Infinite Ergodic Theory and Non-extensive Entropies

We recapitulate results from the infinite ergodic theory that are relevant to the theory of non-extensive entropies. In particular, we recall that the Lyapunov exponent of the corresponding systems is zero and that the deviation between neighboring trajectories does not necessarily grow polynomially. Nonetheless, as we show, no single quantity can describe this subexponential growth, the generalized q-exponential exp _q being, in particular, ruled out. We also revisit a number of dynamical systems preserving nonfinite ergodic measure.     

Dynamical entropy of space translations of CAR and CCR algebras with respect to quasi-free states

We compute the dynamical entropy in the sense of Connes, Narnhofer and Thirring of space translations of the CAR and CCR algebras in -dimensional continuous spaces with respect to invariant quasi-free states. It turns out that the dynamical entropies are equal to the corresponding mean entropies of the systems under consideration. Computing the mean entropies explicitly we derive the entropy formulas for the systems.Research supported in part by the Basic Science Research Program, Ministry of Education, 1992Research supported in part by GARC in 1991–1992     

Entropy and Ergodicity of Boole-Type Transformations

We review some analytic, measure-theoretic and topological techniques for studying ergodicity and entropy of discrete dynamical systems, with a focus on Boole-type transformations and their generalizations. In particular, we present a new proof of the ergodicity of the 1-dimensional Boole map and prove that a certain 2-dimensional generalization is also ergodic. Moreover, we compute and demonstrate the equivalence of metric and topological entropies of the 1-dimensional Boole map employing “compactified”representations and well-known formulas. Several examples are included to illustrate the results. We also introduce new multidimensional Boole-type transformations invariant with respect to higher dimensional Lebesgue measures and investigate their ergodicity and metric and topological entropies.     

Multifractal properties of return time statistics

The global statistics of the return times of a dynamical system can be described by a new spectrum of generalized dimensions. Comparison with the usual multifractal analysis of measures is presented, and the difference between the two corresponding sets of dimensions is established. Theoretical analysis and numerical examples of dynamical systems in the class of iterated functions are presented.     

Symbolic dynamics of one-dimensional maps: Entropies,finite precision,and noise

In the study of nonlinear physical systems, one encounters apparently random or chaotic behavior, although the systems may be completely deterministic. Applying techniques from symbolic dynamics to maps of the interval, we compute two measures of chaotic behavior commonly employed in dynamical systems theory: the topological and metric entropies. For the quadratic logistic equation, we find that the metric entropy converges very slowly in comparison to maps which are strictly hyperbolic. The effects of finite precision arithmetric and external noise on chaotic behavior are characterized with the symbolic dynamics entropies. Finally, we discuss the relationship of these measures of chaos to algorithmic complexity, and use algorithmic information theory as a framework to discuss the construction of models for chaotic dynamics.     

Scaling laws for invariant measures on hyperbolic and nonhyperbolic atractors

The analysis of dynamical systems in terms of spectra of singularities is extended to higher dimensions and to nonhyperbolic systems. Prominent roles in our approach are played by the generalized partial dimensions of the invariant measure and by the distribution of effective Liapunov exponents. For hyperbolic attractors, the latter determines the metric entropies and provides one constraint on the partial dimensions. For nonhyperbolic attractors, there are important modifications. We discuss them for the examples of the logistic and Hénon map. We show, in particular, that the generalized dimensions have singularities with noncontinuous derivative, similar to first-order phase transitions in statistical mechanics.     

Entropy production and time asymmetry in nonequilibrium fluctuations

The time-reversal symmetry of nonequilibrium fluctuations is experimentally investigated in two out-of-equilibrium systems: namely, a Brownian particle in a trap moving at constant speed and an electric circuit with an imposed mean current. The dynamical randomness of their nonequilibrium fluctuations is characterized in terms of the standard and time-reversed entropies per unit time of dynamical systems theory. We present experimental results showing that their difference equals the thermodynamic entropy production in units of Boltzmann's constant.     

Lyapunov Spectrum of Asymptotically Sub-additive Potentials

For general asymptotically sub-additive potentials (resp. asymptotically additive potentials) on general topological dynamical systems, we establish some variational relations between the topological entropy of the level sets of Lyapunov exponents, measure-theoretic entropies and topological pressures in this general situation. Most of our results are obtained without the assumption of the existence of unique equilibrium measures or the differentiability of pressure functions. Some examples are constructed to illustrate the irregularity and the complexity of multifractal behaviors in the sub-additive case and in the case that the entropy map is not upper-semi continuous.     

广义Birkhoff系统的Birkhoff对称性与守恒量

研究广义Birkhoff系统的Birkhoff对称性问题,并给出此情形下相应的守恒量.将力学系统的等效Lagrange函数的一个定理推广到广义Birkhoff系统,证明了在一定条件下与两组动力学函数B,R_μ,Λ_μ和B,R_μ,Λ_μ分别给出的广义Birkhoff方程相关联的矩阵Λ 关键词： 广义Birkhoff系统 Birkhoff对称性 守恒量 矩阵迹     

Legendre structure of the thermostatistics theory based on the Sharma–Taneja–Mittal entropy

The statistical proprieties of complex systems can differ deeply for those of classical systems governed by Boltzmann–Gibbs entropy. In particular, the probability distribution function observed in several complex systems shows a power-law behavior in the tail which disagrees with the standard exponential behavior showed by Gibbs distribution. Recently, a two-parameter deformed family of entropies, previously introduced by Sharma, Taneja and Mittal (STM), has been reconsidered in the statistical mechanics framework. Any entropy belonging to this family admits a probability distribution function with an asymptotic power-law behavior. In the present work we investigate the Legendre structure of the thermostatistics theory based on this family of entropies. We introduce some generalized thermodynamical potentials, study their relationships with the entropy and discuss their main proprieties. Specialization of the results to some one-parameter entropies belonging to the STM family are presented.     

Estimating the errors on measured entropy and mutual information

Information entropy and the related quantity mutual information are used extensively as measures of complexity and to identify nonlinearity in dynamical systems. Expressions for the probability distribution of entropies and mutual informations calculated from finite amounts of data exist in the literature but the expressions have seldom been used in the field of nonlinear dynamics. In this paper formulae for estimating the errors on observed information entropies and mutual informations are derived using the standard error analysis familiar to physicists. Their validity is demonstrated by numerical experiment. For illustration the formulae are then used to evaluate the errors on the time-lagged mutual information of the logistic map.     

Generalized Fokker–Planck equations derived from generalized linear nonequilibrium thermodynamics

Recently, Compte and Jou derived nonlinear diffusion equations by applying the principles of linear nonequilibrium thermodynamics to the generalized nonextensive entropy proposed by Tsallis. In line with this study, stochastic processes in isolated and closed systems characterized by arbitrary generalized entropies are considered and evolution equations for the process probability densities are derived. It is shown that linear nonequilibrium thermodynamics based on generalized entropies naturally leads to generalized Fokker–Planck equations.     

Dynamics of Information Entropies of Atom-Field Entangled States Generated via the Jaynes-Cummings Model

In this paper we have studied the dynamical evolution of Shannon information entropies in position and momentum spaces for two classes of(nonstationary)atom-field entangled states,which are obtained via the JaynesCummings model and its generalization.We have focused on the interaction between two-and(1)-type three-level atoms with the single-mode quantized held.The three-dimensional plots of entropy densities in position and momentum spaces are presented versus corresponding coordinates and time,numerically.It is observed that for particular values of the parameters of the systems,the entropy squeezing in position space occurs.Finally,we have shown that the well-known BBM(Beckner,Bialynicki-Birola and Mycielsky)inequality,which is a stronger statement of the Heisenberg uncertainty relation,is properly satisfied.     

An entropy group and its representation in thermodynamics of nonextensive systems

The Abel entropy group and its matrix representation with the general law of nonextensive entropy composition and quadratic nonlinearity are defined. Four types of matrices for the corresponding parametrical entropies are given and geometries for their measures are determined. New two-parameter entropy is introduced, and a distribution is found for equilibrium thermodynamics of nonextensive systems. Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 2, pp. 79–86, February, 2009.     

Symbolic local information transfer

Recently, the permutation-information theoretic approach has been used in a broad range of research fields. In particular, in the study of high-dimensional dynamical systems, it has been shown that this approach can be effective in characterizing global properties, including the complexity of their spatiotemporal dynamics. Here, we show that this approach can also be applied to reveal local spatiotemporal profiles of distributed computations existing at each spatiotemporal point in the system. J. T. Lizier et al. have recently introduced the concept of local information dynamics, which consists of information storage, transfer, and modification. This concept has been intensively studied with regard to cellular automata, and has provided quantitative evidence of several characteristic behaviors observed in the system. In this paper, by focusing on the local information transfer, we demonstrate that the application of the permutation-information theoretic approach, which introduces natural symbolization methods, makes the concept easily extendible to systems that have continuous states. We propose measures called symbolic local transfer entropies, and apply these measures to two test models, the coupled map lattice (CML) system and the Bak-Sneppen model (BS-model), to show their relevance to spatiotemporal systems that have continuous states. In the CML, we demonstrate that it can be successfully used as a spatiotemporal filter to stress a coherent structure buried in the system. In particular, we show that the approach can clearly stress out defect turbulences or Brownian motion of defects from the background, which gives quantitative evidence suggesting that these moving patterns are the information transfer substrate in the spatiotemporal system. We then show that these measures reveal qualitatively different properties from the conventional approach using the sliding window method, and are also robust against external noise. In the BS-model, we demonstrate that these measures can provide novel insight to the model, featuring how symbolic local information transfer is related to the dynamical properties of the elements involved in a spatiotemporal dynamics.     

Multifractal Analysis of Local Entropies for Expansive Homeomorphisms with Specification

In the present paper we study the multifractal spectrum of local entropies. We obtain results, similar to those of the multifractal analysis of pointwise dimensions, but under much weaker assumptions on the dynamical systems. We assume our dynamical system to be defined by an expansive homeomorphism with the specification property. We establish the variational relation between the multifractal spectrum and other thermodynamical characteristics of the dynamical system, including the spectrum of correlation entropies. Received: 22 September 1998 / Accepted: 11 December 1998     

Generalized entropies of chaotic maps and flows: A unified approach

A thermodynamic study of nonlinear dynamical systems, based on the orbits' return times to the elements of a generating partition, is proposed. Its grand canonical nature makes it suitable for application to both maps and flows, including autonomous ones. When specialized to the evaluation of the generalized entropies K(q), this technique reproduces a well-known formula for the metric entropy K(1) and clarifies the relationship between a flow and the associated Poincare maps, beyond the straightforward case of periodically forced nonautonomous systems. Numerical estimates of the topological and metric entropy are presented for the Lorenz and Rossler systems. The analysis has been carried out exclusively by embedding scalar time series, ignoring any further knowledge about the systems, in order to illustrate its usefulness for experimental signals as well. Approximations to the generating partitions have been constructed by locating the unstable periodic orbits of the systems up to order 9. The results agree with independent estimates obtained from suitable averages of the local expansion rates along the unstable manifolds. (c) 1997 American Institute of Physics.     

On a general concept of multifractality: Multifractal spectra for dimensions,entropies, and Lyapunov exponents. Multifractal rigidity

We introduce the mathematical concept of multifractality and describe various multifractal spectra for dynamical systems, including spectra for dimensions and spectra for entropies. We support the study by providing some physical motivation and describing several nontrivial examples. Among them are subshifts of finite type and one-dimensional Markov maps. An essential part of the article is devoted to the concept of multifractal rigidity. In particular, we use the multifractal spectra to obtain a "physical" classification of dynamical systems. For a class of Markov maps, we show that, if the multifractal spectra for dimensions of two maps coincide, then the maps are differentiably equivalent. (c) 1997 American Institute of Physics.     

Nonlinear mean field Fokker-Planck equations. Application to the chemotaxis of biological populations

We study a general class of nonlinear mean field Fokker-Planck equations in relation with an effective generalized thermodynamical (E.G.T.) formalism. We show that these equations describe several physical systems such as: chemotaxis of bacterial populations, Bose-Einstein condensation in the canonical ensemble, porous media, generalized Cahn-Hilliard equations, Kuramoto model, BMF model, Burgers equation, Smoluchowski-Poisson system for self-gravitating Brownian particles, Debye-Hückel theory of electrolytes, two-dimensional turbulence... In particular, we show that nonlinear mean field Fokker-Planck equations can provide generalized Keller-Segel models for the chemotaxis of biological populations. As an example, we introduce a new model of chemotaxis incorporating both effects of anomalous diffusion and exclusion principle (volume filling). Therefore, the notion of generalized thermodynamics can have applications for concrete physical systems. We also consider nonlinear mean field Fokker-Planck equations in phase space and show the passage from the generalized Kramers equation to the generalized Smoluchowski equation in a strong friction limit. Our formalism is simple and illustrated by several explicit examples corresponding to Boltzmann, Tsallis, Fermi-Dirac and Bose-Einstein entropies among others.