arXiv subjects

Celia Anteneodo

Publications and source records attributed to Celia Anteneodo.

66 records · Page 4Linked to original sources

arXiv2003

Nonextensive scaling in a long-range Hamiltonian system

The nonextensivity of a classical long-range Hamiltonian system is discussed. The system is the so-called $α$-XY model, a lattice of inertial rotators with an adjustable parameter $α$ controlling the range of the interactions. This model has been explored in detail over the last years. For sufficiently long-range interactions, namely $α<d$, where $d$ is the lattice dimension, it was shown to be nonextensive and to exhibit a second order phase transition. However, conclusions in apparent contradiction with the findings above have also been drawn. This picture reveals the fact that there are aspects of the model that remain poorly understood. Here we perform a thorough analysis, essaying an explanation for the origin of the apparent discrepancies.

cond-mat.stat-mech↗

arXiv2003

Statistics of finite-time Lyapunov exponents in the Ulam map

The statistical properties of finite-time Lyapunov exponents at the Ulam point of the logistic map are investigated. The exact analytical expression for the autocorrelation function of one-step Lyapunov exponents is obtained, allowing the calculation of the variance of exponents computed over time intervals of length $n$. The variance anomalously decays as $1/n^2$. The probability density of finite-time exponents noticeably deviates from the Gaussian shape, decaying with exponential tails and presenting $2^{n-1}$ spikes that narrow and accumulate close to the mean value with increasing $n$. The asymptotic expression for this probability distribution function is derived. It provides an adequate smooth approximation to describe numerical histograms built for not too small $n$, where the finiteness of bin size trimmes the sharp peaks.

nlin.CD↗

arXiv2003

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 $S_1\equiv -k \int du p(u) \ln p(u)$). Similarly, other classes of models point toward nonextensive statistical mechanics (based on $S_q \equiv k [1-\int du [p(u)]^q]/[q-1]$, where the value of the entropic index $q\in\Re$ 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 $\dot{u}=f(u)+g(u)ξ(t)+η(t)$, 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 $\partial_t P(u,t) = -\partial_u[f(u) P(u,t)] + M\partial_u\{g(u)\partial_u[g(u)P(u,t)]\} + A\partial_{uu}P(u,t)$. 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, \infty) \propto \bigl\{1-(1-q) β[g(u)]^2 \bigr\}^{\frac{1}{1-q}} $ (with $q \equiv \frac{τ+ 3M}{τ+M}$ and $β=\frac{τ+M}{2A}$). This distribution is precisely the one optimizing $S_q$ with the constraint $<[g(u)]^2 >_q \equiv \{\int du [g(u)]^2[P(u)]^q \}/ \{\int du [P(u)]^q \}= $constant. We also introduce and discuss various characterizations of the width of the distributions.

cond-mat.stat-mech↗

arXiv2003

Risk aversion in financial decisions: A nonextensive approach

The sensitivity to risk that most people (hence, financial operators) feel affects the dynamics of financial transactions. Here we present an approach to this problem based on a current generalization of Boltzmann-Gibbs statistical mechanics.

cond-mat.stat-mech↗

arXiv2003

Lyapunov exponent of many-particle systems: testing the stochastic approach

The stochastic approach to the determination of the largest Lyapunov exponent of a many-particle system is tested in the so-called mean-field XY-Hamiltonians. In weakly chaotic regimes, the stochastic approach relates the Lyapunov exponent to a few statistical properties of the Hessian matrix of the interaction, which can be calculated as suitable thermal averages. We have verified that there is a satisfactory quantitative agreement between theory and simulations in the disordered phases of the XY models, either with attractive or repulsive interactions. Part of the success of the theory is due to the possibility of predicting the shape of the required correlation functions, because this permits the calculation of correlation times as thermal averages.

cond-mat.stat-mech↗

arXiv2003

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 $N\to \infty$ is taken prior to $t\to \infty$, 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 {\em 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. This is the first time that aging is observed in such Hamiltonian systems.

cond-mat.stat-mech↗

arXiv2003

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 accomodate 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 $S_{BG}=-k \sum_i p_i \ln p_i$, and typically provides {\it exponential laws} for describing stationary states and basic time-dependent phenomena, while nonextensive statistical mechanics is instead based on the entropic form $S_q=k(1-\sum_ip_i^q)/(q-1)$ (with $S_1=S_{BG}$), and typically provides, for the same type of description, (asymptotic) {\it power laws}.

cond-mat.stat-mech↗

arXiv2003

Enhancement of thermal entanglement in two-qubit XY models

We analyze conditions leading to enhancement of thermal entanglement in two-qubit XY models. The effect of including cross-product terms, besides the standard XY exchange interactions, in the presence of an external magnetic field, is investigated. We show that entanglement can be yield at elevated temperatures by tuning the orientation of the external magnetic field. The details of the intrinsic exchange interactions determine the optimal orientation.

quant-ph↗

arXiv2002

Theoretical estimates for the largest Lyapunov exponent of many-particle systems

The largest Lyapunov exponent of an ergodic Hamiltonian system is the rate of exponential growth of the norm of a typical vector in the tangent space. For an N-particle Hamiltonian system, with a smooth Hamiltonian of the type p^2 + v(q), the evolution of tangent vectors is governed by the Hessian matrix V of the potential. Ergodicity implies that the Lyapunov exponent is independent of initial conditions on the energy shell, which can then be chosen randomly according to the microcanonical distribution. In this way a stochastic process V(t) is defined, and the evolution equation for tangent vectors can now be seen as a stochastic differential equation. An equation for the evolution of the average squared norm of a tangent vector can be obtained using the standard theory in which the average propagator is written as a cummulant expansion. We show that if cummulants higher than the second one are discarded, the Lyapunov exponent can be obtained by diagonalizing a small-dimension matrix, which, in some cases, can be as small as 3x3. In all cases the matrix elements of the propagator are expressed in terms of correlation functions of the stochastic process. We discuss the connection between our approach and an alternative theory, the so-called geometric method.

cond-mat.stat-mech↗

arXiv2002

Separable-entangled frontier in a bipartite harmonic system

We consider a statistical mixture of two identical harmonic oscillators which is characterized by four parameters, namely, the concentrations (x and y) of diagonal and nondiagonal bipartite states, and their associated thermal-like noises (T/a and T, respectively). The fully random mixture of two spins 1/2 as well as the Einstein-Podolsky-Rosen (EPR) state are recovered as particular instances. By using the conditional nonextensive entropy as introduced by Abe and Rajagopal, we calculate the separable-entangled frontier. Although this procedure is known to provide a necessary but in general not sufficient condition for separability, it does recover, in the particular case x=T=0 (for all a), the 1/3 exact result known as Peres' criterion. This is an indication of reliability of the calculation of the frontier in the entire parameter space. The x=0 frontier remarkably resembles to the critical line associated with standard diluted ferromagnetism where the entangled region corresponds to the ordered one and the separable region to the paramagnetic one. The entangled region generically shrinks for increasing T or increasing a.

quant-ph↗

arXiv2001

Scaling laws for the largest Lyapunov exponent in long-range systems: A random matrix approach

We investigate the laws that rule the behavior of the largest Lyapunov exponent (LLE) in many particle systems with long range interactions. We consider as a representative system the so-called Hamiltonian alpha-XY model where the adjustable parameter alpha controls the range of the interactions of N ferromagnetic spins in a lattice of dimension d. In previous work the dependence of the LLE with the system size N, for sufficiently high energies, was established through numerical simulations. In the thermodynamic limit, the LLE becomes constant for alpha greater than d whereas it decays as an inverse power law of N for alpha smaller than d. A recent theoretical calculation based on Pettini's geometrization of the dynamics is consistent with these numerical results (M.-C. Firpo and S. Ruffo, cond-mat/0108158). Here we show that the scaling behavior can also be explained by a random matrix approach, in which the tangent mappings that define the Lyapunov exponents are modeled by random simplectic matrices drawn from a suitable ensemble.

cond-mat.stat-mech↗

arXiv1999

Long-range interacting rotators: 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 $r^{-α}$ where r is the interparticle distance and $α\ge 0$. Our model contains as particular cases the mean field limit ($α=0$) and the first-neighbor model ($α\to \infty$). By integrating the equations of motion we obtain the microcanonical time averages of both the magnetization and the kinetic energy. Concerning the long-range order, we detect three different regimes at low energies, depending on whether $α$ belongs to the intervals $[0,1)$, $(1,2)$ or $(2,\infty)$. Moreover, for $0 \le α< 1$, the microcanonical averages agree, after a simple scaling, with those obtained in the canonical ensemble for the mean-field XY model. This correspondence offers a mathematically tractable and computationally economic way of dealing with systems governed by slowly decaying long-range interactions.

cond-mat.stat-mech↗