arXiv subjects

G. Paladin

Publications and source records attributed to G. Paladin.

17 recordsLinked to original sources

arXiv1998

An extension of the Lyapunov analysis for the predictability problem

The predictability problem for systems with different characteristic time scales is investigated. It is shown that even in simple chaotic dynamical systems, the leading Lyapunov exponent is not sufficient to estimate the predictability time. This fact is due the saturation of the error on the fast components of the system which therefore do not contribute to the exponential growth of the error at large errors. It is proposed to adopt a generalization of the Lyapunov exponent which is based on the natural concept of error growing time at finite error size. The method is first illustrated on a simple numerical model obtained by coupling two Lorenz systems with different time scales. As a more realistic example, this analysis is then applied to a toy model of Atmospheric circulation recently introduced by Lorenz.

chao-dyn↗

arXiv1996

Predictability in the large: an extension of the concept of Lyapunov exponent

We investigate the predictability problem in dynamical systems with many degrees of freedom and a wide spectrum of temporal scales. In particular, we study the case of $3D$ turbulence at high Reynolds numbers by introducing a finite-size Lyapunov exponent which measures the growth rate of finite-size perturbations. For sufficiently small perturbations this quantity coincides with the usual Lyapunov exponent. When the perturbation is still small compared to large-scale fluctuations, but large compared to fluctuations at the smallest dynamically active scales, the finite-size Lyapunov exponent is inversely proportional to the square of the perturbation size. Our results are supported by numerical experiments on shell models. We find that intermittency corrections do not change the scaling law of predictability. We also discuss the relation between finite-size Lyapunov exponent and information entropy.

chao-dyn↗

arXiv1996

Earthquake statistics and fractal faults

We introduce a Self-affine Asperity Model (SAM) for the seismicity that mimics the fault friction by means of two fractional Brownian profiles (fBm) that slide one over the other. An earthquake occurs when there is an overlap of the two profiles representing the two fault faces and its energy is assumed proportional to the overlap surface. The SAM exhibits the Gutenberg-Richter law with an exponent $β$ related to the roughness index of the profiles. Apart from being analytically treatable, the model exhibits a non-trivial clustering in the spatio-temporal distribution of epicenters that strongly resembles the experimentally observed one. A generalized and more realistic version of the model exhibits the Omori scaling for the distribution of the aftershocks. The SAM lies in a different perspective with respect to usual models for seismicity. In this case, in fact, the critical behaviour is not Self-Organized but stems from the fractal geometry of the faults, which, on its turn, is supposed to arise as a consequence of geological processes on very long time scales with respect to the seismic dynamics. The explicit introduction of the fault geometry, as an active element of this complex phenomenology, represents the real novelty of our approach.

cond-mat↗

arXiv1996

Beyond the Mean Field Approximation for Spin Glasses

We study the d-dimensional random Ising model using a Bethe-Peierls approximation in the framework of the replica method. We take into account the correct interaction only inside replicated clusters of spins. Our ansatz is that the interaction of the borders of the clusters with the external world can be described via an effective interaction among replicas. The Bethe-Peierls model is mapped into a single Ising model with a random gaussian field, whose strength (related to the effective coupling between two replicas) is determined via a self-consistency equation. This allows us to obtain analytic estimates of the internal energy and of the critical temperature in d dimensions.

cond-mat↗

arXiv1996

Growth of non-infinitesimal perturbations in turbulence

We discuss the effects of finite perturbations in fully developed turbulence by introducing a measure of the chaoticity degree associated to a given scale of the velocity field. This allows one to determine the predictability time for non-infinitesimal perturbations, generalizing the usual concept of maximum Lyapunov exponent. We also determine the scaling law for our indicator in the framework of the multifractal approach. We find that the scaling exponent is not sensitive to intermittency corrections, but is an invariant of the multifractal models. A numerical test of the results is performed in the shell model for the turbulent energy cascade.

chao-dyn↗

arXiv1995

Characterization of chaos in random maps

We discuss the characterization of chaotic behaviours in random maps both in terms of the Lyapunov exponent and of the spectral properties of the Perron-Frobenius operator. In particular, we study a logistic map where the control parameter is extracted at random at each time step by considering finite dimensional approximation of the Perron-Frobenius operator

chao-dyn↗

arXiv1995

Bethe-Peierls Approximation for the 2D Random Ising Model

The partition function of the 2d Ising model with random nearest neighbor coupling is expressed in the dual lattice made of square plaquettes. The dual model is solved in the the mean field and in different types of Bethe-Peierls approximations, using the replica method.

cond-mat↗

arXiv1995

Mean Field Solution of the Random Ising Model on the Dual Lattice

We perform a duality transformation that allows one to express the partition function of the d-dimensional Ising model with random nearest neighbor coupling in terms of new spin variables defined on the square plaquettes of the lattice. The dual model is solved in the mean field approximation.

cond-mat↗

arXiv1995

Self-affine Asperity Model for earthquakes

A model for fault dynamics consisting of two rough and rigid brownian profiles that slide one over the other is introduced. An earthquake occurs when there is an intersection between the two profiles. The energy release is proportional to the overlap interval. Our model exhibits some specific features which follow from the fractal geometry of the fault: (1) non-universality of the exponent of the Gutenberg-Richter law for the magnitude distribution; (2) presence of local stress accumulation before a large seismic event; (3) non-trivial space-time clustering of the epicenters. These properties are in good agreement with various observations and lead to specific predictions that can be experimentally tested.

cond-mat↗

arXiv1995

On the concept of complexity in random dynamical systems

We introduce a measure of complexity in terms of the average number of bits per time unit necessary to specify the sequence generated by the system. In random dynamical system, this indicator coincides with the rate K of divergence of nearby trajectories evolving under two different noise realizations. The meaning of K is discussed in the context of the information theory, and it is shown that it can be determined from real experimental data. In presence of strong dynamical intermittency, the value of K is very different from the standard Lyapunov exponent computed considering two nearby trajectories evolving under the same randomness. However, the former is much more relevant than the latter from a physical point of view as illustrated by some numerical computations for noisy maps and sandpile models.

cond-mat↗

arXiv1995

Predictability in Systems with Many Characteristic Times: The Case of Turbulence

In chaotic dynamical systems, an infinitesimal perturbation is exponentially amplified at a time-rate given by the inverse of the maximum Lyapunov exponent $λ$. In fully developed turbulence, $λ$ grows as a power of the Reynolds number. This result could seem in contrast with phenomenological arguments suggesting that, as a consequence of `physical' perturbations, the predictability time is roughly given by the characteristic life-time of the large scale structures, and hence independent of the Reynolds number. We show that such a situation is present in generic systems with many degrees of freedom, since the growth of a non-infinitesimal perturbation is determined by cumulative effects of many different characteristic times and is unrelated to the maximum Lyapunov exponent. Our results are illustrated in a chain of coupled maps and in a shell model for the energy cascade in turbulence.

chao-dyn↗

arXiv1994

Constrained annealing for spin glasses

The quenched free energy of spin glasses is estimated by means of annealed averages where the frustration is constrained to its average value. We discuss the case of d-dimensional Ising models with random nearest neighbour coupling, and we introduce a new method to obtain constrained annealed averages without recurring to Lagrange multipliers. It requires to perform quenched averages either on small volumes in an analytic way, or on finite size strips via products of random transfer matrices. We thus give a sequence of converging lower bounds for the quenched free energy of 2d spin glasses.

cond-mat↗

arXiv1994

Stochastic Resonance in Deterministic Chaotic Systems

We propose a mechanism which produces periodic variations of the degree of predictability in dynamical systems. It is shown that even in the absence of noise when the control parameter changes periodically in time, below and above the threshold for the onset of chaos, stochastic resonance effects appears. As a result one has an alternation of chaotic and regular, i.e. predictable, evolutions in an almost periodic way, so that the Lyapunov exponent is positive but some time correlations do not decay.

chao-dyn↗

arXiv1994

Statistical Mechanics of Shell Models for 2D-Turbulence

We study shell models that conserve the analogues of energy and enstrophy, hence designed to mimic fluid turbulence in 2D. The main result is that the observed state is well described as a formal statistical equilibrium, closely analogous to the approach to two-dimensional ideal hydrodynamics of Onsager, Hopf and Lee. In the presence of forcing and dissipation we observe a forward flux of enstrophy and a backward flux of energy. These fluxes can be understood as mean diffusive drifts from a source to two sinks in a system which is close to local equilibrium with Lagrange multipliers (``shell temperatures'') changing slowly with scale. The dimensional predictions on the power spectra from a supposed forward cascade of enstrophy, and from one branch of the formal statistical equilibrium, coincide in these shell models at difference to the corresponding predictions for the Navier-Stokes and Euler equations in 2D. This coincidence have previously led to the mistaken conclusion that shell models exhibit a forward cascade of enstrophy.

chao-dyn↗

arXiv1994

Transition to Chaos in a Shell Model of Turbulence

We study a shell model for the energy cascade in three dimensional turbulence at varying the coefficients of the non-linear terms in such a way that the fundamental symmetries of Navier-Stokes are conserved. When a control parameter $ε$ related to the strength of backward energy transfer is enough small, the dynamical system has a stable fixed point corresponding to the Kolmogorov scaling. This point becomes unstable at $ε=0.3843...$ where a stable limit cycle appears via a Hopf bifurcation. By using the bi-orthogonal decomposition, the transition to chaos is shown to follow the Ruelle-Takens scenario. For $ε> 0.3953..$ the dynamical evolution is intermittent with a positive Lyapunov exponent. In this regime, there exists a strange attractor which remains close to the Kolmogorov (now unstable) fixed point, and a local scaling invariance which can be described via a intermittent one-dimensional map.

cond-mat↗

arXiv1994

Transition to Chaos in a Shell Model of Turbulence

chao-dyn↗

arXiv1993

Lack of self-average in weakly disordered one dimensional systems

We introduce a one dimensional disordered Ising model which at zero temperature is characterized by a non-trivial, non-self-averaging, overlap probability distribution when the impurity concentration vanishes in the thermodynamic limit. The form of the distribution can be calculated analytically for any realization of disorder. For non-zero impurity concentration the distribution becomes a self-averaging delta function centered on a value which can be estimated by the product of appropriate transfer matrices.

cond-mat↗