Weak Mixing Properties Research Articles

Weak mixing and anomalous kinetics along filamented surfaces.

We consider chaotic properties of a particle in a square billiard with a horizontal bar in the middle. Such a system can model field-line windings of the merged surfaces. The system has weak-mixing properties with zero Lyapunov exponent and entropy, and it can be also interesting as an example of a system with intermediate chaotic properties, between the integrability and strong mixing. We show that the transport is anomalous and that its properties can be linked to the ergodic properties of continued fractions. The distribution of Poincare recurrences, distribution of the displacements, and the moments of the truncated distribution of the displacements are obtained. Connections between different exponents are found. It is shown that the distribution function of displacements and its truncated moments as a function of time exhibit log-periodic oscillations (modulations) with a universal period T(log)=pi(2)/12 ln 2. We note that similar results are valid for a family of billiard, particularly for billiards with square-in-square geometry. (c) 2001 American Institute of Physics.

A class of generalized Ornstein transformations with the weak mixing property

It is shown that there is a class of Ornstein transformations, distinct from the Ornstein mixing class, which are almost surely weak mixing

Optimizing the SDS Sedimentation Test for End‐Use Quality Selection in a Soft White and Club Wheat Breeding Program

ABSTRACTSoft white and club wheat (Triticum aestivum L.) market subclasses have specific end‐use characteristics. Among the most important of these characteristics are weak dough mixing and handling properties as a result of weak gluten. The SDS sedimentation test has gained wide acceptance as a useful, small‐scale test in bread wheat breeding programs to predict gluten strength and baking quality. To optimize its use for soft white or club wheat breeding, variations of the SDS sedimentation test were performed on grain from winter wheats grown at eight locations in the U.S. Pacific Northwest, and the effects of lines, environment, and their interactions on SDS sedimentation volumes were determined. Using different sample weights and substituting whole meal for flour did not affect the ability of the SDS sedimentation test to differentiate among lines. Changes in protein concentration and sample weight caused proportional changes in SDS sedimentation volumes; however, the response was not consistent among all lines. Line had a greater effect on the SDS sedimentation volumes than any other source of variation. If differential effects of protein to SDS sedimentation among lines are taken into account, the SDS sedimentation test should be an effective small‐scale test for end‐use quality assessment in soft white and club wheat breeding programs.

ON DYNAMICAL MODEL OF DRILLING WELLS

A family of one-dimensional mappings and their stochastic perturbations, related to the determining of cogged bits efficiency, is studied. A sufficient condition for the existence of absolutely continuous invariant probability measure is given, and the weak mixing property of this measure is established. We formulate and discuss the results concerning the stochastic stability of the invariant measure and its continuous dependence on the dimensionless parameter of the model. Some new problems are also outlined.

Asymptotic behavior of eigenfunctions and eigenvalues for ergodic and periodic systems

The purpose of this article is to study the influence of dynamical behavior of classical Hamilton systems with ergodic and periodic properties on asymptotic behavior of eigenfunctions and eigenvalues of the corresponding positive elliptic operator on a compact Riemannian manifold, and conversely, to investigate the asymptotic properties of eigenfunctions or eigenvalues which make the corresponding classical mechanics ergodic or periodic. We will give an estimate of the off-diagonal asymptotics of quantum observables for quantum ergodic systems and a regularity result on limit measures associated with quantum observables for systems with homogeneous Lebesgue spectrum. We will also give necessary and sufficient conditions for ergodicity and weak-mixing property of the classical Hamilton systems, which are obtained by a reduction procedure with symmetry, in terms of semi-classical asymptotic properties of eigenfunctions. Finally, a result on the structure of the set of cluster points for the differences of eigenvalues in a certain semi-classical sense is given, which is considered as a semi-classical analogy of Helton’s theorem.

Shift endomorphisms and compact Lie extensions

We consider skew-products with an arbitrary compact Lie group, when the base map is a one-sided shift of finite type endowed with an equilibrium state of a Holder continuous function. First we show that the weak-mixing property of the skew-product implies exactness and exponential mixing. Then we address the problem of classification under measure-theoretic isomorphisms. We show that for a generic set of equilibrium states the isomorphism class of the skew-products corresponds essentially to the cohomology classes of the defining skewing function and the isomorphism is essentially a homeomorphism.

Singular continuous spectrum and quantitative rates of weak mixing

We prove that for a dense $G_{\delta}$ of shift-invariant measures on $A^{\ZZ^d}$, all $d$ shifts have purely singular continuous spectrum and give a new proof that in the weak topology of measure preserving $\ZZ^d$ transformations, a dense $G_{\delta}$ is generated by transformations with purely singular continuous spectrum. We also give new examples of smooth unitary cocycles over an irrational rotation which have purely singular continuous spectrum. Quantitative weak mixing properties are related by results of Strichartz and Last to spectral properties of the unitary Koopman operators.

A constructive mixing condition for 2-D Gibbs measures with random interactions

For spin systems with random, finite-range interactions, we define an analog of the usual weak mixing property, which we call “weak mixing in expectation” (WME). This property implies (almost sure) uniqueness of the Gibbs measure. We concentrate on the two-dimensional case, for which we present finite-volume conditions which are sufficient for WME. We also show the reverse: if the system is WME, then the condition is satisfied for some (sufficiently large) volume. Simultaneously, we obtain an extension (to random interactions) of the result by Martinelli, Olivieri and Schonmann that weak mixing implies strong mixing. Our method is based on a rescaled version of the disagreement percolation approach of van den Berg and Maes, combined with ideas and techniques of Gielis and Maes, and Martinelli, Olivieri and Schonmann. However, apart from some general results on coupling, stated in Section 2, this paper is self-contained.

Large deviation probabilities for sums of heavy-tailed dependent random vectors*

Necessary and sufficient conditions are given for multidimensional p – stable limit theorems (i.e. theorems on convergence of normalized partial sums of a stationary sequence of random vectors to a non-degenerate strictly p-stable limiting law μ with -regularly varying normalizing sequence ). It is proved that similarly as in the one-dimensional case the conditions for consist of two parts: one responsible for (very weak) mixing properties and another, describing asymptotics of probabilities of large deviations (with a minor additional condition for ). The paper focuses on effective methods of proving such large deviation results

Index and dynamics of quantized contact transformations

Quantized contact transformations are Toeplitz operators over a contact manifold (X,α) of the form U χ =ΠAχΠ, where Π:H 2 (X)→L 2 (X) is a Szegö projector, where χ is a contact transformation and where A is a pseudodifferential operator over X. They provide a flexible alternative to the Kähler quantization of symplectic maps, and encompass many of the examples in the physics literature, e.g. quantized cat maps and kicked rotors. The index problem is to determine ind (U χ ) when the principal symbol is unitary, or equivalently to determine whether A can be chosen so that U χ is unitary. We show that the answer is yes in the case of quantized symplectic torus automorphisms g—by showing that U g duplicates the classical transformation laws on theta functions. Using the Cauchy-Szegö kernel on the Heisenberg group, we calculate the traces on theta functions of each degree N. We also study the quantum dynamics generated by a general q.c.t. U χ , i.e. the quasi-classical asymptotics of the eigenvalues and eigenfunctions under various ergodicity and mixing hypotheses on χ. Our principal results are proofs of equidistribution of eigenfunctions φ Nj and weak mixing properties of matrix elements (Bφ Ni ,φ Nj ) for quantizations of mixing symplectic maps.

Intramolecular Vibrational Relaxation in Electronically Excited States

A density matrix approach for describing intramolecular dynamics, with special application to vibrational relaxation in excited electronic states, is presented. We derive the master equations governing intramolecular transfer of excitation energy between the states in a zeroth-order basis defined by considering the excitation and detection conditions in the time-resolved experiments. It is shown that, in this formalism, the memory function plays a central role. We note that the form of intramolecular memory is determined by the importance of quantum mechanical mixing of the zeroth-order states. A distinction is made between the subsystems of states with strong and weak mixing properties; while the former account for quasiperiodic character of coherent motion, the latter display a Markovian behavior. The physical conditions fixing the relative importance of quasiperiodic and statistical dynamics in individual systems are discussed. In the application to time resolved intramolecular vibrational relaxation, special consideration is given to the nature of the initially excited doorway states and the intermode couplings. The symmetry restrictions and the possible role of rotational motion in vibrational relaxation are also discussed, before considering the recent results obtained by Zewail et al. with anthracene in the first excited singlet state.