Universal Behaviour of Extreme Value Statistics for Selected Observables of Dynamical Systems

The main results of the extreme value theory developed for the investigation of the observables of dynamical systems rely, up to now, on the block maxima approach. In this framework, extremes are identified with the block maxima of the time series of the chosen observable, in the limit of infinitely long blocks. It has been proved that, assuming suitable mixing conditions for the underlying dynamical systems, the extremes of a specific class of observables are distributed according to the so called Generalised Extreme Value (GEV) distribution. Direct calculations show that in the case of quasi-periodic dynamics the block maxima are not distributed according to the GEV distribution. In this paper we show that considering the exceedances over a given threshold instead of the block-maxima approach it is possible to obtain a Generalised Pareto Distribution also for extremes computed in systems which do not satisfy mixing conditions. Requiring that the invariant measure locally scales with a well defined exponent—the local dimension—, we show that the limiting distribution for the exceedances of the observables previously studied with the block maxima approach is a Generalised Pareto distribution where the parameters depend only on the local dimensions and the values of the threshold but not on the number of observations considered. We also provide connections with the results obtained with the block maxima approach. In order to provide further support to our findings, we present the results of numerical experiments carried out considering the well-known Chirikov standard map.     

The dynamical instability of nonrelativistic many-body systems

From computations in an exactly solvable many-body dynamical model we argue that, quite generally, a nonrelativistic quantum mechanics of infinitely many interacting particles must admit states without a global time evolution; equivalently, that the (quasi-local) observables of any such theory are not preserved in time by the Heisenberg dynamics. Our analysis is based on a dynamical instability common to interacting finite-particle systems.Work supported in part by the National Science Foundation     

Quantum Reality and Measurement: A?Quantum?Logical Approach

The recently established universal uncertainty principle revealed that two nowhere commuting observables can be measured simultaneously in some state, whereas they have no joint probability distribution in any state. Thus, one measuring apparatus can simultaneously measure two observables that have no simultaneous reality. In order to reconcile this discrepancy, an approach based on quantum logic is proposed to establish the relation between quantum reality and measurement. We provide a language speaking of values of observables independent of measurement based on quantum logic and we construct in this language the state-dependent notions of joint determinateness, value identity, and simultaneous measurability. This naturally provides a contextual interpretation, in which we can safely claim such a statement that one measuring apparatus measures one observable in one context and simultaneously it measures another nowhere commuting observable in another incompatible context.     

An Algebraic Characterization of Vacuum States in Minkowski Space. II. Continuity Aspects

An algebraic characterization of vacuum states in Minkowski space is given which relies on recently proposed conditions of geometric modular action and modular stability for algebras of observables associated with wedge-shaped regions. In contrast to previous work, continuity properties of these algebras are not assumed but derived from their inclusion structure. Moreover, a unique continuous unitary representation of spacetime translations is constructed from these data. Thus, the dynamics of relativistic quantum systems in Minkowski space is encoded in the observables and state and requires no prior assumption about any action of the spacetime symmetry group upon these quantities.     

Classical Properties of Infinite Quantum Open Systems

Long time asymptotic properties of a class of environmentally induced dynamical semigroups on arbitrary von Neumann algebras are discussed. Such a semigroup selects observables, called effective observables, which are immune to the process of decoherence and so evolve in a reversible automorphic way. In particular, it is shown that effective observables of the quantum system in the thermodynamic limit, subjected to a specific interaction with another quantum system, obey classical dynamics.This work was supported by the KBN research grant no 5P03B 081 21     

States and automorphism groups of operator algebras

Suppose that a group of automorphisms of a von Neumann algebraM, fixes the center elementwise. We show that if this group commutes with the modular (KMS) automorphism group associated with a normal faithful state onM, then this state is left invariant by the group of automorphisms. As a result we obtain a “noncommutative” ergodic theorem. The discrete spectrum of an abelian unitary group acting as automorphisms ofM is completely characterized by elements inM. We discuss the KMS condition on the CAR algebra with respect to quasi-free automorphisms and gauge invariant generalized free states. We also obtain a necessary and sufficient condition for the CAR algebra and a quasi-free automorphism group to be η-abelian.     

Investigating nonlinear dynamics from time series: The influence of symmetries and the choice of observables

When a dynamical system is investigated from a time series, one of the most challenging problems is to obtain a model that reproduces the underlying dynamics. Many papers have been devoted to this problem but very few have considered the influence of symmetries in the original system and the choice of the observable. Indeed, it is well known that there are usually some variables that provide a better representation of the underlying dynamics and, consequently, a global model can be obtained with less difficulties starting from such variables. This is connected to the problem of observing the dynamical system from a single time series. The roots of the nonequivalence between the dynamical variables will be investigated in a more systematic way using previously defined observability indices. It turns out that there are two important ingredients which are the complexity of the coupling between the dynamical variables and the symmetry properties of the original system. As will be mentioned, symmetries and the choice of observables also has important consequences in other problems such as synchronization of nonlinear oscillators. (c) 2002 American Institute of Physics.     

Dynamics and potentials

A dynamics (i.e. a one-parameter group of automorphisms) of a system described by a C*-algebra with a local structure in terms of C*-subalgebras A(I) for local domains I of the physical space (a discrete lattice) is normally constructed out of potentials P(I), each of which is a self-adjoint element of the subalgebra A(I), such that the the first time derivative of the dynamical change of any local observable A is i times the convergent sum of the commutator [P(I), A] over all finite regions I. We will invert this relation under the assumption (obviously assumed in the usual approach) that local observables all have the first time derivative, i.e. we prove the existence of potentials for any given dynamics satisfying the above-stated condition. Furthermore, by imposing a further condition for the potential P(I) to be chosen for each I that it does not have a portion which can be shifted to potentials for any proper subset of I, we also show (1) the existence, (2) uniqueness, (3) an automatic convergence property for the sum over I, and (4) a quite convenient property for the chosen potential. The so-obtained properties (3) and (4) are not assumed and are very useful, though they were never noticed nor used before.     

Dynamics of charged anisotropic spherical collapse in energy-momentum squared gravity

This paper investigates the dynamics of charged spherical collapse with anisotropic matter configuration in the context of energy-momentum squared gravity. This newly developed proposal resolves the big-bang singularity and yields the physically viable cosmological results in the early time universe. We establish dynamical equations through Misner-Sharp technique and analyze the effects of charge, anisotropy, effective matter variables and dark source terms on the collapse rate. A relation between Weyl scalar, fluid parameters and dark source terms is also established. The spacetime is not conformally flat due to the presence of anisotropic pressure, multivariate functions and their derivatives. In order to obtain conformally flat spacetime, we consider a specific model of this gravity, neglect the impact of charge and assume the isotropic matter distribution which yields homogeneity of the energy density and conformally flat spacetime. We conclude that positive dark source terms, anisotropy and charge yield the action of a repulsive force which enhances the stability of the system and hence diminishes the collapse rate.     

Spectral Automorphisms in Quantum Logics

In quantum mechanics, the Hilbert space formalism might be physically justified in terms of some axioms based on the orthomodular lattice (OML) mathematical structure (Piron in Foundations of Quantum Physics, Benjamin, Reading, 1976). We intend to investigate the extent to which some fundamental physical facts can be described in the more general framework of OMLs, without the support of Hilbert space-specific tools. We consider the study of lattice automorphisms properties as a “substitute” for Hilbert space techniques in investigating the spectral properties of observables. This is why we introduce the notion of spectral automorphism of an OML. Properties of spectral automorphisms and of their spectra are studied. We prove that the presence of nontrivial spectral automorphisms allow us to distinguish between classical and nonclassical theories. We also prove, for finite dimensional OMLs, that for every spectral automorphism there is a basis of invariant atoms. This is an analogue of the spectral theorem for unitary operators having purely point spectrum.     

Critical dynamics of the three-dimensional Ising model: A Monte Carlo study

Extensive Monte Carlo simulations have been performed to analyze the dynamical behavior of the three-dimensional Ising model with local dynamics. We have studied the equilibrium correlation functions and the power spectral densities of odd and even observables. The exponential relaxation times have been calculated in the asymptotic one-exponential time region. We find that the critical exponentz=2.09 ±0.02 characterizes the algebraic divergence with lattice size for all observables. The influence of scaling corrections has been analyzed. We have determined integrated relaxation times as well. Their dynamical exponentz _int agrees withz for correlations of the magnetization and its absolute value, but it is different for energy correlations. We have applied a scaling method to analyze the behavior of the correlation functions. This method verifies excellent scaling behavior and yields a dynamical exponentz _scal which perfectly agrees withz.     

Phase synchronization of bursting neural networks with electrical and delayed dynamic chemical couplings

Diffusive electrical connections in neuronal networks are instantaneous, while excitatoryor inhibitory couplings through chemical synapses contain a transmission time-delay.Moreover, chemical synapses are nonlinear dynamical systems whose behavior can bedescribed by nonlinear differential equations. In this work, neuronal networks withdiffusive electrical couplings and time-delayed dynamic chemical couplings are considered.We investigate the effects of distributed time delays on phase synchronization of burstingneurons. We observe that in both excitatory and Inhibitory chemical connections, the phasesynchronization might be enhanced when time-delay is taken into account. This distributedtime delay can induce a variety of phase-coherent dynamical behaviors. We also study thecollective dynamics of network of bursting neurons. The network model presents theso-called Small-World property, encompassing neurons whose dynamics have two time scales(fast and slow time scales). The neuron parameters in such Small-World network, aresupposed to be slightly different such that, there may be synchronization of the bursting(slow) activity if the coupling strengths are large enough. Bounds for the criticalcoupling strengths to obtain burst synchronization in terms of the network structure aregiven. Our studies show that the network synchronizability is improved, as itsheterogeneity is reduced. The roles of synaptic parameters, more precisely those of thecoupling strengths and the network size are also investigated.     

Endomorphism semigroups and lightlike translations

Borchers and Wiesbrock have studied the one-parameter semigroups of endomorphisms of von Neumann algebras that appear as lightlike translations in the theory of algebras of local observables, showing that they automatically transform under the appropriate modular automorphisms as under velocity transformations. Here, these results are abstracted and analyzed as essentially operator-theoretic. Criteria are then established for a spatial derivation of a von Neumann algebra to generate a one-parameter semigroup of endomorphisms, and all of this is combined to establish a von Neumann-algebraic converse to the Borchers and Wiesbrock results. This sort of analysis is then applied to questions of isotony and covariance for local algebras, to show that Poincaré covariance together with a domain condition for the translations can imply isotony.This research was partly supported by a fellowship from the Consiglio Nazionale delle Ricerche.     

Ergodic Theory of Generic Continuous Maps

We study the ergodic properties of generic continuous dynamical systems on compact manifolds. As a main result we prove that generic homeomorphisms have convergent Birkhoff averages under continuous observables at Lebesgue almost every point. In spite of this, when the underlying manifold has dimension greater than one, generic homeomorphisms have no physical measures—a somewhat strange result which stands in sharp contrast to current trends in generic differentiable dynamics. Similar results hold for generic continuous maps. To further explore the mysterious behaviour of C ⁰ generic dynamics, we also study the ergodic properties of continuous maps which are conjugated to expanding circle maps. In this context, generic maps have divergent Birkhoff averages along orbits starting from Lebesgue almost every point.