Analytic solutions of simple flows and analysis of nonslip boundary conditions for the lattice Boltzmann BGK model

In this paper we analytically solve the velocity of the lattice Boltzmann BGK equation (LBGK) for several simple flows. The analysis provides a framework to theoretically analyze various boundary conditions. In particular, the analysis is used to derive the slip velocities generated by various schemes for the nonslip boundary condition. We find that the slip velocity is zero as long as fe=0 at boundaries, no matter what combination of distributions is chosen. The schemes proposed by Nobleet al. and by Inamuroet al. yield the correct zeroslip velocity, while some other schemes, such as the bounce-back scheme and the equilibrium distribution scheme, would inevitably generate a nonzero slip velocity. The bounce-back scheme with the wall located halfway between a flow node and a bounce-back node is also studied for the simple flows considered and is shown to produce results of second-order accuracy. The momentum exchange at boundaries seems to be highly related to the slip velocity at boundaries. To be specific, the slip velocity is zero only when the momentum dissipated by boundaries is equal to the stress provided by fluids.     

Large Deviations for Quadratic Functionals of Gaussian Processes

The Large Deviation Principle (LDP) is derived for several quadratic additive functionals of centered stationary Gaussian processes. For example, the rate function corresponding to is the Fenchel-Legendre transform of where X _t is a continuous time process with the bounded spectral density f(s). This spectral density condition is strictly weaker than the one necessary for the LDP to hold for all bounded continuous functionals. Similar results are obtained for the energy of multivariate discrete-time Gaussian processes and in the regime of moderate deviations, the latter yielding the corresponding Central Limit Theorems.     

Tail estimates for one-dimensional random walk in random environment

Suppose that the integers are assigned i.i.d. random variables { _x } (taking values in the unit interval), which serve as an environment. This environment defines a random walk {X _k } (called a RWRE) which, when atx, moves one step to the right with probability _x , and one step to the left with probability 1- _x . Solomon (1975) determined the almost-sure asymptotic speed (=rate of escape) of a RWRE. For certain environment distributions where the drifts 2 _x -1 can take both positive and negative values, we show that the chance of the RWRE deviating below this speed has a polynomial rate of decay, and determine the exponent in this power law; for environments which allow only positive and zero drifts, we show that these large-deviation probabilities decay like exp(–Cn ^1/3). This differs sharply from the rates derived by Greven and den-Hollander (1994) for large deviation probabilities conditioned on the environment. As a by product we also provide precise tail and moment estimates for the total population size in a Branching Process with Random Environment.Partially supported by NSF DMS-9209712 and DMS-9403553 grants, by a US-ISRAEL BSF grant and by the S. and N. Grand research fund.Research partially supported by NSF grant # DMS-9404391 and a Junior Faculty Fellowship from the Regents of the University of California.Partially supported by NSF grant # DMS-9302709, by a US-Israel BSF grant and by the fund for promotion of research at the Technion.     

Quenching of giant hysteresis effects in La(1-z)Y(z)Hx switchable mirrors

Hydrogen atoms adsorbed on TiO2(110)-(1x1) surfaces have been characterized by scanning tunneling microscopy (STM) combined with electron stimulated desorption (ESD) technique. Certain amounts of H atoms are unexpectedly found on the TiO2 surfaces annealed at 900 K. Two forms of adsorption were discriminated in STM images from the different sensitivity to ESD and tentatively assigned to hydroxyl-type (O-H) and hydride-type (Ti-H) species.     

Ordered overlaps in disordered mean-field models

Let M(N) be a sequence of integers with M→∞ as N→∞ and M=o(N). For bounded i.i.d. r.v. ξ _i ^k and bounded i.i.d. r.v. σ _i , we study the large deviation of the family of (ordered) scalar products X ^k =N ⁻¹∑ _{i =1} ^N σ _i ξ _i ^k ,k≤M, under the distribution conditioned on the ξ _i ^k 's. To get a full large deviation principle, it is necessary to specify also the total norm(∑ _{k ≤ M} (X ^k )²)^1/2, which turns to be associated with some extra Gaussian distribution. Our results apply to disordered, mean-field systems, including generalized Hopfield models in the regime of a sublinear number of patterns. We build also a class of examples where this norm is the crucial order parameter. Received: 6 April 1999 / Revised version: 29 May 2000 /?Published online: 24 July 2001     

Regular microstructures in gel–surfactant complexes: Influence of water content and comparison with the surfactant structure in water

Interaction of slightly crosslinked hydrogels of poly(diallyldimethylammonium chloride) (PDADMACI) and of copolymer DADMACI/acrylamide (AAm) with sodium dodecylsulfate (SDS) and sodium dodecylbenzenesulfate (SDBS) results in significant shrinking of the gels due to the formation of polymer-surfactant complexes. Jump-wise transitions in the collapsed state were observed for the networks with the content of cationic groups 100 and 75 mol %. The structure of complexes was studied by means of X-ray scattering method. The scattering curves for collapsed gels, where most chloride anions were replaced by anions of SDS, show a set of well-pronounced narrow diffraction maxima. Fully charged “wet” complexes studied at the equilibrium swelling conditions exhibit high degree of ordering, which diminishes upon drying with the simultaneous transition from hexagonal to lamellar type of ordering. In contrast to this, for DADMACl/AAm copolymer gels (75 mol % of DADMACl monomers in the initial polymerization mixture) the ordering is less pronounced in the “wet” state and becomes more perfect upon drying. The SDS aqueous solutions of the same concentration in the absence of gel do not show such high degree of ordering, while the system of SDS/neutral AAm gel exhibits lamellar ordering typical for low-temperature phases of SDS solutions. © 1996 John Wiley & Sons, Inc.     

A comparative study of immunoglobulin IgM and rheumatoid factor IgMRF in solution by small-angle X-ray scattering

The shapes of rheumatoid factor molecules calculated from X-ray small-angle solution scattering data suggest that the peripheral Fab regions of rheumatoid factor in solution are more flocculent because of a higher flexibility of Fab domains.     

Large deviations for random walks on Galton–Watson trees: averaging and uncertainty

 In the study of large deviations for random walks in random environment, a key distinction has emerged between quenched asymptotics, conditional on the environment, and annealed asymptotics, obtained from averaging over environments. In this paper we consider a simple random walk {X _n } on a Galton–Watson tree T, i.e., on the family tree arising from a supercritical branching process. Denote by |X _n | the distance between the node X _n and the root of T. Our main result is the almost sure equality of the large deviation rate function for |X _n |/n under the “quenched measure” (conditional upon T), and the rate function for the same ratio under the “annealed measure” (averaging on T according to the Galton–Watson distribution). This equality hinges on a concentration of measure phenomenon for the momentum of the walk. (The momentum at level n, for a specific tree T, is the average, over random walk paths, of the forward drift at the hitting point of that level). This concentration, or certainty, is a consequence of the uncertainty in the location of the hitting point. We also obtain similar results when {X _n } is a λ-biased walk on a Galton–Watson tree, even though in that case there is no known formula for the asymptotic speed. Our arguments rely at several points on a “ubiquity” lemma for Galton–Watson trees, due to Grimmett and Kesten (1984). Received: 15 November 2000 / Revised version: 27 February 2001 / Published online: 19 December 2001