenever a meta-equilibrium state is considered, the BG statistics is strongly violated, hence another formalism must be needed, (ii) For a (multi)fractal space, BG formalism is strongly violated again and a new formalism is needed. The way out from these problems seems to be Nonextensive Statistical Thermodynamics which must be a generalization of the BG statistics in a manner that allows a correct description of the nonextensive physical systems as well. Recently a nonextensive thermostatistics has been proposed by C.Tsallis to handle the nonextensive physical systems and up to now, besides the generalization of some of the conventional concepts, the formalism has been prosperous in some of the physical applications. In this study, our effort is to introduce Tsallis thermostatistics in some details and to emphasize its achievements on physical systems by noting the recent developments on this line. © Tübi·tak.Öğe Nonadditive entropy and nonextensive statistical mechanics - Some central concepts and recent applications(Institute of Physics Publishing, 2010) Tsallis C.; Tirnakli U.We briefly review central concepts concerning nonextensive statistical mechanics, based on the nonadditive entropy Sq = k 1-?ip iq/q-1(q ? R; S1 = k ? ipiIn piAmong 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-rangeinteracting Hamiltonian systems. © 2010 IOP Publishing Ltd.Öğe Nonextensive statistical approach to non-coding human DNA(2008) Oikonomou Th.; Provata A.; Tirnakli U.We use q-exponential distributions, which maximize the nonextensive entropy Sq (defined as Sq ? (1 - ?i piq) / (q - 1)), to study the size distributions of non-coding DNA (including introns and intergenic regions) in all human chromosomes. We show that the value of the exponent q describing the non-coding size distributions is similar for all chromosomes and varies between 2 ? q ? 2.3 with the exception of chromosomes X and Y. © 2007 Elsevier Ltd. All rights reserved.Öğe On the relevance of q-distribution functions: The return time distribution of restricted random walker(Institute of Physics Publishing, 2015) Zand J.; Tirnakli U.; Jensen H.J.There exists a large literature on the application of q-statistics to the out-of-equilibrium non-ergodic systems in which some degree of strong correlations exists. Here we study the distribution of first return times to zero, PR (0, t), of a random walk on the set of integers {0, 1, 2,..., L} with a position dependent transition probability given by . We find that for all values of PR(0, t) can be fitted by q-exponentials, but only for a = 1 is PR (0, t) given exactly by a q-exponential in the limit . This is a remarkable result since the exact analytical solution of the corresponding continuum model represents PR (0, t) as a sum of Bessel functions with a smooth dependence on a from which we are unable to identify a = 1 as of special significance. However, from the high precision numerical iteration of the discrete master equation, we do verify that only for a = 1 is PR(0, t) exactly a q-exponential and that a tiny departure from this parameter value makes the distribution deviate from q-exponential. Further research is certainly required to identify the reason for this result and also the applicability of q-statistics and its domain. � 2015 IOP Publishing Ltd.Öğe Self-organization in nonadditive systems with external noise(World Scientific Publishing Co. Pte Ltd, 2009) Bagci G.B.; Tirnakli U.A nonadditive generalization of Klimontovich's S-theorem [Bagci, 2008] has recently been obtained by employing Tsallis entropy. This general version allows one to study physical systems whose stationary distributions are of the inverse power law in contrast to the original S-theorem, which only allows exponential stationary distributions. The nonadditive S-theorem has been applied to the modified Van der Pol oscillator with inverse power law stationary distribution. By using nonadditive S-theorem, it is shown that the entropy decreases as the system is driven out of equilibrium, indicating self-organization in the system. The allowed values of the nonadditivity index q are found to be confined to the regime (0.5,1]. © 2009 World Scientific Publishing Company.Öğe Short-time dynamics of isotropic and anisotropic Bak-Sneppen model: Extensive simulation results(2004) Tirnakli U.; Lyra M.L.In this work, the short-time dynamics of the isotropic and anisotropic versions of the Bak-Sneppen (BS) model has been investigated using the standard damage spreading technique. Since the system sizes attained in our simulations are larger than the ones employed in previous studies, our results for the dynamic scaling exponents are expected to be more accurate than the results of the existing literature. The obtained scaling exponents of both versions of the BS model are found to be greater than the ones given in previous works. These findings are in agreement with the recent claim of Cafiero et al. (Eur. Phys. J. B7 (1999) 505). Moreover, it is found that the short-time dynamics of the anisotropic model is only slightly affected by finite-size effects and the reported estimate of ??0.53 can be considered as a good estimate of the true exponent in the thermodynamic limit. © 2004 Elsevier B.V. All rights reserved.Öğe Two-dimensional dissipative maps at chaos threshold: Sensitivity to initial conditions and relaxation dynamics(2004) Borges E.P.; Tirnakli U.The sensitivity to initial conditions and relaxation dynamics of two-dimensional maps are analyzed at the edge of chaos, along the lines of nonextensive statistical mechanics. We verify the dual nature of the entropic index for the Henon map, one (qsen<1) related to its sensitivity to initial condition properties, and the other, graining-dependent (q rel(W) >1), related to its relaxation dynamics towards its stationary state attractor. We also corroborate a scaling law between these two indices, previously found for z-logistic maps. Finally, we perform a preliminary analysis of a linearized version of the Henon map (the smoothed Lozi map). We find that the sensitivity properties of all these z-logistic, Henon and Lozi maps are the same, qsen=0.2445.... © 2004 Elsevier B.V. All rights reserved.
