Statistics of Poincaré recurrences for maps with integrable and ergodic components

Recurrence gives powerful tools to investigate the statistical properties of dynamical systems. We present in this paper some applications of the statistics of first return times to characterize the mixed behavior of dynamical systems in which chaotic and regular motion coexist. Our analysis is local: we take a neighborhood A of a point x and consider the conditional distribution of the points leaving A and for which the first return to A, suitably normalized, is bigger than t. When the measure of A shrinks to zero the distribution converges to the exponential e(-t) for almost any point x, if the system is mixing and the set A is a ball or a cylinder. We consider instead a system, a skew integrable map of the cylinder, which is not ergodic and has zero entropy. This map describes a shear flow and has a local mixing property. We rigorously prove that the statistics of first return is of polynomial type around the fixed points and we generalize around other points with numerical computations. The result could be extended to quasi-integrable area preserving maps such as the standard map for small coupling. We then analyze the distribution of return times in a region which is composed by two invariants subdomains: one with a mixing dynamics and the other with an integrable dynamics given by our shear flow. We show that the statistics of first return in this mixed region is asymptotically given by the exponential law, but this limit is attained by an intermediate regime where exponential and polynomial laws are linearly superposed and weighted by some factors which are proportional to the relative sizes of the chaotic and regular regions. The result on the statistics of first return times for mixed regions in the phase space can provide a basis to analyze such a property for area preserving maps in mixed regions even when a rigorous result is not available. To this end we present numerical investigations on the standard map which confirm the results of the model.     

The Photon-Number Distribution of the Lasers with Binomial Pump Model

A method on how to take into account pump statistics and spontaneous atomic decays when calculating the photon-number distribution of lasers in photon-number representation is presented. The intracavity phonon-number and extracavity photocount distributions with binomial pump and spontaneous atomic decays are calculated. In two limit cases, the results of calculations transit respectively to the photon-number distributions of systems with common Poissonian pump and regular pump, and both with the spontaneous atomic decays being suppressed.     

On codings and dynamics of planar piecewise rotations

In this paper, we investigate codings and dynamics of the piecewise rotations belonging to a sub-class of piecewise isometries. We show that the irrational set is empty for some piecewise rational rotations under some assumptions, while a piecewise irrational rotation has at least one irrational coding by extending the definition of the coding map onto the entire phase space. We further prove that the cell corresponding to irrational coding is a single point set for a piecewise irrational rotation.     

Regular-to-chaotic tunneling rates using a fictitious integrable system

We derive a formula predicting dynamical tunneling rates from regular states to the chaotic sea in systems with a mixed phase space. Our approach is based on the introduction of a fictitious integrable system that resembles the regular dynamics within the island. For the standard map and other kicked systems we find agreement with numerical results for all regular states in a regime where resonance-assisted tunneling is not relevant.     

Statistical universality and the method of Poissonian randomizations

In this paper we demonstrate the remarkable effectiveness of Poissonian randomizations in the generation of statistical universality. We do so via a highly versatile spatio-statistical model in which points are randomly scattered, according to a Poisson process, across a general metric space. The points have general independent and identically distributed random physical characteristics. A probe is positioned in space, and is affected by the points. The effect of a given point on the probe is a function of the physical characteristic of the point and the distance of the point from the probe. We determine the classes of Poissonian randomizations – i.e., the spatial Poissonian scatterings of the points – that render the effects of the points invariant with respect to the physical characteristics of the points. These Poissonian randomizations have intrinsic power-law structures, yield statistical robustness, and generate universal statistics including Lévy distributions and extreme-value distributions. In effect, our results establish how “fractal” spatial geometries lead to statistical universality.     

Scale-invariance of random populations: From Paretian to Poissonian fractality

Random populations represented by stochastically scattered collections of real-valued points are abundant across many fields of science. Fractality, in the context of random populations, is conventionally associated with a Paretian distribution of the population's values.Using a Poissonian approach to the modeling of random populations, we introduce a definition of “Poissonian fractality” based on the notion of scale-invariance. This definition leads to the characterization of four different classes of Fractal Poissonian Populations—three of which being non-Paretian objects. The Fractal Poissonian Populations characterized turn out to be the unique fixed points of natural renormalizations, and turn out to be intimately related to Extreme Value distributions and to Lévy Stable distributions.     

Theoretical derivation of 1/f noise in quantum chaos

It was recently conjectured that 1/f noise is a fundamental characteristic of spectral fluctuations in chaotic quantum systems. This conjecture is based on the power spectrum behavior of the excitation energy fluctuations, which is different for chaotic and integrable systems. Using random matrix theory, we derive theoretical expressions that explain without free parameters the universal behavior of the excitation energy fluctuations power spectrum. The theory gives excellent agreement with numerical calculations and reproduces to a good approximation the 1/f (1/f(2)) power law characteristic of chaotic (integrable) systems. Moreover, the theoretical results are valid for semiclassical systems as well.     

Universality of level spacing distributions in classical chaos

We suggest that random matrix theory applied to a matrix of lengths of classical trajectories can be used in classical billiards to distinguish chaotic from non-chaotic behavior. We consider in 2D the integrable circular and rectangular billiard, the chaotic cardioid, Sinai and stadium billiard as well as mixed billiards from the Limaçon/Robnik family. From the spectrum of the length matrix we compute the level spacing distribution, the spectral auto-correlation and spectral rigidity. We observe non-generic (Dirac comb) behavior in the integrable case and Wignerian behavior in the chaotic case. For the Robnik billiard close to the circle the distribution approaches a Poissonian distribution. The length matrix elements of chaotic billiards display approximate GOE behavior. Our findings provide evidence for universality of level fluctuations—known from quantum chaos—to hold also in classical physics.     

New class of level statistics in correlated disordered chains

We study the properties of the level statistics of 1D disordered systems with long-range spatial correlations. We find a threshold value in the degree of correlations below which in the limit of large system size the level statistics follows a Poisson distribution (as expected for 1D uncorrelated-disordered systems), and above which the level statistics is described by a new class of distribution functions. At the threshold, we find that with increasing system size, the standard deviation of the function describing the level statistics converges to the standard deviation of the Poissonian distribution as a power law. Above the threshold we find that the level statistics is characterized by different functional forms for different degrees of correlations.     

Averaged number of visits

We introduce a new indicator for dynamical systems, namely the averaged number of visits, to estimate the frequency of visits in small regions when a map is iterated up to the inverse of the measure of this region. We compute this quantity analytically and numerically for various systems and we show that it depends on the ergodic properties of the systems and on their topological properties, such as the presence of periodic points.     

Levelstatistics and stochasticity in a driven quantum system

We investigate quantum properties of one anisotropic spin driven by an external time-dependent magnetic field which shows a transition from regular to irregular dynamics with increasing field strength in the classical limit. In particular we study the statistical properties of the quasi-spectrum. Our results support the conjecture that Poisson- and GOE-statistics are to be associated with integrable and nonintegrable systems resp. in the semiclassical limit. Approaching the quantum case we observe significant deviations from GOE statistics.     

Intersection properties of invariant manifolds in certain twist maps

We consider the spaceN ofC ² twist maps that satisfy the following requirements. The action is the sum of a purely quadratic term and a periodic potential times a constantk (hereafter called the nonlinearity). The potential restricted to the unit circle is bimodal, i.e. has one local minimum and one local maximum. The following statements are proven for maps inN with nonlinearityk large enough. The intersection of the unstable and stable invariant manifolds to the hyperbolic minimizing periodic points contains minimizing homoclinic points. Consider two finite pieces of these manifolds that connect two adjacent homoclinic minimizing points (hereafter called fundamental domains). We prove that all such fundamental domains have precisely one point in their intersection (the Single Intersection theorem). In addition, we show that limit points of minimizing points are recurrent, which implies that Aubry Mather sets (with irrational rotation number) are contained in diamonds formed by local stable and unstable manifolds of nearby minimizing periodic orbits (the Diamond Configuration theorem). Another corollary concerns the intersection of the minimax orbits with certain symmetry lines of the map.     

Bäcklund transformations for finite-dimensional integrable systems: a geometric approach

We present a geometric construction of Bäcklund transformations and discretizations for a large class of algebraic completely integrable systems. To be more precise, we construct families of Bäcklund transformations, which are naturally parameterized by the points on the spectral curve(s) of the system. The key idea is that a point on the curve determines, through the Abel–Jacobi map, a vector on its Jacobian which determines a translation on the corresponding level set of the integrals (the generic level set of an algebraic completely integrable systems has a group structure). Globalizing this construction we find (possibly multi-valued, as is very common for Bäcklund transformations) maps which preserve the integrals of the system, they map solutions to solutions and they are symplectic maps (or, more generally, Poisson maps). We show that these have the spectrality property, a property of Bäcklund transformations that was recently introduced. Moreover, we recover Bäcklund transformations and discretizations which have up to now been constructed by ad hoc methods, and we find Bäcklund transformations and discretizations for other integrable systems. We also introduce another approach, using pairs of normalizations of eigenvectors of Lax operators and we explain how our two methods are related through the method of separation of variables.     

Geodesics on the ellipsoid and monodromy

After reviewing the properties of the geodesic flow on the three-dimensional ellipsoid with distinct semi-axes, we investigate the three-dimensional ellipsoid with the two middle semi-axes equal, corresponding to a Hamiltonian invariant under rotations. The system is Liouville integrable, and symmetry reduction leads to a (singular) system on a two-dimensional ellipsoid with an additional potential and with a hard billiard wall inserted in the middle coordinate plane. We show that the regular part of the image of the energy–momentum map is not simply connected and there is an isolated critical value for zero angular momentum. The singular fibre of the isolated singular value is a doubly pinched torus multiplied by a circle. This circle is not a group orbit of the symmetry group, and thus analysis of this fibre is non-trivial. Finally we show that the system has a non-trivial monodromy, and consequently does not admit single-valued globally smooth action variables.     

Continuum percolation conductivity exponents in restricted domains

Conductivity behavior of continuum percolation in restricted two-dimensional domains is simulated by considering systems of randomly distributed disks. The domain is restricted in that conducting objects are permitted to lie in only a portion of the domain. Such a restricted domain might better approximate some natural systems. Simulations of two-dimensional systems, based on three distributions of local conductances, are examined and found to demonstrate a power-law behavior with conductivity exponents smaller than those arising in regular lattice and continuum percolation     

QUATERNIONIC INTEGRABILITY

Standard (Arnold–Liouville) integrable systems are intimately related to complex rotations. One can define a generalization of these, sharing many of their properties, where complex rotations are replaced by quaternionic ones, and more generally by the action of a Clifford group. Such a generalization is not limited to integrable systems but — in the quaternionic case — goes over to a generalization of standard Hamilton dynamics.     

Universal Hitting Time Statistics for Integrable Flows

The perceived randomness in the time evolution of “chaotic” dynamical systems can be characterized by universal probabilistic limit laws, which do not depend on the fine features of the individual system. One important example is the Poisson law for the times at which a particle with random initial data hits a small set. This was proved in various settings for dynamical systems with strong mixing properties. The key result of the present study is that, despite the absence of mixing, the hitting times of integrable flows also satisfy universal limit laws which are, however, not Poisson. We describe the limit distributions for “generic” integrable flows and a natural class of target sets, and illustrate our findings with two examples: the dynamics in central force fields and ellipse billiards. The convergence of the hitting time process follows from a new equidistribution theorem in the space of lattices, which is of independent interest. Its proof exploits Ratner’s measure classification theorem for unipotent flows, and extends earlier work of Elkies and McMullen.     

Poissonian and non-Poissonian Voronoi diagrams with application to the aggregation of molecules

The distributions that regulate the spatial domains of the Poissonian Voronoi diagrams are discussed adopting the sum of gamma variate of argument two. The distributions that arise from the product and quotient of two gamma variates of argument two are also derived. Three examples of non-Poissonian seeds for the Voronoi Diagrams are discussed. The developed algorithm allows the simulation of an aggregation of methanol and water.