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

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.     

Sensitivity to initial conditions in coherent noise models

Sensitivity to initial conditions in the coherent noise model of biological evolution, introduced by Newman, is studied by making use of damage spreading technique. A power-law behavior has been observed, the associated exponent α and the dynamical exponent z are calculated. Using these values a clear data collapse has been obtained.     

Sensitivity to initial conditions in the Bak-Sneppen model of biological evolution

We consider biological evolution as described within the Bak and Sneppen 1993 model. We exhibit, at the self-organized critical state, a power-law sensitivity to the initial conditions, calculate the associated exponent, and relate it to the recently introduced nonextensive thermostatistics. The scenario which here emerges without tuning strongly reminds of that of the tuned onset of chaos in say logistic-like one-dimensional maps. We also calculate the dynamical exponent z. Received: 5 November 1997 / Received in final form: 11 November 1997 / Accepted: 19 November 1997     

Sensitivity to initial conditions and lyapunov exponent of a quasiperiodic system

The dynamics of a quasiperiodic map is analyzed both in the presence and in the absence of weak noise. It is shown that, in the presence of weak noise, a strange chaotic attractor with a negative Lyapunov exponent and sensitive dependence of trajectories on the initial conditions can exist in the system. This means that the types of motion of a fluctuating system cannot be classified only by the sign of the leading Lyapunov exponent.     

Intermittency at critical transitions and aging dynamics at the onset of chaos

We recall that at both the intermittency transitions and the Feigenbaum attractor, in unimodal maps of non-linearity of order ζ > 1, the dynamics rigorously obeys the Tsallis statistics. We account for theq-indices and the generalized Lyapunov coefficients λ^q that characterize the universality classes of the pitchfork and tangent bifurcations. We identify the Mori singularities in the Lyapunov spectrum at the onset of chaos with the appearance of a special value for the entropic indexq. The physical area of the Tsallis statistics is further probed by considering the dynamics near criticality and glass formation in thermal systems. In both cases a close connection is made with states in unimodal maps with vanishing Lyapunov coefficients.     

Minimum entropy production and logarithmic rate equations

A system interacting with a heat bath and radiation is considered. It is assumed that the steady state is exactly characterized by the principle of minimum entropy production. From this, the general form of the equations for the time rate of change of the probabilities of the states is derived and the rate equations are shown to be nonlinear and to involve the differences of the logarithms of the probabilities. Some properties of these equations are discussed and the specific cases of two- and three-state subsystems are considered and compared with results obtained from the usual linear rate equations.     

Symbolic dynamics and topological entropy at the onset of pruning

The topological entropy and pruning rules are investigated for two-dimensional smooth maps at the onset of pruning. Typically the difference of the parameter-dependent topological entropy from its maximum value increases with a power law. Superimposed on this decrease, there are periodic or quasiperiodic oscillations on a logarithmic scale. Both, the scaling exponent and the periodicity are determined by the Lyapunov exponents of the first pruned orbit and the minimal number of letters in the alphabet of the symbolic dynamics. If, at the onset of pruning, the averaged Lyapunov exponent is sufficiently large and the first pruned orbit is homoclinic, the entropy function of area-preserving maps exhibits a series of plateaux. On the plateaux, the symbolic dynamics can be described by finitely many finite forbidden words. There is a series of plateaux which, in different systems, can be described by the same type of forbidden words.     

Sensitivity to initial conditions of the wave-packet dynamics in diluted Anderson chains

We study the one-electron wave-packet dynamics in the one-dimensional diluted Anderson model which is composed of two interpenetrating chains with pure and random on-site potentials, respectively. This model presents extended states at a particular resonance energy. Starting with one electron fully localized at the site closer to the chain center, we solve the set of coupled motion equations and calculate the time evolution of the wave-packet width. We report on a long-time memory effect which is reflected by distinct asymptotic dynamics governing the wave-function spread for electrons initially localized at random or pure sites. This anomalous behavior is discussed under the light of the Bloch character of the extended resonant state.     

Fluctuations and the onset of chaos

We consider the role of fluctuations on the onset and characteristics of chaotic behavior associated with period doubling subharmonic bifurcations. By studying the problem of forced dissipative motion of an anharmonic oscillator we show that the effect of noise is to produce a bifurcation gap in the set of available states. We discuss the possible experimental observation of this gap in many systems which display turbulent behavior.     

Space-time renormalization at the onset of spatio-temporal chaos in coupled maps

The transition regime to spatio-temporal chaos via the quasiperiodic route as well as the period-doubling route is examined for coupled-map lattices. Space-time renormalization-group analysis is carried out and the scaling exponents for the coherence length, the Lyapunov exponent, and the size of the phase fluctuations are determined. Universality classes for the different types of coupling at various routes to chaos are identified.     

Decoherence and the rate of entropy production in chaotic quantum systems

We show that for an open quantum system which is classically chaotic (a quartic double well with harmonic driving coupled to a sea of harmonic oscillators) the rate of entropy production has, as a function of time, two relevant regimes: For short times it is proportional to the diffusion coefficient (fixed by the system-environment coupling strength). For longer times (but before equilibration) there is a regime where the entropy production rate is fixed by the Lyapunov exponent. The nature of the transition time between both regimes is investigated.     

Critical behavior of the Lyapunov number at the period-doubling onset of chaos

For period-doubling bifurcations of 1 d-maps the Lyapunov number λ is calculated explicitly using a renormalization procedure. We find that its slope diverges like (δ/2)^k. In the chaotic regime the unstable cycles yield a continuous curve as an upper bound for the Lyapunov exponent of chaotic bands. The bound has a critical exponent t = 0.449 80…, which is the same as for the chaotic bands.     

Logistic混沌系统的熵特性研究

作为形式上相对较为简单的一维混沌函数,Logistic系统在很多领域有着重要的应用.本文主要分析了Logistic系统的熵稳定特性,对不同参数μ和系统初值形成的Logistic序列,进行了统计分类,得到了一系列的熵值,并详细分析了熵的分布情况.数值仿真结果表明,Logistic系统的熵由参数μ决定,而与系统初值基本无关,且当参数μ取值接近上界(μ=4)时,序列分布越趋于均匀,熵也接近理论极限值.     

Entropy, entropy flux and entropy rate of granular materials

The aim of this work is to analyze the entropy, entropy flux and entropy rate of granular materials within the frameworks of the Boltzmann equation and continuum thermodynamics. It is shown that the entropy inequality for a granular gas that follows from the Boltzmann equation differs from the one of a simple fluid due to the presence of a term which can be identified as the entropy density rate. From the knowledge of a non-equilibrium distribution function-valid for processes closed to equilibrium-it is obtained that the entropy density rate is proportional to the internal energy density rate divided by the temperature, while the entropy flux is equal to the heat flux vector divided by the temperature. A thermodynamic theory of a granular material is also developed whose objective is the determination of the basic fields of mass density, momentum density and internal energy density. The constitutive laws are restricted by the principle of material frame indifference and by the entropy principle. Through the exploitation of the entropy principle with Lagrange multipliers, it is shown that the results obtained from the kinetic theory for granular gases concerning the entropy density rate and entropy flux are valid in general for processes close to equilibrium of granular materials, where linearized constitutive equations hold.     

Strict entropy production bounds and stability of the rate of convergence to equilibrium for the Boltzmann equation

We first consider the Boltzmann equation with a collision kernel such that all kinematically possible collisions are run at equal rates. This is the simplest Boltzmann equation having the compressible Euler equations as a scaling limit. For it we prove a stability result for theH-theorem which says that when the entropy production is small, the solution of the spatially homogeneous Boltzmann equation is necessarily close to equilibrium in the entropie sense, and therefore strongL ¹ sense. We use this to prove that solutions to the spatially homogeneous Boltzmann equation converge to equilibrium in the entropie sense with a rate of convergence which is uniform in the initial condition for all initial conditions belonging to certain natural regularity classes. Every initial condition with finite entropy andp ^th velocity moment for some p>2 belongs to such a class. We then extend these results by a simple monotonicity argument to the case where the collision rate is uniformly bounded below, which covers a wide class of slightly modified physical collision kernels. These results are the basis of a study of the relation between scaling limits of solutions of the Boltzmann equation and hydrodynamics which will be developed in subsequent papers; the program is described here.On leave from School of Mathematics, Georgia Institute of Technology, Atlanta, Georgia 30332.On leave from C.F.M.C. and Departamento de Matemática da Faculdade de Ciencias de Lisboa, 1700 Lisboa codex, Portugal.     

Saddle pattern resonance and onset of crisis to spatiotemporal chaos

We observe a novel phenomenon which we call "pattern resonance." It occurs in a theoretical model when the realized wave solution evolves to about the same shape of a saddle steady-wave solution. Here the unstable saddle solution behaves like a "potential." We find that the resonance triggers the onset of a crisis, leading to a transition from temporal to spatiotemporal chaos. We also show that in our case the pattern resonance is essentially a nonlinear frequency resonance.     

Efficient procedure to compute the microcanonical volume of initial conditions that lead to escape trajectories from a multidimensional potential well

A procedure is presented for computing the phase space volume of initial conditions for trajectories that escape or "react" from a multidimensional potential well. The procedure combines a phase space transition state theory, which allows one to construct dividing surfaces that are free of local recrossing and that minimize the directional flux, and a classical spectral theorem. The procedure gives the volume of reactive initial conditions in terms of a sum over each entrance channel of the well of the product of the phase space flux across the dividing surface associated with the channel and the mean residence time in the well of trajectories which enter through the channel. This approach is illustrated for HCN isomerization in three dimensions, for which the method is several orders of magnitude more efficient than standard Monte Carlo sampling.