Fulvio Baldovin

Nonextensive Pesin identity: exact renormalization group analytical results for the dynamics at the edge of chaos of the logistic map.

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.

Quasi-stationary states in low-dimensional Hamiltonian systems

Abstract We address a simple connection between results of Hamiltonian non-linear dynamical theory and thermostatistics. Using a properly defined dynamical temperature in low-dimensional symplectic maps, we display and characterize long-standing quasi-stationary states that eventually cross over to a Boltzmann–Gibbs-like regime. As time evolves, the geometrical properties (e.g., fractal dimension) of the phase space change sensibly, and the duration of the anomalous regime diverges with decreasing chaoticity. The scenario that emerges is consistent with the non-extensive statistical mechanics one.

Mixing and equilibration: protagonists in the scene of nonextensive statistical mechanics

After a brief review of the present status of nonextensive statistical mechanics, we present a conjectural scenario where mixing (characterized by the entropic index qmix⩽1) and equilibration (characterized by the entropic index qeq⩾1) play central and inter-related roles, and appear to determine a priori the values of the relevant indices of the formalism. Boltzmann–Gibbs statistical mechanics is recovered as the qmix=qeq=1 particular case.

Linear instability and statistical laws of physics

We show that a meaningful statistical description is possible in conservative and mixing systems with zero Lyapunov exponent in which the dynamical instability is only linear in time. More specifically, i) the sensitivity to initial conditions is given by ξ = [1 + (1 − q)λqt]1/(1 − q) with q = 0; ii) the statistical entropy Sq = (1 − ∑i piq)/(q − 1) (S1 = − ∑i piln pi) in the infinitely fine graining limit (i.e., W ≡ number of cells into which the phase space has been partitioned → ∞), increases linearly with time only for q = 0; iii) a nontrivial, q-generalized, Pesin-like identity is satisfied, namely the limt → ∞limW → ∞ S0(t)/t = max {λ0}. These facts (which are in analogy to the usual behaviour of strongly chaotic systems with q = 1), seem to open the door for a statistical description of conservative many-body nonlinear systems whose Lyapunov spectrum vanishes.

Ubiquity of metastable-to-stable crossover in weakly chaotic dynamical systems

We present a comparative study of several dynamical systems of increasing complexity, namely, the logistic map with additive noise, one, two and many globally coupled standard maps, and the Hamiltonian mean field model (i.e., the classical inertial infinitely ranged ferromagnetically coupled XY spin model). We emphasize the appearance, in all of these systems, of metastable states and their ultimate crossover to the equilibrium state. We comment on the underlying mechanisms responsible for these phenomena (weak chaos) and compare common characteristics. We point out that this ubiquitous behavior appears to be associated to the features of the nonextensive generalization of the Boltzmann–Gibbs statistical mechanics.

Microcanonical quasistationarity of long-range interacting systems in contact with a heat bath

On the basis of analytical results and molecular dynamics simulations, we clarify the nonequilibrium dynamics of a long-range interacting system in contact with a heat bath. For small couplings with the bath, we show that the system can first be trapped in a Vlasov quasistationary state, then a microcanonical one follows, and finally canonical equilibrium is reached at the bath temperature. We demonstrate that, even out of equilibrium, Hamiltonian reservoirs microscopically coupled with the system and Langevin thermostats provide equivalent descriptions. Our identification of the key parameters determining the quasistationary lifetimes could be exploited to control experimental systems such as the free-electron laser, in the presence of external noise or inherent imperfections.

Central limit theorem for anomalous scaling due to correlations

We derive a central limit theorem for the probability distribution of the sum of many critically correlated random variables. The theorem characterizes a variety of different processes sharing the same asymptotic form of anomalous scaling and is based on a correspondence with the Lévy-Gnedenko uncorrelated case. In particular, correlated anomalous diffusion is mapped onto Lévy diffusion. Under suitable assumptions, the nonstandard multiplicative structure used for constructing the characteristic function of the total sum allows us to determine correlations of partial sums exclusively on the basis of the global anomalous scaling.

Hamiltonian dynamics reveals the existence of quasistationary states for long-range systems in contact with a reservoir.

We introduce a Hamiltonian dynamics for the description of long-range interacting systems in contact with a thermal bath (i.e., in the canonical ensemble). The dynamics confirms statistical mechanics equilibrium predictions for the Hamiltonian mean field model and the equilibrium ensemble equivalence. We find that long-lasting quasistationary states persist in the presence of the interaction with the environment. Our results indicate that quasistationary states are indeed reproducible in real physical experiments.

Anomalous sensitivity to initial conditions and entropy production in standard maps: Nonextensive approach

Abstract.We perform a throughout numerical study of the average sensitivity to initial conditions and entropy production for two symplectically coupled standard maps focusing on the control-parameter region close to regularity. Although the system is ultimately strongly chaotic (positive Lyapunov exponents), it first stays lengthily in weak-chaotic regions (zero Lyapunov exponents). We argue that the nonextensive generalization of the classical formalism is an adequate tool in order to get nontrivial information about the first stage of this crossover phenomenon. Within this context we analyze the relation between the power-law sensitivity to initial conditions and the entropy production.

Scaling and efficiency determine the irreversible evolution of a market

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.