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 qindices that characterize these universality classes and perform for the first time the calculation of the q-generalized Lyapunov coefficient λq. 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 a priori numerical calculations. *
We show that the dynamical and entropic properties at the chaos threshold of the logistic map are naturally linked through the nonextensive expressions for the sensitivity to initial conditions and for the entropy. We corroborate analytically, with the use of the Feigenbaum renormalization group transformation, the equality between the generalized Lyapunov coefficient lambda(q) and the rate of entropy production, K(q), given by the nonextensive statistical mechanics. Our results advocate the validity of the q -generalized Pesin identity at critical points of one-dimensional nonlinear dissipative maps.
We uncover the dynamics at the chaos threshold µ∞ 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∞. 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 λq. 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.Critical points of nonlinear maps offer a suitable playground for testing the validity of the non-extensive generalization of the Botzmann-Gibbs (BG) statistical mechanics proposed by Tsallis over a decade ago [1,2]. Here we describe universal properties related to the dynamics of iterates at the onset of chaos in unimodal maps [3], that provide a literal confirmation of the generalized nonextensive theory. To this end we employ the celebrated one-dimensional logistic map, f µ (x) = 1 − µ |x| 2 , −1 ≤ x ≤ 1, and the properties of its renormalization group (RG) fixed point, to present evidence of previously unexposed scaling properties at the onset of chaos µ = µ ∞ . At this state, the most prominent of the map critical points, the trajectories of the iterates exhibit an intricate structure, that we describe and show is governed by the Feigenbaum's RG transformation [3].The domain of validity of BG statistical mechanics has been implicitly challenged by the proposal of its nonextensive generalization. Subsequent studies have offered experimental and numerical evidence that point out both the inadequacy of the standard BG statistics and the plausible competence of the generalized theory in describing various types of phenomena and systems. This theoretical development represents an exceptional event in the long and trustworthy history of BG statistical mechanics. However, it is still in the process of being converted into a rigorously corroborated and fully understood fact. The suggested circumstances under which the generalized theory is believed to be applicable, at least with regards to non-linear dynamical systems, are those associated to a phase space with power-law sensitivity to initial conditions, to the consequent vanishing of the largest Lyapunov exponent, and to a fractal, or multifractal geometrical structure [2]. Here we show that our results for the dynamics at the onset of chaos in unimodal maps constitute an unequivocal proof of the universal validity of the non-extensive statistics at such critical points.In fact, at the chaos threshold (as well as at other critical points of the map) the Lyapunov exponent λ 1 vanishes, and the sensitivity to initial conditions ξ t , for large iteration time t, ceases to obey exponential behavior, exhibiting instead power-law behavior [4]. In order to describe the dynamics at such critical points, the q...
In setting up a stochastic description of the time evolution of a financial index, the challenge consists in devising a model compatible with all stylized facts emerging from the analysis of financial time series and providing a reliable basis for simulating such series. Based on constraints imposed by market efficiency and on an inhomogeneous-time generalization of standard simple scaling, we propose an analytical model which accounts simultaneously for empirical results like the linear decorrelation of successive returns, the power law dependence on time of the volatility autocorrelation function, and the multiscaling associated to this dependence. In addition, our approach gives a justification and a quantitative assessment of the irreversible character of the index dynamics. This irreversibility enters as a key ingredient in a novel simulation strategy of index evolution which demonstrates the predictive potential of the model. complex systems ͉ finance ͉ stochastic processes F or over a century, it has been recognized (1) that the unpredictable time evolution of a financial index is inherently a stochastic process. However, despite many efforts (2-11), a unified framework for simultaneously understanding empirical facts (12-19), such as the non-Gaussian form and multiscaling in time of the distribution of returns, the linear decorrelation of successive returns, and volatility clustering, has been elusive. This situation occurs in many natural phenomena, when strong correlations determine various forms of anomalous scaling (20)(21)(22)(23)(24)(25)(26)(27). Here, by employing mathematical tools at the basis of a generalization of the central limit theorem to strongly correlated variables (28), we propose a model of index evolution and a corresponding simulation strategy which account for all robust features revealed by the empirical analysis.Let S(t) be the value of a given asset at time t. The logarithmic return over the interval [t, t ϩ T] is defined as r(t, T) ϵ ln S(t ϩ T) Ϫ ln S(t), where t ϭ 0, 1, . . . and T ϭ 1, 2, . . . , in some unit (e.g., day). From a sufficiently long historical series, one can sample the empirical probability density function (PDF) of r over a time T, p T (r), and the joint PDF of two successive returns r 1 ϵ r(t, T) and r 2 ϵ r(t ϩ T, T), denoted by p 2T(2) (r 1 , r 2 ). This joint PDF contains the information on the correlation between r 1 and r 2 in the sampling. A well established property (13-16) is that, if T is longer than tens of minutes, the linear correlation vanishes: ͐p 2T(2) (r 1 , r 2 )r 1 r 2 dr 1 dr 2 ϵ ͗r 1 r 2 ͘ p 2T (2) ϭ 0. This is a consequence of the efficiency of the market (3), which quickly suppresses any arbitrage opportunity. Another remarkable feature is that, within specific T ranges, p T approximately assumes a simple scaling formwhere g and D are the scaling function and exponent, respectively. Eq. 1 manifests self-similarity, a symmetry often met in natural phenomena (20)(21)(22)(23)27): plots of T D p T vs. r/T D for different T values collapse o...
We use a Hamiltonian dynamics to discuss the statistical mechanics of long-lasting quasistationary states particularly relevant for long-range interacting systems. Despite the presence of an anomalous single-particle velocity distribution, we find that the Central Limit Theorem implies the Boltzmann expression in Gibbs' Γ-space. We identify the nonequilibrium sub-manifold of Γ-space characterizing the anomalous behavior and show that by restricting the Boltzmann-Gibbs approach to this sub-manifold we obtain the statistical mechanics of the quasi-stationary states.PACS numbers: 05.70.Ln, In comparison with its equilibrium counterpart, nonequilibrium statistical mechanics does not rely on universal notions, like the ensembles ones, through which one can handle large classes of physical systems [1]. Incomplete (or partial) equilibrium states [2,3] are in this respect a remarkable exception, since in these cases concepts of equilibrium statistical mechanics can be used to describe nonequilibrium situations. Incomplete equilibrium states arise when different parts of the system themselves reach a state of equilibrium long before they equilibrate with each other [2]. The classical understanding on how a system approaches equilibrium is based on the short time-scale collisions mechanism which links any initial condition to the statistical equilibrium. For longrange interacting systems, this picture is not valid anymore since the time-scale for microscopic collisions diverges with the range of the interactions. This implies that the Boltzmann equation must be substituted with other approximations such as the Vlasov or the BalescuLenard equations [4], where the interparticle correlations are negligible or almost negligible and a nonequilibrium initial configuration could stay frozen or almost frozen for a very long time. This applies, e.g., to gravitational systems, Bose-Einstein condensates and plasma physics [5]. Due to the physical relevance of long-range interacting systems and to the privileged position of incomplete equilibrium states in nonequilibrium statistical mechanics, it is important to investigate whether the notion of incomplete equilibrium plays an important role in understanding the nonequilibrium properties of these systems.Recently we showed [6] that nonequilibrium states in which the value of macroscopic quantities remains stationary or quasi-stationary for reasonably long time (quasi-stationary states -QSSs) are important, e.g., for experiments, since they appear even when the long-range system exchanges energy with a thermal bath (TB). Using the same paradigmatic long-range interacting system of Ref.[6], the Hamiltonian Mean Field (HMF) model [7], here we discuss the Gibbs' Γ-space statistical mechan- * Electronic address: baldovin@pd.infn.it, orlandin@pd.infn.it ics description of the QSSs in a canonical ensemble perspective. We identify the nonequilibrium sub-manifold of Γ-space within which the quasi-stationary dynamics is confined and we show that the Boltzmann-Gibbs (BG) approach, restricted t...
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