Almost Sure Invariance Principle for Nonuniformly Hyperbolic Systems

We prove an almost sure invariance principle that is valid for general classes of nonuniformly expanding and nonuniformly hyperbolic dynamical systems. Discrete time systems and flows are covered by this result. In particular, the result applies to the planar periodic Lorentz flow with finite horizon.Statistical limit laws such as the central limit theorem, the law of the iterated logarithm, and their functional versions, are immediate consequences.     

Central Limit Theorems and Invariance Principles¶for Time-One Maps of Hyperbolic Flows

We give a general method for deducing statistical limit laws in situations where rapid decay of correlations has been established. As an application of this method, we obtain new results for time-one maps of hyperbolic flows. In particular, using recent results of Dolgopyat, we prove that many classical limit theorems of probability theory, such as the central limit theorem, the law of the iterated logarithm, and approximation by Brownian motion (almost sure invariance principle), are typically valid for such time-one maps. The central limit theorem for hyperbolic flows goes back to Ratner 1973 and is always valid, irrespective of mixing hypotheses. We give examples which demonstrate that the situation for time-one maps is more delicate than that for hyperbolic flows, illustrating the need for rapid mixing hypotheses. Received: 4 January 2002 / Accepted: 16 February 2002?Published online: 24 July 2002     

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.     

Deterministic Approach to the Kinetic Theory of Gases

In the so-called Bernoulli model of the kinetic theory of gases, where (1) the particles are dimensionless points, (2) they are contained in a cube container, (3) no attractive or exterior forces are acting on them, (4) there is no collision between the particles, (5) the collision against the walls of the container are according to the law of elastic reflection, we deduce from Newtonian mechanics two local probabilistic laws: a Poisson limit law and a central limit theorem. We also prove some global law of large numbers, justifying that “density” and “pressure” are constant. Finally, as a byproduct of our research, we prove the surprising super-uniformity of the typical billiard path in a square.     

The universal macroscopic statistics and phase transitions of rank distributions

We establish a “Central Limit Theorem” for rank distributions, which provides a detailed characterization and classification of their universal macroscopic statistics and phase transitions. The limit theorem is based on the statistical notion of Lorenz curves, and is termed the “Lorenzian Limit Law” (LLL). Applications of the LLL further establish: (i) a statistical explanation for the universal emergence of Pareto’s law in the context of rank distributions; (ii) a statistical classification of universal macroscopic network topologies; (iii) a statistical classification of universal macroscopic socioeconomic states; (iv) a statistical classification of Zipf’s law, and a characterization of the “self-organized criticality” it manifests.     

Recurrence and return times of the Sierpinski carpet

We compute the limit distribution of the recurrence and of the normalizedk th return times to small sets of the Sierpinski carpet with respect to a natural measure defined on it. It is proved that this dynamical system follows the Poisson law, as one could have expected for such schemes. We study different sequences which converge in finite distribution to the Poisson point process. This limit in law is very interesting in ergodic theory, and it seems to appear for chaotic dynamical systems such as the one we study.     

Extreme Value Laws in Dynamical Systems for Non-smooth Observations

We prove the equivalence between the existence of a non-trivial hitting time statistics law and Extreme Value Laws in the case of dynamical systems with measures which are not absolutely continuous with respect to Lebesgue. This is a counterpart to the result of the authors in the absolutely continuous case. Moreover, we prove an equivalent result for returns to dynamically defined cylinders. This allows us to show that we have Extreme Value Laws for various dynamical systems with equilibrium states with good mixing properties. In order to achieve these goals we tailor our observables to the form of the measure at hand.     

Hamiltonian description and quantization of dissipative systems

Dissipative systems are described by a Hamiltonian, combined with a “dynamical matrix” which generalizes the simplectic form of the equations of motion. Criteria for dissipation are given and the examples of a particle with friction and of the Lotka-Volterra model are presented. Quantization is first introduced by translating generalized Poisson brackets into commutators and anticommutators. Then a generalized Schrödinger equation expressed by a dynamical matrix is constructed and discussed.     

Long-wavelength excitations and the Goldstone theorem in many-particle systems with “broken symmetries”

Long-wavelength excitations in ferro-, antiferromagnets, superfluids and crystals are studied in connection with the nonrelativistic Goldstone theorem. It is shown that, except in the example of superfluids, the statement of “broken symmetry” together with the local conservation laws implies the existence of gapless excitations in the limit of vanishing momenta (magnons, phonons). For this purpose a generalized version of Bogoliubov's 1/k²-theorem is derived. This theorem is valid for all examples considered in this paper and yields long-range order in certain correlation-functions which are specific for condensed systems with broken symmetry. It is stressed that the longwavelength density-excitation spectra in quantum liquids are insensitive to whether gauge symmetry is broken or not. They are controlled by the range of the interactionpotential. Therefore these density-excitations do not necessarily provide an argument pro or contra the nonrelativistic Goldstone theorem.     

Limit Theorems for Dispersing Billiards with Cusps

Dispersing billiards with cusps are deterministic dynamical systems with a mild degree of chaos, exhibiting “intermittent” behavior that alternates between regular and chaotic patterns. Their statistical properties are therefore weak and delicate. They are characterized by a slow (power-law) decay of correlations, and as a result the classical central limit theorem fails. We prove that a non-classical central limit theorem holds, with a scaling factor of \({\sqrt{n\log n}}\) replacing the standard \({\sqrt{n}}\) . We also derive the respective Weak Invariance Principle, and we identify the class of observables for which the classical CLT still holds.     

Nonlinear canonical transformations in the coupling between degenerate high and low energy excitations

In microscopic many-body physics the coupling between the motion of fast particles (electrons) and slow particles (nuclei) is universal. The standard Born-Oppenheimer decoupling procedure breaks down, if the energy separation in the “fast” system is of the same order as the elementary excitation in the “slow” system. In this case “dynamical resonance” effects are to be expected. In the present investigation a model system of a coupling between a doubly degenerate high energy excitation and doubly degenerate low energy oscillator is handled by a non-linear canonical transformation which is shown to be quasi-exact in the sense that it diagonalizes the Hamiltonian in both extremal coupling cases. The transformation has some flexibility, so that the diagonalization regions can be enlarged. It is employed to calculate the “zero-phonon” optical response, which indeed displays aresonance effect. Likewise, another nonlinear transformation is devised, which only in the strong coupling limit yields diagonalization. This latter transformation in a natural way leads to the conventional semi-classical approaches to the dynamical Jahn-Teller problem. The results gotten with it are identical with those from our transformation in the strong coupling limit. On the basis of our results some remarks are made concerning the possible impact of the breakdown of the adiabatic approximation in other regions.     

Complete Hamiltonian Description of Wave-Like Features in Classical and Quantum Physics

The analysis of the Helmholtz equation is shown to lead to an exact Hamiltonian system describing in terms of ray trajectories, for a stationary refractive medium, a very wide family of wave-like phenomena (including diffraction and interference) going much beyond the limits of the geometrical optics (“eikonal”) approximation, which is contained as a simple limiting case. Due to the fact, moreover, that the time independent Schrödinger equation is itself a Helmholtz-like equation, the same mathematics holding for a classical optical beam turns out to apply to a quantum particle beam moving in a stationary force field, and leads to a system of Hamiltonian equations providing exact and deterministic particle trajectories and dynamical laws, and containing the laws of Classical Mechanics in the eikonal limit.     

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     

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.     

Quantentheorie der Zeitmessung

As a consequence of its dynamical motion a quantum mechanical system may be considered as a quantum mechanical clock. If one demands that the time be an observable which corresponds to a hypermaximal time operator in Hilbert space, then, for systems having a continuous energy spectrum with a lower limit, in the framework of the nonrelativistic theory to be discussed here there must exist an upper limit of energy, too. Furthermore the time operator is not defined on the whole Hilbert space, but only on state functions satisfying a certain condition. Therefrom it results that a quantum mechanical clock of this kind can be read off only in a sequence of equidistant times separated by a “minimal time”. The beginning of the time measurement being arbitrary the scale of time may be shifted according to the homogenity in time. Especially for a free particle beside the minimal time also a minimal length is obtained. The equidistant scale in space is not absolute either, but permits an arbitrary choice of the point of reference according to homogenity in space. The modificated spreading of the probability distribution of particle position is discussed.     

Deduction of the Quasi-linear Transport Equations of Hydro-thermodynamics from the Gyarmati Principle

From the universal form of Gyarmati's variational principle of thermodynamics the differential equations governing the internal energy and impulse transport of one component hydro-thermodynamic systems are derived. In our particular case Gyarmati's “supplementary theorem” is confirmed, by which the validity of the universal form of Gyarmati's variational principle is guaranted also in non-linear cases. Finally some problems of the Gyarmati principle and of non-linear thermodynamics are discussed.     

Multiple returns for some regular and mixing maps

We study the distributions of the number of visits for some noteworthy dynamical systems, considering whether limit laws exist by taking domains that shrink around points of the phase space. It is well known that for highly mixing systems such limit distributions exhibit a Poissonian behavior. We analyze instead a skew integrable map defined on a cylinder that models a shear flow. Since almost all fibers are given by irrational rotations, we at first investigate the distributions of the number of visits for irrational rotations on the circle. In this last case the numerical results strongly suggest the existence of limit laws when the shrinking domain is chosen in a descending chain of renormalization intervals. On the other hand, the numerical analysis performed for the skew map shows that limit distributions exist even if we take domains shrinking in an arbitrary way around a point, and these distributions appear to follow a power law decay of which we propose a theoretical explanation. It is interesting to note that we observe a similar behavior for domains wholly contained in the integrable region of the standard map. We also consider the case of two or more systems coupled together, proving that the distributions of the number of visits for domains intersecting the boundary between different regions are a linear superposition of the distributions characteristic of each region. Using this result we show that the real limit distributions can be hidden by some finite-size effects. In particular, when a chaotic and a regular region are glued together, the limit distributions follow a Poisson-like law, but as long as the measure of the shrinking domain is not zero, the polynomial behavior of the regular component dominates for large times. Such an analysis seems helpful to understand the dynamics in the regions where ergodic and regular motions are intertwined, as it may occur for the standard map. Finally, we study the distributions of the number of visits around generic and periodic points of the dissipative Henon map. Although this map is not uniformly hyperbolic, the distributions computed for generic points show a Poissonian behavior, as usually occurs for systems with highly mixing dynamics, whereas for periodic points the distributions follow a different law that is obtained from the statistics of first return times by assuming that subsequent returns are independent. These results are consistent with a possible rapid decay of the correlations for the Henon map.     

Gibbs processes and generalized Bernoulli flows for hard-core one-dimensional systems

We consider a class of infinite-range potentials for which phase transitions are absent, and prove by the Ornstein-Friedman theorem, that they generate dynamical systems that are Bernoulli flows in a generalized sense.     

Conserved quantities and symmetries related to stochastic Hamiltonian systems

In this paper symmetries and conservation laws for stochastic dynamical systems are discussed in detail. Determining equations for infinitesimal approximate symmetries of Ito and Stratonovich dynamical systems are derived. It shows how to derive conserved quantities for stochastic dynamical systems by using their symmetries without recourse to Noether's theorem.