Random walk statistics on fractal structures

We consider some statistical properties of simple random walks on fractal structures viewed as networks of sites and bonds: range, renewal theory, mean first passage time, etc. Asymptotic behaviors are shown to be controlled by the fractal (¯d) and spectral (¯d) dimensionalities of the considered structure. A simple decimation procedure giving the value of (¯d) is outlined and illustrated in the case of the Sierpinski gaskets. Recent results for the trapping problem, the self-avoiding walk, and the true-self-avoiding walk are briefly reviewed. New numerical results for diffusion on percolation clusters are also presented.     

Adsorption on model fractal surfaces

A simple model for the adsorption of gas molecules on fractal surfaces is formulated. For surface reactions where adsorption is the rate determining step, considerations based on the probability of adsorption are sufficient to describe the kinetics. Different values of fractal dimension correspond to different values of rate constant for adsorption. A linear relationship between the dynamics and the geometric properties of a solid surface as well as a statistical mechanical relationship between internal partition functions of the gas molecule and the adsorption complex are obtained.     

Regular and irregular geodesics on spherical harmonic surfaces

The behavior of geodesic curves on even seemingly simple surfaces can be surprisingly complex. In this paper we use the Hamiltonian formulation of the geodesic equations to analyze their integrability properties. In particular, we examine the behavior of geodesics on surfaces defined by the spherical harmonics. Using the Morales-Ramis theorem and Kovacic algorithm we are able to prove that the geodesic equations on all surfaces defined by the sectoral harmonics are not integrable, and we use Poincaré sections to demonstrate the breakdown of regular motion.     

Non-abelian vortices on surfaces and their statistics

We discuss the physics of topological vortices moving on an arbitrary surface M in a YangMills-Higgs theory in which the gauge group G breaks to a finite subgroup H. We concentrate on the case where M is compact and/or non-orientable. Interesting new features arise which have no analog on the plane. The consequences for the quantum statistics of vortices are discussed, particularly when H is non-abelian.     

Universal first-passage properties of discrete-time random walks and Lévy flights on a line: Statistics of the global maximum and records

In these lecture notes I will discuss the universal first-passage properties of a simple correlated discrete-time sequence {x₀=0,x₁,x₂,…,x_n} up to n steps where x_i represents the position at step i of a random walker hopping on a continuous line by drawing independently, at each time step, a random jump length from an arbitrary symmetric and continuous distribution (it includes, e.g., the Lévy flights). I will focus on the statistics of two extreme observables associated with the sequence: (i) its global maximum and the time step at which the maximum occurs and (ii) the number of records in the sequence and their ages. I will demonstrate how the universal statistics of these observables emerge as a consequence of Pollaczek-Spitzer formula and the associated Sparre Andersen theorem.     

Brownian surfaces with boundary and Deligne cohomology

We define differentiable random surfaces, which realize a kind of generalized parallel transport for line bundles over the loop space. This gives a realization of one of the axioms of Segal of conformal field theory.     

Microgels and fractal structures at interfaces and surfaces

The behavior of microgels near surfaces and their adsorption is studied by simple scaling theory. Two different types of microgels can be studied, i.e., fractal type microgels and randomly crosslinked polymer chains. In the first case the gel can be described mainly by introducing a spectral dimension. The second type requires more attention and uses the number of crosslinks as parameter. The main result is that soft gels with weakly coupled crosslinks and a low number of crosslinks adsorb much better than hard gels, with many crosslinks. Similar results for fractal gels and branched polymer are presented. Fractal gels with low connectivity adsorb easier than gels with a large connectivity dimension. We discuss also consequences on surface protection by microgels. Received: 11 August 1997 / Received in final form: 20 November 1997 / Accepted 22 January 1998     

Scattering of electromagnetic radiation by surfaces with a fractal relief

The scattering indicatrix of electromagnetic waves for different types of rough surfaces and angles of incidence is calculated using the Kirchhoff scalar theory. The rough surfaces are modelled by a two dimensional Weierstrass function. The scattering index for a rough surface with a fractal relief is found to have a complicated structure with intensity bursts in directions quite far from the direction of mirror reflection.     

Time-evolving statistics of cavitation damage on metallic surfaces

The statistics of surface damage on polycrystalline aluminium plates caused by acoustic cavitation is studied experimentally as a function of time. Cavitation is shown to produce a uniform distribution of crater-like holes with different depth, area and eccentricity. Most notably, the size distribution of such craters evolves with time from a gamma function into a power law. By contrast, on the surface of a martensitic Cu–Ni–Al crystal cavitation damage generates ramified patterns, reminiscent of a fractal object.     

Approach to an irregular time series on the basis of the fractal theory

We present a technique to measure the fractal dimension of the set of points (t, f(t)) forming the graph of a function f defined on the unit interval. First we apply it to a fractional Brownian function [1] which has a property of self-similarity for all scales, and we can get the stable and precise fractal dimension. This technique is also applied to the observational data of natural phenomena. It does not show self-similarity all over the scale but has a different self-similarity across the characteristic time scale. The present method gives us a stable characteristic time scale as well as the fractal dimension.     

A three-dimensional Brownian path reflected on a Brownian path is a free Brownian path

Three-dimensional Brownian path reflected on Brownian path is a free Brownian path.Research partially supported by NSF grant DMS 91-00244.     

Stable superhydrophobic surfaces over a wide pH range

A stable superhydrophobic surface was fabricated by solidifying poly(epoxy-terminated polydimethylsiloxane-co-bisphenol A) [P(ETPDMS-co-BPA)] copolymer on a rough substrate. The low surface energy of the copolymer and the geometric structure at micrometer scale of the surface contribute to the superhydrophobic property. The as-prepared surface shows stable superhydrophobicity over a wide pH range (1-14) and the wettability is excellent stable to heating, water, corrosive solution and organic solvent treatments. The procedure is simple and time-saving as well as utilizing non-fluorine-containing compounds.     

Brownian motion on a manifold

The question of the existence and correct form of equations describing Brownian motion on a manifold cannot be answered by mathematics alone, but requires a study of the underlying physics. As in classical mechanics, manifolds enter through the transformation of variables needed to account for the presence of constraints. The constraints are either due to a physical agency that forces the motion to remain on a manifold, or they represent conserved quantities of the equation of motion themselves. Also the Brownian motion is described either by a Smoluchowski diffusion equation or by a Kramers equation. The four cases lead to the following conclusions, (i) Smoluchowski diffusion with a conserved quantity reduces to a diffusion equation on the manifold; (ii) The same is true for diffusion with a physical constraint in three dimensions, but in more dimensions it may happen thatno autonomous equation on the manifold results; (iii) A Kramers equation with a conserved quantity reduces to an equation on the manifold, but in general not of the form of a Kramers equation; (iv) The Kramers equation with a physical constraint reduces to an autonomous Kramers equation on the manifold only for a special shape of that constraint. Throughout, only a certain type of physical constraints has been envisaged, and global questions are ignored. Finally, the customary heuristic construction of a Fokker-Planck equation for a mechanical system on a manifold is demonstrated for the case of Brownian rotation of a rigid body, and its shortcomings are emphasized.     

Parabolic equation formulation via a singular perturbation technique and its application to scattering from irregular surfaces

Abstract

This paper consists of two parts. In the first part, the solution of the Helmholtz equation under forward-scattering or propagation conditions is sought as a uniform asymptotic perturbation expansion using the method of multiple scales. It is then shown that the parabolic wave equation (PWE) solution is the zeroth-order term in this expansion. In the second part, the electric-field integral equation and the magnetic-field integral equation, derived under the PWE approximation, are solved for surface currents induced on a sinusoidal surface. The scattered fields produced by these currents are then calculated using the appropriate radiation integrals. Results are compared to those obtained using the method of ordered multiple interactions developed by Kapp and Brown.     

Parabolic equation formulation via a singular perturbation technique and its application to scattering from irregular surfaces

This paper consists of two parts. In the first part, the solution of the Helmholtz equation under forward-scattering or propagation conditions is sought as a uniform asymptotic perturbation expansion using the method of multiple scales. It is then shown that the parabolic wave equation (PWE) solution is the zeroth-order term in this expansion. In the second part, the electric-field integral equation and the magnetic-field integral equation, derived under the PWE approximation, are solved for surface currents induced on a sinusoidal surface. The scattered fields produced by these currents are then calculated using the appropriate radiation integrals. Results are compared to those obtained using the method of ordered multiple interactions developed by Kapp and Brown.