Celia Anteneodo

Nonextensive statistical mechanics and economics

Ergodicity, this is to say, dynamics whose time averages coincide with ensemble averages, naturally leads to Boltzmann–Gibbs (BG) statistical mechanics, hence to standard thermodynamics. This formalism has been at the basis of an enormous success in describing, among others, the particular stationary state corresponding to thermal equilibrium. There are, however, vast classes of complex systems which accommodate quite badly, or even not at all, within the BG formalism. Such dynamical systems exhibit, in one way or another, nonergodic aspects. In order to be able to theoretically study at least some of these systems, a formalism was proposed 14 years ago, which is sometimes referred to as nonextensive statistical mechanics. We briefly introduce this formalism, its foundations and applications. Furthermore, we provide some bridging to important economical phenomena, such as option pricing, return and volume distributions observed in the financial markets, and the fascinating and ubiquitous concept of risk aversion. One may summarize the whole approach by saying that BG statistical mechanics is based on the entropy SBG=−k∑ipilnpi, and typically provides exponential laws for describing stationary states and basic time-dependent phenomena, while nonextensive statistical mechanics is instead based on the entropic form Sq=k(1−∑ipiq)/(q−1) (with S1=SBG), and typically provides, for the same type of description, (asymptotic) power laws.

Multiplicative noise: A mechanism leading to nonextensive statistical mechanics

A large variety of microscopic or mesoscopic models lead to generic results that accommodate naturally within Boltzmann–Gibbs statistical mechanics [based on S1≡−k∫du p(u)ln p(u)]. Similarly, other classes of models point toward nonextensive statistical mechanics [based on Sq≡k[1−∫du[p(u)]q]/[q−1], where the value of the entropic index q∈R depends on the specific model]. We show here a family of models, with multiplicative noise, which belongs to the nonextensive class. More specifically, we consider Langevin equations of the type u=f(u)+g(u)ξ(t)+η(t), where ξ(t) and η(t) are independent zero-mean Gaussian white noises with respective amplitudes M and A. This leads to the Fokker–Planck equation ∂tP(u,t)=−∂u[f(u)P(u,t)]+M∂u{g(u)∂u[g(u)P(u,t)]}+A∂uuP(u,t). Whenever the deterministic drift is proportional to the noise induced one, i.e., f(u)=−τg(u)g′(u), the stationary solution is shown to be P(u,∞)∝{1−(1−q)β[g(u)]2}1/(1−q) [with q≡(τ+3M)/(τ+M) and β=(τ+M/2A)]. This distribution is precisely the one optimiz...

Rotators with long-range interactions: connection with the mean-field approximation

We analyze the equilibrium properties of a chain of ferromagnetically coupled rotators which interact through a force that decays as r(-alpha) where r is the interparticle distance and alpha>/=0. By integrating the equations of motion we obtain the microcanonical time averages of both the magnetization and the kinetic energy. We detect three different regimes depending on whether alpha belongs to the intervals [0,1), (1,2), or (2,infinity). For 0<alpha<1, the microcanonical averages agree, after a scaling, with those obtained in the canonical ensemble for the mean-field case (alpha = 0). This correspondence offers a mathematically tractable way of dealing with systems governed by slowly decaying long-range interactions.

Two-dimensional turbulence in pure-electron plasma: A nonextensive thermostatistical description

Huang and Driscoll (1994) studied the two-dimensional turbulent metaequilibrium state that appears in an experiment in which pure-electron plasma evolves in the interior of a conducting cylinder (of radius Rw) in the presence of an external axial magnetic field. They measured the electron radial distribution and compared their data with the profiles resulting from four different phenomenological theories developed by themselves. Two among these theories are based on the optimization of the standard (Boltzmann-Gibbs) entropy, the other two being based on the optimization of the enstrophy. Only one of the latter (Restricted Minimum Enstrophy theory, where restricted stands for the fact that a cut-off radius Rc 12, q = 12 and q < 12, respectively.

MAXIMUM ENTROPY APPROACH TO STRETCHED EXPONENTIAL PROBABILITY DISTRIBUTIONS

We introduce a nonextensive entropy functionalS whose optimization under simple constraints (mean values of some standard quantities) yields stretched exponential probability distributions, which occur in many complex systems. The new entropy functional is characterized by a parameter (the stretching exponent) such that for D 1 the standard logarithmic entropy is recovered. We study its mathematical properties, showing that the basic requirements for a well-behaved entropy functional are verified, i.e. S possesses the usual properties of positivity, equiprobability, concavity and irreversibility and verifies Khinchin axioms except the one related to additivity since S is nonextensive. The entropyS is shown to be superadditive for 1.

Analytical results for coupled-map lattices with long-range interactions

We obtain exact analytical results for lattices of maps with couplings that decay with distance as r(-alpha). We analyze the effect of the coupling range on the system dynamics through the Lyapunov spectrum. For lattices whose elements are piecewise linear maps, we get an algebraic expression for the Lyapunov spectrum. When the local dynamics is given by a nonlinear map, the Lyapunov spectrum for a completely synchronized state is analytically obtained. The critical line characterizing the synchronization transition is determined from the expression for the largest transversal Lyapunov exponent. In particular, it is shown that in the thermodynamical limit, such transition is only possible for sufficiently long-range interactions, namely, for alpha

Aging in an infinite-range Hamiltonian system of coupled rotators.

We analyze numerically the out-of-equilibrium relaxation dynamics of a long-range Hamiltonian system of N fully coupled rotators. For a particular family of initial conditions, this system is known to enter a particular regime in which the dynamic behavior does not agree with thermodynamic predictions. Moreover, there is evidence that in the thermodynamic limit, when N--> infinity is taken prior to t--> infinity, the system will never attain true equilibrium. By analyzing the scaling properties of the two-time autocorrelation function we find that, in that regime, a very complex dynamics unfolds, in which aging phenomena appear. The scaling law strongly suggests that the system behaves in a complex way, relaxing towards equilibrium through intricate trajectories. The present results are obtained for conservative dynamics, where there is no thermal bath in contact with the system.

Additive-multiplicative stochastic models of financial mean-reverting processes.

We investigate a generalized stochastic model with the property known as mean reversion, that is, the tendency to relax towards a historical reference level. Besides this property, the dynamics is driven by multiplicative and additive Wiener processes. While the former is modulated by the internal behavior of the system, the latter is purely exogenous. We focus on the stochastic dynamics of volatilities, but our model may also be suitable for other financial random variables exhibiting the mean reversion property. The generalized model contains, as particular cases, many early approaches in the literature of volatilities or, more generally, of mean-reverting financial processes. We analyze the long-time probability density function associated to the model defined through an Itô-Langevin equation. We obtain a rich spectrum of shapes for the probability function according to the model parameters. We show that additive-multiplicative processes provide realistic models to describe empirical distributions, for the whole range of data.

Diffusive anomalies in a long-range Hamiltonian system.

We scrutinize the anomalies in diffusion observed in an extended long-range system of classical rotors, the HMF model. Under suitable preparation, the system falls into long-lived quasi-stationary states for which superdiffusion of rotor phases has been reported. In the present work, we investigate diffusive motion by monitoring the evolution of full distributions of unfolded phases. After a transient, numerical histograms can be fitted by the q -Gaussian form P(x) proportional to {1+(q-1)[x/beta]2}(1/(1-q)) , with parameter q increasing with time before reaching a steady value q approximately 32 (squared Lorentzian). From the analysis of the second moment of numerical distributions, we also discuss the relaxation to equilibrium and show that diffusive motion in quasistationary trajectories depends strongly on system size.

Thermodynamical fingerprints of fractal spectra

We investigate the thermodynamics of model systems exhibiting two-scale fractal spectra. In particular, we present both analytical and numerical studies on the temperature dependence of the vibrational and electronic specific heats. For phonons, and for bosons in general, we show that the average specific heat can be associated to the average (power law) density of states. The corrections to this average behavior are log-periodic oscillations, which can be traced back to the self-similarity of the spectral staircase. In the electronic problem, even if the thermodynamical quantities exhibit a strong dependence on the particle number, regularities arise when special cases are considered. Applications to substitutional and hierarchical structures are discussed.