Statistics of close visits to the indifferent fixed point of an interval map

We study a dynamical system defined by a map of the interval [0, 1] which has 0 as an indifferent fixed point but is otherwise expanding. We prove that the sequence of successive entrance times in a small neighborhood [0,a] converges in law when suitably normalized to a homogeneous Poisson point process.     

Statistics of Return Times:¶A General Framework and New Applications

In this paper we provide general estimates for the errors between the distribution of the first, and more generally, the K ^th return time (suitably rescaled) and the Poisson law for measurable dynamical systems. In the case that the system exhibits strong mixing properties, these bounds are explicitly expressed in terms of the speed of mixing. Using these approximations, the Poisson law is finally proved to hold for a large class of non hyperbolic systems on the interval. Received: 4 August 1998 / Accepted: 9 March 1999     

Spectral Form Factors of Rectangle Billiards

The Berry–Tabor conjecture asserts that local statistical measures of the eigenvalues λ _j of a “generic” integrable quantum system coincide with those of a Poisson process. We prove that, in the case of a rectangle billiard with random ratio of sides, the sum behaves for τ random and N large like a random walk in the complex plane with a non-Gaussian limit distribution. The expectation value of the distribution is zero; its variance, which is essentially the average pair correlation function, is one, in accordance with the Berry–Tabor conjecture, but all higher moments (≥ 4) diverge. The proof of the existence of the limit distribution uses the mixing property of a dynamical system defined on a product of hyperbolic surfaces. The Berry–Tabor conjecture and the existence of the limit distribution for a fixed generic rectangle are related to an equidistribution conjecture for long horocycles on this product space. Received: 16 February 1998 / Accepted: 24 April 1998     

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.     

Benford’s law: A Poisson perspective

Benford’s law is a counterintuitive statistical law asserting that the distribution of leading digits, taken from a large ensemble of positive numerical values that range over many orders of scale, is logarithmic rather than uniform (as intuition suggests). In this paper we explore Benford’s law from a Poisson perspective, considering ensembles of positive numerical values governed by Poisson-process statistics. We show that this Poisson setting naturally accommodates Benford’s law and: (i) establish a Poisson characterization and a Poisson multidigit-extension of Benford’s law; (ii) study a system-invariant leading-digit distribution which generalizes Benford’s law, and establish a Poisson characterization and a Poisson multidigit-extension of this distribution; (iii) explore the universal emergence of the system-invariant leading-digit distribution, couple this universal emergence to the universal emergence of the Weibull and Fréchet extreme-value distributions, and distinguish the special role of Benford’s law in this universal emergence; (iv) study the continued-fractions counterpart of the system-invariant leading-digit distribution, and establish a Poisson characterization of this distribution; and (v) unveil the elemental connection between the system-invariant leading-digit distribution and its continued-fractions counterpart. This paper presents a panoramic Poisson approach to Benford’s law, to its system-invariant generalization, and to its continued-fractions counterpart.     

Asymptotic dynamics,non-critical and critical fluctuations for a geometric long-range interacting model

We study the dynamics of geometric spin system on the torus with long-range interaction. As the number of particles goes to infinity, the process converges to a deterministic, dynamical magnetization field that satisfies an Euler equation (law of large numbers). Its stable steady states are related to the limits of the equilibrium measures (Gibbs states) of the finite particle system. A related equation holds for the magnetization densities, for which the property of propagation of chaos also is established. We prove a dynamical central limit theorem with an infinite-dimensional Ornstein-Uhlenbeck process as a limiting fluctuation process. At the critical temperature of a ferromagnetic phase transition, both a tighter quantity scaling and a time scaling is required to obtain convergence to a one-dimensional critical fluctuation process with constant magnetization fields, which has a non-Gaussian invariant distribution. Similarly, at the phase transition to an antiferromagnetic state with frequencyp ₀, the fluctuation process with critical scaling converges to a two-dimensional critical fluctuation process, which consists of fields with frequencyp ₀ and has a non-Gaussian invariant distribution on these fields. Finally, we compute the critical fluctuation process in the infinite particle limit at a triple point, where a ferromagnetic and an antiferromagnetic phase transition coincide.Work supported by Deutsche Forschungsgemeinschaft     

Underlying probability distributions of Planck's radiation law

The derivation of Planck's radiation law can be considered as a transformation of a thermodynamic relation for black-body radiation into a fundamental relation in which the error law is the negative binomial distribution. In both limiting frequency ranges it transforms into Poisson distributions; in the Wien limit, it is the distribution of the number of photons, whose most probable value is given by Boltzmann's expression, while in the Rayleigh-Jeans limit, it is the distribution of the number of Planck oscillators. In the general case, they are Bernoullian random variables. In the Rayleigh-Jeans limit, the probability of determining the number of oscillators in a given frequency interval for a fixed value of the energy can be inverted to determining the probability of the energy for a fixed number of oscillators. The probability density is that of the canonical ensemble.     

Theory of tensor invariants of integrable Hamiltonian systems. I. Incompatible Poisson structures

This paper develops a new theory of tensor invariants of a completely integrable non-degenerate Hamiltonian system on a smooth manifoldM ⁿ. The central objects in this theory are supplementary invariant Poisson structuresP _c which are incompatable with the original Poisson structureP ₁ for this Hamiltonian system. A complete classification of invariant Poisson structures is derived in a neighbourhood of an invariant toroidal domain. This classification resolves the well-known Inverse Problem that was brought into prominence by Magri's 1978 paper deveoted to the theory of compatible Poisson structures. Applications connected with the KAM theory, with the Kepler problem, with the basic integrable problem of celestial mechanics, and with the harmonic oscillator are pointed out. A cohomology is defined for dynamical systems on smooth manifolds. The physically motivated concepts of dynamical compatibility and strong dynamical compatibility of pairs of Poisson structures are introduced to study the diversity of pairs of Poisson structures incompatible in Magri's sense. It is proved that if a dynamical systemV preserves two strongly dynamically compatible Poisson structuresP ₁ andP ₂ in a general position then this system is completely integrable. Such a systemV generates a hierarchy of integrable dynamical systems which in general are not Hamiltonian neither with respect toP ₁ nor with respect toP ₂. Necessary conditions for dynamical compatibility and for strong dynamical compatibility are derived which connect these global properties with new local invariants of an arbitrary pair of incompatible Poisson structures.Supported by NSERC grant OGPIN 337.     

Nonequilibrium Phase Transition in a Model of Diffusion, Aggregation, and Fragmentation

We study the nonequilibrium phase transition in a model of aggregation of masses allowing for diffusion, aggregation on contact, and fragmentation. The model undergoes a dynamical phase transition in all dimensions. The steady-state mass distribution decays exponentially for large mass in one phase. In the other phase, the mass distribution decays as a power law accompanied, in addition, by the formation of an infinite aggregate. The model is solved exactly within a mean-field approximation which keeps track of the distribution of masses. In one dimension, by mapping to an equivalent lattice gas model, exact steady states are obtained in two extreme limits of the parameter space. Critical exponents and the phase diagram are obtained numerically in one dimension. We also study the time-dependent fluctuations in an equivalent interface model in (1+1) dimension and compute the roughness exponent and the dynamical exponent z analytically in some limits and numerically otherwise. Two new fixed points of interface fluctuations in (1+1) dimension are identified. We also generalize our model to include arbitrary fragmentation kernels and solve the steady states exactly for some special choices of these kernels via mappings to other solvable models of statistical mechanics.     

Burstiness and fractional diffusion on complex networks

Many dynamical processes on real world networks display complex temporal patterns as, forinstance, a fat-tailed distribution of inter-events times, leading to heterogeneouswaiting times between events. In this work, we focus on distributions whose averageinter-event time diverges, and study its impact on the dynamics of random walkers onnetworks. The process can naturally be described, in the long time limit, in terms ofRiemann-Liouville fractional derivatives. We show that all the dynamical modes possess, inthe asymptotic regime, the same power law relaxation, which implies that the dynamics doesnot exhibit time-scale separation between modes, and that no mode can be neglected versusanother one, even for long times. Our results are then confirmed by numericalsimulations.     

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.     

Continuum Limit of the Volterra Model, Separation of Variables and Non-Standard Realizations of the Virasoro Poisson Bracket

The classical Volterra model, equipped with the Faddeev-Takhtajan Poisson bracket provides a lattice version of the Virasoro algebra. The Volterra model being integrable, we can express the dynamical variables in terms of the so-called separated variables. Taking the continuum limit of these formulae, we obtain the Virasoro generators written as determinants of infinite matrices, the elements of which are constructed with a set of points lying on an infinite genus Riemann surface. The coordinates of these points are separated variables for an infinite set of Poisson commuting quantities including L ₀. The scaling limit of the eigenvector can also be calculated explicitly, so that the associated Schroedinger equation is in fact exactly solvable.     

The nature of bursting noises,stochastic resonance and deterministic chaos in excitable neurons

We study the Hindmarsh-Rose model of excitable neurons and show that in the asymptotic limit this monostable model can possess some kind of dynamical bistability: small-amplitude quasiharmonic and large-amplitude relaxational oscillations can be simultaneously excited and their formation is accompanied by a narrow hysteresis. We show that bursting noises, stochastic resonance and deterministic chaos are determined by random transitions between these two dynamical states under slow and small changes of one of the model variables (z). We find that these effects take place even for such model parameters when hysteresis transforms into a step and they disappear when this step is smoothed out enough. We analyze some characteristics and conditions of formation of the deterministic chaos. We emphasize that such dynamical bistability and the effects related to it are universal phenomena and occur in a wide class of dynamical systems of different nature including brusselator.     

Feedback write scheme for memristive switching devices

In nanoscale memristive switching devices, the statistical distribution of resistance values and other relevant parameters for device operation often exhibits a lognormal distribution, causing large fluctuations of memristive analog state variables after each switching event, which may be problematic for digital nonvolatile memory applications. The state variable w in such devices has been proposed to be the length of an undoped semiconductor region along the thickness of the thin film that acts as a tunnel barrier for electronic transport across it. The dynamical behavior of w is governed by the drift diffusion of ionized dopants such as oxygen vacancies. Making an analogy to scanning tunneling microscopes (STM), a closed-loop write scheme using current feedback is proposed to switch the memristive devices in a controlled manner. An integrated closed-loop current driver circuit for switching a bipolar memristive device is designed and simulated. The estimated upper limit of the feedback loop bandwidth is in the order of 100 MHz. We applied a SPICE model built upon the TiO₂ memristive switching dynamics to simulate the single-device write operation and found the closed-loop write scheme caused a narrowing of the statistical distribution of the state variable w.     

Free Path Lengths in Quasi Crystals

The Lorentz gas is a model for a cloud of point particles (electrons) in a distribution of scatterers in space. The scatterers are often assumed to be spherical with a fixed diameter d, and the point particles move with constant velocity between the scatterers, and are specularly reflected when hitting a scatterer. There is no interaction between point particles. An interesting question concerns the distribution of free path lengths, i.e. the distance a point particle moves between the scattering events, and how this distribution scales with scatterer diameter, scatterer density and the distribution of the scatterers. It is by now well known that in the so-called Boltzmann–Grad limit, a Poisson distribution of scatterers leads to an exponential distribution of free path lengths, whereas if the scatterer distribution is periodic, the free path length distribution asymptotically behaves as a power law.     

A model for the multiplicity distribution in pp-collisions from 5 to 300 GeV/c

The multiplicity distribution of charged particles in pp-collisions in the range 5–300 GeV/c is explained on the basis of a dynamical model which leads to a specific mixture of Poisson distributions.     

Poisson Involutions, Spin Calogero-Moser Systems Associated with Symmetric Lie Subalgebras and the Symmetric Space Spin Ruijsenaars-Schneider Models

We develop a general scheme to construct integrable systems starting from realizations in symmetric coboundary dynamical Lie algebroids and symmetric coboundary dynamical Poisson groupoids. The method is based on the successive use of Dirac reduction and Poisson reduction. Then we show that certain spin Calogero-Moser systems associated with symmetric Lie subalgebras can be studied in this fashion. We also consider some spin-generalized Ruisjenaars-Schneider equations which correspond to the N-soliton solutions of affine Toda field theory. In this case, we show how the equations are obtained from the Dirac reduction of some Hamiltonian system on a symmetric coboundary dynamical Poisson groupoid.     

The Compound Poisson Limit Ruling Periodic Extreme Behaviour of Non-Uniformly Hyperbolic Dynamics

We prove that the distributional limit of the normalised number of returns to small neighbourhoods of periodic points of certain non-uniformly hyperbolic dynamical systems is compound Poisson. The returns to small balls around a fixed point in the phase space correspond to the occurrence of rare events, or exceedances of high thresholds, so that there is a connection between the laws of Return Times Statistics and Extreme Value Laws. The fact that the fixed point in the phase space is a repelling periodic point implies that there is a tendency for the exceedances to appear in clusters whose average sizes is given by the Extremal Index, which depends on the expansion of the system at the periodic point. We recall that for generic points, the exceedances, in the limit, are singular and occur at Poisson times. However, around periodic points, the picture is different: the respective point processes of exceedances converge to a compound Poisson process, so instead of single exceedances, we have entire clusters of exceedances occurring at Poisson times with a geometric distribution ruling its multiplicity. The systems to which our results apply include: general piecewise expanding maps of the interval (Rychlik maps), maps with indifferent fixed points (Manneville-Pomeau maps) and Benedicks-Carleson quadratic maps.     

Quantization of Classical Dynamical <Emphasis Type="Italic">r</Emphasis>-Matrices with Nonabelian Base

We construct some classes of dynamical r-matrices over a nonabelian base, and quantize some of them by constructing dynamical (pseudo)twists in the sense of Xu. This way, we obtain quantizations of r-matrices obtained in earlier work of the second author with Schiffmann and Varchenko. A part of our construction may be viewed as a generalization of the Donin-Mudrov nonabelian fusion construction. We apply these results to the construction of equivariant star-products on Poisson homogeneous spaces, which include some homogeneous spaces introduced by De Concini.Acknowledgement The authors thank J. Donin and A. Mudrov for references. P.E. is grateful to C. De Concini for a very useful discussion on Poisson homogeneous spaces related to automorphisms, which was crucial for writing Sect. 6 of this paper. P.E. is indebted to IRMA (Strasbourg) for hospitality. The work of P.E. was partially supported by the NSF grant DMS-9988796.     

Unambiguous Quantization from the Maximum Classical Correspondence that Is Self-consistent: The Slightly Stronger Canonical Commutation Rule Dirac Missed

Dirac’s identification of the quantum analog of the Poisson bracket with the commutator is reviewed, as is the threat of self-inconsistent overdetermination of the quantization of classical dynamical variables which drove him to restrict the assumption of correspondence between quantum and classical Poisson brackets to embrace only the Cartesian components of the phase space vector. Dirac’s canonical commutation rule fails to determine the order of noncommuting factors within quantized classical dynamical variables, but does imply the quantum/classical correspondence of Poisson brackets between any linear function of phase space and the sum of an arbitrary function of only configuration space with one of only momentum space. Since every linear function of phase space is itself such a sum, it is worth checking whether the assumption of quantum/classical correspondence of Poisson brackets for all such sums is still self-consistent. Not only is that so, but this slightly stronger canonical commutation rule also unambiguously determines the order of noncommuting factors within quantized dynamical variables in accord with the 1925 Born-Jordan quantization surmise, thus replicating the results of the Hamiltonian path integral, a fact first realized by E.H. Kerner. Born-Jordan quantization validates the generalized Ehrenfest theorem, but has no inverse, which disallows the disturbing features of the poorly physically motivated invertible Weyl quantization, i.e., its unique deterministic classical “shadow world” which can manifest negative densities in phase space.