Asymptotic scaling in the two-dimensional O(3) Nonlinear sigma model: a Monte Carlo study on parallel computers

We investigate the 2d O(3) model with the standard action by Monte Carlo simulation at couplings β up to 2.05. We measure the energy density, mass gap and susceptibility of the model, and gather high statistics on lattices of size L ≤ 1024 using the Floating Point Systems T-series vector hypercube and the Thinking Machines Corp.'s Connection Machine 2. Asymptotic scaling does not appear to set in for this action, even at β = 2.10, where the correlation length is 420. We observe a 20% difference between our estimate m/Λ^─_(Ms) = 3.52(6) at this β and the recent exact analytical result . We use the overrelaxation algorithm interleaved with Metropolis updates and show that decorrelation time scales with the correlation length and the number of overrelaxation steps per sweep. We determine its effective dynamical critical exponent to be z' = 1.079(10); thus critical slowing down is reduced significantly for this local algorithm that is vectorizable and parallelizable.

We also use the cluster Monte Carlo algorithms, which are non-local Monte Carlo update schemes which can greatly increase the efficiency of computer simulations of spin models. The major computational task in these algorithms is connected component labeling, to identify clusters of connected sites on a lattice. We have devised some new SIMD component labeling algorithms, and implemented them on the Connection Machine. We investigate their performance when applied to the cluster update of the two dimensional Ising spin model.

Finally we use a Monte Carlo Renormalization Group method to directly measure the couplings of block Hamiltonians at different blocking levels. For the usual averaging block transformation we confirm the renormalized trajectory (RT) observed by Okawa. For another improved probabilistic block transformation we find the RT, showing that it is much closer to the Standard Action. We then use this block transformation to obtain the discrete β-function of the model which we compare to the perturbative result. We do not see convergence, except when using a rescaled coupling β_E to effectively resum the series. For the latter case we see agreement for m/ Λ^─_(Ms) at , β = 2.14, 2.26, 2.38 and 2.50. To three loops m/Λ^─_(Ms) = 3.047(35) at β = 2.50, which is very close to the exact value m/ Λ^─_(Ms) = 2.943. Our last point at β = 2.62 disagrees with this estimate however.

Analysis and optimization of stress wave propagation in two-dimensional granular crystals with defects

Granular crystals are compact periodic assemblies of elastic particles in Hertzian contact whose dynamic response can be tuned from strongly nonlinear to linear by the addition of a static precompression force. This unique feature allows for a wide range of studies that include the investigation of new fundamental nonlinear phenomena in discrete systems such as solitary waves, shock waves, discrete breathers and other defect modes. In the absence of precompression, a particularly interesting property of these systems is their ability to support the formation and propagation of spatially localized soliton-like waves with highly tunable properties. The wealth of parameters one can modify (particle size, geometry and material properties, periodicity of the crystal, presence of a static force, type of excitation, etc.) makes them ideal candidates for the design of new materials for practical applications. This thesis describes several ways to optimally control and tailor the propagation of stress waves in granular crystals through the use of heterogeneities (interstitial defect particles and material heterogeneities) in otherwise perfectly ordered systems. We focus on uncompressed two-dimensional granular crystals with interstitial spherical intruders and composite hexagonal packings and study their dynamic response using a combination of experimental, numerical and analytical techniques. We first investigate the interaction of defect particles with a solitary wave and utilize this fundamental knowledge in the optimal design of novel composite wave guides, shock or vibration absorbers obtained using gradient-based optimization methods.

Suspension mechanics: I. Inertial and non-Newtonian migration of neutrally buoyant rigid spheres in two-dimensional unidirectional flows; II. The effect of viscoelasticity on the creeping motion of a train of neutrally buoyant Newtonian drops through a circular tube

The lateral migration of neutrally buoyant rigid spheres in two-dimensional unidirectional flows was studied theoretically. The cases of both inertia-induced migration in a Newtonian fluid and normal stress-induced migration in a second-order fluid were considered. Analytical results for the lateral velocities were obtained, and the equilibrium positions and trajectories of the spheres compared favorably with the experimental data available in the literature. The effective viscosity was obtained for a dilute suspension of spheres which were simultaneously undergoing inertia-induced migration and translational Brownian motion in a plane Poiseuille flow. The migration of spheres suspended in a second-order fluid inside a screw extruder was also considered.

The creeping motion of neutrally buoyant concentrically located Newtonian drops through a circular tube was studied experimentally for drops which have an undeformed radius comparable to that of the tube. Both a Newtonian and a viscoelastic suspending fluid were used in order to determine the influence of viscoelasticity. The extra pressure drop due to the presence of the suspended drops, the shape and velocity of the drops, and the streamlines of the flow were obtained for various viscosity ratios, total flow rates, and drop sizes. The results were compared with existing theoretical and experimental data.

Mean-speed measurements in two-dimensional, incompressible, fully-developed turbulent channel flow

Mean velocity profiles were measured in the 5” x 60” wind channel of the turbulence laboratory at the GALCIT, by the use of a hot-wire anemometer. The repeatability of results was established, and the accuracy of the instrumentation estimated. Scatter of experimental results is a little, if any, beyond this limit, although some effects might be expected to arise from variations in atmospheric humidity, no account of this factor having been taken in the present work. Also, slight unsteadiness in flow conditions will be responsible for some scatter.

Irregularities of a hot-wire in close proximity to a solid boundary at low speeds were observed, as have already been found by others.

That Kármán’s logarithmic law holds reasonably well over the main part of a fully developed turbulent flow was checked, the equation u/u_t = 6.0 + 6.25 log₁₀ y^ut/v being obtained, and, as has been previously the case, the experimental points do not quite form one straight line in the region where viscosity effects are small. The values of the constants for this law for the best over-all agreement were determined and compared with those obtained by others.

The range of Reynolds numbers used (based on half-width of channel) was from 20,000 to 60,000.

Thermopower in Two-Dimensional Electron Systems

The subject of this thesis is the measurement and interpretation of thermopower in high-mobility two-dimensional electron systems (2DESs). These 2DESs are realized within state-of-the-art GaAs/AlGaAs heterostructures that are cooled to temperatures as low as T = 20 mK. Much of this work takes place within strong magnetic fields where the single-particle density of states quantizes into discrete Landau levels (LLs), a regime best known for the quantum Hall effect (QHE). In addition, we review a novel hot-electron technique for measuring thermopower of 2DESs that dramatically reduces the influence of phonon drag.

Early chapters concentrate on experimental materials and methods. A brief overview of GaAs/AlGaAs heterostructures and device fabrication is followed by details of our cryogenic setup. Next, we provide a primer on thermopower that focuses on 2DESs at low temperatures. We then review our experimental devices, temperature calibration methods, as well as measurement circuits and protocols.

Latter chapters focus on the physics and thermopower results in the QHE regime. After reviewing the basic phenomena associated with the QHE, we discuss thermopower in this regime. Emphasis is given to the relationship between diffusion thermopower and entropy. Experimental results demonstrate this relationship persists well into the fractional quantum Hall (FQH) regime.

Several experimental results are reviewed. Unprecedented observations of the diffusion thermopower of a high-mobility 2DES at temperatures as high as T = 2 K are achieved using our hot-electron technique. The composite fermion (CF) effective mass is extracted from measurements of thermopower at LL filling factor ν = 3/2. The thermopower versus magnetic field in the FQH regime is shown to be qualitatively consistent with a simple entropic model of CFs. The thermopower at ν = 5/2 is shown to be quantitatively consistent with the presence of non-Abelian anyons. An abrupt collapse of thermopower is observed at the onset of the reentrant integer quantum Hall effect (RIQHE). And the thermopower at temperatures just above the RIQHE transition suggests the existence of an unconventional conducting phase.

Variable-angle electron spectroscopic studies of various polyatomic molecular systems

The complementary techniques of low-energy, variable-angle electron-impact spectroscopy and ultraviolet variable-angle photoelectron spectroscopy have been used to study the electronic spectroscopy and structure of several series of molecules. Electron-impact studies were performed at incident beam energies between 25 eV and 100 eV and at scattering angles ranging from 0° to 90°. The energy-loss regions from 0 eV to greater than 15 eV were studied. Photoelectron spectroscopic studies were conducted using a HeI radiation source and spectra were measured at scattering angles from 45° to 90°. The molecules studied were chosen because of their spectroscopic, chemical, and structural interest. The operation of a new electron-impact spectrometer with multiple-mode target source capability is described. This spectrometer has been used to investigate the spin-forbidden transitions in a number of molecular systems.

The electron-impact spectroscopy of the six chloro-substituted ethylenes has been studied over the energy-loss region from 0-15 eV. Spin-forbidden excitations corresponding to the π → π*, N → T transition have been observed at excitation energies ranging from 4.13 eV in vinyl chloride to 3.54 eV in tetrachloroethylene. Symmetry-forbidden transitions of the type π → np have been oberved in trans-dichloroethyene and tetrachlor oethylene. In addition, transitions to many states lying above the first ionization potential were observed for the first time. Many of these bands have been assigned to Rydberg series converging to higher ionization potentials. The trends observed in the measured transition energies for the π → π*, N → T, and N → V as well as the π → 3s excitation are discussed and compared to those observed in the methyl- and fluoro- substituted ethylenes.

The electron energy-loss spectra of the group VIb transition metal hexacarbonyls have been studied in the 0 eV to 15 eV region. The differential cross sections were obtained for several features in the 3-7 eV energy-loss region. The symmetry-forbidden nature of the ¹A_1g → ¹A_1g, 2t_2g(π) → 3t_2g(π*) transition in these compounds was confirmed by the high-energy, low-angle behavior of their relative intensities. Several low lying transitions have been assigned to ligand field transitions on the basis of the energy and angular behavior of the differential cross sections for these transitions. No transitions which could clearly be assigned to singlet → triplet excitations involving metal orbitals were located. A number of states lying above the first ionization potential have been observed for the first time. A number of features in the 6-14 eV energy-loss region of the spectra of these compounds correspond quite well to those observed in free CO.

A number of exploratory studies have been performed. The π → π*, N → T, singlet → triplet excitation has been located in vinyl bromide at 4.05 eV. We have also observed this transition at approximately 3.8 eV in a cis-/trans- mixture of the 1,2-dibromoethylenes. The low-angle spectrum of iron pentacarbonyl was measured over the energy-loss region extending from 2-12 eV. A number of transitions of 8 eV or greater excitation energy were observed for the first time. Cyclopropane was also studied at both high and low angles but no clear evidence for any spin- forbidden transitions was found. The electron-impact spectrum of the methyl radical resulting from the pyrolysis of tetramethyl tin was obtained at 100 eV incident energy and at 0° scattering angle. Transitions observed at 5.70 eV and 8.30 eV agree well with the previous optical results. In addition, a number of bands were observed in the 8-14 eV region which are most likely due to Rydberg transitions converging to the higher ionization potentials of this molecule. This is the first reported electron-impact spectrum of a polyatomic free radical.

Variable-angle photoelectron spectroscopic studies were performed on a series of three-membered-ring heterocyclic compounds. These compounds are of great interest due to their highly unusual structure. Photoelectron angular distributions using HeI radiation have been measured for the first time for ethylene oxide and ethyleneimine. The measured anisotropy parameters, β, along with those measured for cyclopropane were used to confirm the orbital correlations and photoelectron band assignments. No high values of β similar to those expected for alkene π orbitals were observed for the Walsh or Forster-Coulson-Moffit type orbitals.

I. Exact solution of some nonlinear evolution equations. II. The similarity solution for the Korteweg-De Vries equation and the related Painleve Transcendent

In Part I, a method for finding solutions of certain diffusive dispersive nonlinear evolution equations is introduced. The method consists of a straightforward iteration procedure, applied to the equation as it stands (in most cases), which can be carried out to all terms, followed by a summation of the resulting infinite series, sometimes directly and other times in terms of traces of inverses of operators in an appropriate space.

We first illustrate our method with Burgers' and Thomas' equations, and show how it quickly leads to the Cole-Hopft transformation, which is known to linearize these equations.

We also apply this method to the Korteweg and de Vries, nonlinear (cubic) Schrödinger, Sine-Gordon, modified KdV and Boussinesq equations. In all these cases the multisoliton solutions are easily obtained and new expressions for some of them follow. More generally we show that the Marcenko integral equations, together with the inverse problem that originates them, follow naturally from our expressions.

Only solutions that are small in some sense (i.e., they tend to zero as the independent variable goes to ∞) are covered by our methods. However, by the study of the effect of writing the initial iterate u_1 = u_(1)(x,t) as a sum u_1 = ^∼/u_1 + ^≈/u_1 when we know the solution which results if u_1 = ^∼/u_1, we are led to expressions that describe the interaction of two arbitrary solutions, only one of which is small. This should not be confused with Backlund transformations and is more in the direction of performing the inverse scattering over an arbitrary “base” solution. Thus we are able to write expressions for the interaction of a cnoidal wave with a multisoliton in the case of the KdV equation; these expressions are somewhat different from the ones obtained by Wahlquist (1976). Similarly, we find multi-dark-pulse solutions and solutions describing the interaction of envelope-solitons with a uniform wave train in the case of the Schrodinger equation.

Other equations tractable by our method are presented. These include the following equations: Self-induced transparency, reduced Maxwell-Bloch, and a two-dimensional nonlinear Schrodinger. Higher order and matrix-valued equations with nonscalar dispersion functions are also presented.

In Part II, the second Painleve transcendent is treated in conjunction with the similarity solutions of the Korteweg-de Vries equat ion and the modified Korteweg-de Vries equation.