Quantum field-theory models on fractal spacetime

Continuing the analysis of a previous paper, the present work applies rigorous renormalization group methods to the hierarchical models to establish the existence of field theories with non-Gaussian ultraviolet renormalization group fixed points in 4- dimensions.     

Besov spaces and the multifractal hypothesis

Parisi and Frisch proposed some time ago an explanation for multiscaling of turbulent velocity structure functions in terms of a multifractal hypothesis, i.e., they conjecture that the velocity field has local Hölder exponents in a range [h _min,h _max], with exponents <h occurring on a setS(h) with a fractal dimensionD(h). Heuristic reasoning led them to an expression for the scaling exponentz _p ofpth order as the Legendre transform of the codimensiond-D(h). We show here that a part of the multifractal hypothesis is correct under even weaker assumptions: namely, if the velocity field hasL ^p -mean Hölder indexs, i.e., if it lies in the Besov spaceB _p ^s, , then local Hölder regularity is satisfied. Ifs<d/p, then the hypothesis is true in a generalized sense of Hölder space with negative exponents and we discuss the proper definition of local Hölder classes of negative index. Finally, if a certain box-counting dimension exists, then the Legendre transform of its codimension gives the scaling exponentz _p , and, more generally, the maximal Besov index of order,p, ass _p =z _p /p. Our method of proof is derived from a recent paper of S. Jaffard using compactly-supported, orthonormal wavelet bases and gives an extension of his results. We discuss implications of the theorems for ensemble-average scaling and fluid turbulence.     

Intermittency in the joint cascade of energy and helicity

The statistics of the energy and helicity fluxes in isotropic turbulence are studied using high resolution direct numerical simulation. The scaling exponents of the energy flux agree with those of the transverse velocity structure functions through refined similarity hypothesis, consistent with Kraichnan's prediction. The helicity flux is even more intermittent than the energy flux. Consistent with this observation, the spatial helicity-flux structures are finer than those of energy flux and more tubelike in geometry.     

A Maximum Entropy Method for Particle Filtering

Standard ensemble or particle filtering schemes do not properly represent states of low priori probability when the number of available samples is too small, as is often the case in practical applications. We introduce here a set of parametric resampling methods to solve this problem. Motivated by a general H-theorem for relative entropy, we construct parametric models for the filter distributions as maximum-entropy/minimum-information models consistent with moments of the particle ensemble. When the prior distributions are modeled as mixtures of Gaussians, our method naturally generalizes the ensemble Kalman filter to systems with highly non-Gaussian statistics. We apply the new particle filters presented here to two simple test cases: a one-dimensional diffusion process in a double-well potential and the three-dimensional chaotic dynamical system of Lorenz.     

A model of dispenser cathode activity

A semiquantitative model of dispenser cathode activity based on recent work on the co-adsorption of Ba and O onto W surfaces is presented. The co-adsorption studies have determined the shape of a three-dimensional surface of work function as a function of θ_O and θ_Ba, the surface coverages of O and Ba, respectively. Compositions of a variety of pedigreed dispenser cathodes were fitted to this surface and their composition changes during lifetime were modeled. Changes of surface composition with temperature and of workfunction, φ, with temperature were also found to fit these curves. The concept of a patchy surface implied by the co-adsorption measurements was used to explain earlier results on the shape of the X-ray excited Ba MNN Auger feature. Finally, SIMS measurements under UHV conditions was found to provide an extremely sensitive measurement of surface composition in the region of surface coverages of interest in the study of cathode phenomena. Extensions of this work to other types of cathodes such as M-types, and rhenium substrate cathodes is also discussed.     

Quantum field-theory models on fractal spacetime

The present work explores the possibility of giving a non-perturbative definition of the quantum field-theory models in non-integer dimensions, which have been previously studied by Wilson and others using analytic continuation of dimension in perturbation integrals. The method employed here is to base the models on fractal point-sets of non-integer Hausdorff-Besicovitch dimension. Two types of scalar-field models are considered: the one has its propagator (=covariance operator kernel) given by a proper-time or heat-kernel representation and the other has a hierarchical propagator. The fractal lattice version of the proper-time propagator is shown to be reflection-positive. The hierarchical models are introduced and their properties discussed on an informal basis.This paper is based largely on the thesis presented by the author for the degree doctor of philosophy in physics at the Ohio State University in 1987