Points In Parameter Space Research Articles

Reduction of the sign problem using the meron-cluster approach.

The sign problem in quantum Monte Carlo calculations is analyzed using the meron-cluster solution. A meron is a loop that alters the sign of the configuration, and the concept of merons can be used to solve the sign problem for a limited class of models. Here we show that the method can be used to reduce the sign problem in a wider class of models. We investigate how the meron solution evolves between a point in parameter space where it eliminates the sign problem and a point where it does not affect the sign problem at all. In this intermediate regime, the merons can be used to reduce the sign problem. The average sign still decreases exponentially with system size and inverse temperature, but with a different prefactor. The sign exhibits the slowest decrease in the vicinity of points where the meron-cluster solution eliminates the sign problem. We have used stochastic series expansion quantum Monte Carlo combined with the concept of directed loops.

Combinatorial investigations of interfacial failure

AbstractConventional measurements of interfacial strength focus on a single variable, whereas many variables couple nontrivially and simultaneously to define this property. We present a combinatorial methodology that allows the effects of multivariable environments on interfacial strength to be investigated in a high‐throughput, parallel, and quantitative manner. This technique is largely based on the theory of Johnson, Kendall, and Roberts that quantifies adhesion through the contact and separation of a spherical lens and flat substrate. For our experiments, we fabricated a combinatorial library consisting of a two‐dimensional array of spherical caps and a complementary substrate. The array of spherical caps was brought into contact and subsequently separated from the substrate, whereas the relative displacement and contact area of the individual lenses were recorded. With gradient library‐fabrication methods, two adhesion‐controlling parameters can be continuously varied along the orthogonal axes of the array. In this manner, each lens quantifies the interfacial strength at a unique point in parameter space. We demonstrate this multilens contact‐adhesion test by measuring the effect of temperature and coating thickness on the self‐adhesion of polystyrene thin films. © 2003 Wiley Periodicals, Inc. J Polym Sci Part B: Polym Phys 41: 883–891, 2003

Computational Modeling of Pool Games: Sensitivity of Outcomes to Initial Conditions

We present a study of the sensitivity of trajectories of pool balls to initial conditions. In the first component of the study our simulations include all sixteen balls. Variables include cue ball initial velocity and position on the “table”. We find that in a certain regime of initial conditions the system seems to show self-similarity, but as the range of initial cue ball angle and initial velocity is restricted, the system exhibits an interesting evolution towards a single point in parameter space, with the ball landing in only one pocket. We also examine the effects of varying the number of balls on the table, and how their dynamics may be interpreted using various plots and maps. Finally, the trajectory of a single cue ball is examined while it moves through the table space. Starting with the cue ball placed in the middle of the right wall of the table (traditional and rectangular in shape) and fired directly downward the system exhibits a two-cycle pattern. Then as the angle of fire is increased the system exhibits a four cycle, a three cycle and finally a two cycle all separated by noisy patterns. Effects of numerical artificialities are briefly discussed.

Scaling in ordered and critical random boolean networks.

Random Boolean networks, originally invented as models of genetic regulatory networks, are simple models for a broad class of complex systems that show rich dynamical structures. From a biological perspective, the most interesting networks lie at or near a critical point in parameter space that divides "ordered" from "chaotic" attractor dynamics. We study the scaling of the average number of dynamically relevant nodes and the median number of distinct attractors in such networks. Our calculations indicate that the correct asymptotic scalings emerge only for very large systems.

Stark resonances for a double quantum well: crossing scenarios, exceptional points and geometric phases

The complex energy resonances of a double δ potential well in a constant (Stark) field are studied. Varying the two system parameters (well distance and field strength) we investigate the behaviour of the resonance energies and wavefunctions both analytically and numerically. Different crossing scenarios for the real and imaginary parts of two resonance energies are observed and compared with a simple two-state model. In addition, a point in parameter space where both the real and imaginary parts of the two energies degenerate, an exceptional point, is found. Varying the system parameters around this exceptional point, the behaviour of energies and wavefunctions is discussed and the corresponding geometric phases, or Berry phases, for this non-Hermitian system are considered.

S-duality of the Leigh-Strassler Deformation via Matrix Models

We investigate an exactly marginal N=1 supersymmetric deformation of SU(N) N=4 supersymmetric Yang-Mills theory discovered by Leigh and Strassler. We use a matrix model to compute the exact superpotential for a further massive deformation of the U(N) Leigh-Strassler theory. We then show how the exact superpotential and eigenvalue spectrum for the SU(N) theory follows by a process of integrating-in. We find that different vacua are related by an action of the SL(2,Z) modular group on the bare couplings of the theory extending the action of electric-magnetic duality away from the N=4 theory. We perform non-trivial tests of the matrix model results against semiclassical field theory analysis. We also show that there are interesting points in parameter space where condensates can diverge and vacua disappear. Based on the matrix model results, we propose an exact elliptic superpotential to describe the theory compactified on a circle of finite radius.

Quantum Monte Carlo with directed loops.

We introduce the concept of directed loops in stochastic series expansion and path-integral quantum Monte Carlo methods. Using the detailed balance rules for directed loops, we show that it is possible to smoothly connect generally applicable simulation schemes (in which it is necessary to include backtracking processes in the loop construction) to more restricted loop algorithms that can be constructed only for a limited range of Hamiltonians (where backtracking can be avoided). The "algorithmic discontinuities" between general and special points (or regions) in parameter space can hence be eliminated. As a specific example, we consider the anisotropic S=1/2 Heisenberg antiferromagnet in an external magnetic field. We show that directed-loop simulations are very efficient for the full range of magnetic fields (zero to the saturation point) and anisotropies. In particular, for weak fields and anisotropies, the autocorrelations are significantly reduced relative to those of previous approaches. The back-tracking probability vanishes continuously as the isotropic Heisenberg point is approached. For the XY model, we show that back tracking can be avoided for all fields extending up to the saturation field. The method is hence particularly efficient in this case. We use directed-loop simulations to study the magnetization process in the two-dimensional Heisenberg model at very low temperatures. For LxL lattices with L up to 64, we utilize the step structure in the magnetization curve to extract gaps between different spin sectors. Finite-size scaling of the gaps gives an accurate estimate of the transverse susceptibility in the thermodynamic limit: chi( perpendicular )=0.0659+/-0.0002.

Entrainment and Decoupling Relations for Cloudy Boundary Layers

An idealized model of the relationship between entrainment in cloud-topped boundary layers, circulation structure, and the degree of decoupling between the cloud and subcloud layers is developed based on simple turbulent flux distributions and the premise that the entrainment rate, both at cloud top and across cloud base when some stability exists there, is controlled by the large-eddy structure for quasi-steady buoyantly driven conditions. Layers are classified in three regimes depending on whether the cloud-top entrainment rate is ultimately limited by the transport of eddies spanning the full boundary layer (I), the cloud layer (II), or the subcloud layer (III). Algebraic relations are derived for the boundaries between, and entrainment fluxes in, each regime as a function of a convenient set of physical input parameters. The transition from regime II into III, representing the decoupling transition leading to a cumulus coupled layer, is emphasized. The model predicts that decoupling is promoted by a decrease in Bowen ratio or increase in cloud-top humidity to temperature jump ratio, and, depending on the point in parameter space, either promoted or inhibited by an increase in cloud-top radiative cooling or an increase in cloud depth. In spite of the complex cloud-layer dynamics involving cumulus plumes, a simple prediction is given for the quasi-steady cloud-top entrainment rate in regime III based on the subcloud dynamics. The model is compared with results from an extensive set of large-eddy simulations varying surface heat and moisture fluxes, cloud-top humidity and temperature jumps, and relative cloud depth. Good agreement is found with the predicted entrainment rates, the qualitative layer structure, and location of the decoupling boundary in the parameter space varied.

A Mathematical and Computational Framework for Reliable Real-Time Solution of Parametrized Partial Differential Equations

We present in this article two components: these components can in fact serve various goals independently, though we consider them here as an ensemble. The first component is a technique for the rapid and reliable evaluation prediction of linear functional outputs of elliptic (and parabolic) partial differential equations with affine parameter dependence. The essential features are (i )( prov- ably) rapidly convergent global reduced-basis approximations — Galerkin projection onto a space WN spanned by solutions of the governing partial differential equation at N selected points in parameter space; (ii) a posteriori error estimation — relaxations of the error-residual equation that provide inex- pensive yet sharp and rigorous bounds for the error in the outputs of interest; and (iii) off-line/on-line computational procedures — methods which decouple the generation and projection stages of the ap- proximation process. This component is ideally suited — considering the operation count of the online stage — for the repeated and rapid evaluation required in the context of parameter estimation, design, optimization, and real-time control. The second component is a framework for distributed simulations. This framework comprises a library providing the necessary abstractions/concepts for distributed sim- ulations and a small set of tools — namely SimTEX and SimLaB— allowing an easy manipulation of those simulations. While the library is the backbone of the framework and is therefore general, the various interfaces answer specific needs. We shall describe both components and present how they interact.

Random coupling of chaotic maps leads to spatiotemporal synchronization.

We investigate the spatiotemporal dynamics of a network of coupled chaotic maps, with varying degrees of randomness in coupling connections. While strictly nearest neighbor coupling never allows spatiotemporal synchronization in our system, randomly rewiring some of those connections stabilizes entire networks at x*, where x* is the strongly unstable fixed point solution of the local chaotic map. In fact, the smallest degree of randomness in spatial connections opens up a window of stability for the synchronized fixed point in coupling parameter space. Further, the coupling epsilon(bifr) at which the onset of spatiotemporal synchronization occurs, scales with the fraction of rewired sites p as a power law, for 0.1<p<1. We also show that the regularizing effect of random connections can be understood from stability analysis of the probabilistic evolution equation for the system, and approximate analytical expressions for the range and epsilon(bifr) are obtained.

Bifurcation analysis of a class of first-order nonlinear delay-differential equations with reflectional symmetry

We consider a general class of first-order nonlinear delay-differential equations (DDEs) with reflectional symmetry, and study completely the bifurcations of the trivial equilibrium under some generic conditions on the Taylor coefficients of the DDE. Our analysis reveals a Hopf bifurcation curve terminating on a pitchfork bifurcation line at a codimension two Takens–Bogdanov point in parameter space. We compute the normal form coefficients of the reduced vector field on the centre manifold in terms of the Taylor coefficients of the original DDE, and in contrast to many previous bifurcation analyses of DDEs, we also compute the unfolding parameters in terms of these coefficients. For application purposes, this is important since one can now identify the possible asymptotic dynamics of the DDE near the bifurcation points by computing quantities which depend explicitly on the Taylor coefficients of the original DDE. We illustrate these results using simple model systems relevant to the areas of neural networks and atmospheric physics, and show that the results agree with numerical simulations.

Classification of stringlike solutions in dilaton gravity

The static stringlike solutions of the Abelian Higgs model coupled to dilaton gravity are analyzed and compared to the nondilatonic case. Except for a special coupling between the Higgs Lagrangian and the dilaton, the solutions are flux tubes that generate a nonasymptotically flat geometry. Any point in parameter space corresponds to two branches of solutions with two different asymptotic behaviors. Unlike the nondilatonic case, where one branch is always asymptotically conic, in the present case the asymptotic behavior changes continuously along each branch.

A PosterioriError Estimation for Reduced-Basis Approximation of Parametrized Elliptic Coercive Partial Differential Equations: “Convex Inverse” Bound Conditioners

We present a technique for the rapid and reliable prediction of linear-functional outputs of elliptic coercive partial differential equations with affine parameter dependence. The essential components are (i ) (provably) rapidly convergent global reduced-basis approximations – Galerkin projection onto a space W N spanned by solutions of the governing partial differential equation at N selected points in parameter space; (ii ) a posteriori error estimation – relaxations of the error-residual equation that provide inexpensive bounds for the error in the outputs of interest; and ( iii ) off-line/on-line computational procedures – methods which decouple the generation and projection stages of the approximation process. The operation count for the on-line stage – in which, given a new parameter value, we calculate the output of interest and associated error bound – depends only on N (typically very small) and the parametric complexity of the problem; the method is thus ideally suited for the repeated and rapid evaluations required in the context of parameter estimation, design, optimization, and real-time control. In our earlier work we develop a rigorous a posteriori error bound framework for reduced-basis approximations of elliptic coercive equations. The resulting error estimates are, in some cases, quite sharp: the ratio of the estimated error in the output to the true error in the output, or effectivity , is close to (but always greater than) unity. However, in other cases, the necessary “bound conditioners” – in essence, operator preconditioners that (i ) satisfy an additional spectral “bound” requirement, and (ii ) admit the reduced-basis off-line/on-line computational stratagem – either can not be found, or yield unacceptably large effectivities. In this paper we introduce a new class of improved bound conditioners: the critical innovation is the direct approximation of the parametric dependence of the inverse of the operator (rather than the operator itself); we thereby accommodate higher-order (e.g. , piecewise linear) effectivity constructions while simultaneously preserving on-line efficiency. Simple convex analysis and elementary approximation theory suffice to prove the necessary bounding and convergence properties.

Reliable Real-Time Solution of Parametrized Partial Differential Equations: Reduced-Basis Output Bound Methods

We present a technique for the rapid and reliable prediction of linear-functional outputs of elliptic (and parabolic) partial differential equations with affine parameter dependence. The essential components are (i) (provably) rapidly convergent global reduced-basis approximations—Galerkin projection onto a space WN spanned by solutions of the governing partial differential equation at N selected points in parameter space; (ii) a posteriori error estimation—relaxations of the error-residual equation that provide inexpensive yet sharp and rigorous bounds for the error in the outputs of interest; and (iii) off-line/on-line computational procedures methods which decouple the generation and projection stages of the approximation process. The operation count for the on-line stage in which, given a new parameter value, we calculate the output of interest and associated error bound, depends only on N (typically very small) and the parametric complexity of the problem; the method is thus ideally suited for the repeated and rapid evaluations required in the context of parameter estimation, design, optimization, and real-time control.

Five regimes of the quasi-cnoidal, steadily translating waves of the rotation-modified Korteweg-de Vries (“Ostrovsky”) equation

The rotation-modified Korteweg-de Vries (RMKdV) equation differs from the ordinary KdV equation only through an extra undifferentiated term due to Coriolis force. This article describes the steadily travelling, spatially periodic solutions which have peaks of identical size. These generalize the “cnoidal” waves of the KdV equation. There are five overlapping regimes in the parameter space. We derive four different analytical approximations to interpret them. There is also a narrow region where the solution folds over so that there are three distinct shapes at a given point in parameter space. The shortest of these shapes is approximated everywhere in space by a parabola except for a thin interior layer at the crest. A low-order Fourier–Galerkin algorithm, usually thought of as a numerical method, also yields an explicit analytic approximation, too. We illustrate the usefulness of these approximations through comparisons with pseudospectral Fourier numerical computations over the whole parameter space.

Classification of behaviour in a steady plug-flow model of catalytic combustion

We take a model for steady-state plug-flow in a single channel of a catalytic monolith reactor and then solve the appropriate boundary value problem for a range of parameter values, using continuation methods adapted from dynamical systems theory. The problem is solved using either of two methods, shooting or relaxation. We show the effect of varying different parameters on the positions of the turning points in parameter space, and discuss the practical implications of these variations in qualitative behaviour for the design and operation of such a system.

Numerical unfoldings of codimension-three resonant homoclinic flipbifurcations

Resonant homoclinic flip bifurcations arecodimension-three phenomena that act as organizingcentres for codimension-two inclination flip, orbit flipand eigenvalue-resonance bifurcations of homoclinicorbits to a real saddle. In a recent paper by Homburg andKrauskopf unfoldings for several cases of resonanthomoclinic flip bifurcations were proposed asbifurcation diagrams on a sphere around the centralsingularity.This paper presents a comprehensive numerical investigation into theseunfoldings in a specific three-dimensional vector field, which wasconstructed by Sandstede to explicitly contain inclination flip and orbitflip bifurcations. For both orbit and inclination flips, different cases canbe classified according to the eigenvalues of the saddle point. All possiblecases are treated including complicated ones involvinghomoclinic-doubling cascades and chaos. In each case, by choosing asufficiently small sphere around the codimension-three point inparameter space, the conjectured unfoldings are largely confirmed.However, for larger spheres interesting new codimension-threebifurcations occur, leading to a more complicated bifurcation structure.The results suggest an important trade-off between finding bifurcationcurves numerically and introducing new bifurcations by enlarging thesphere too much.

Complex periodic orbits, renormalization, and scaling for quasiperiodic golden-mean transition to chaos.

At the critical point of the golden-mean quasiperiodic transition to chaos we show the presence of an infinite sequence of unstable orbits in complex domain with periods given by the Fibonacci numbers. The Floquet eigenvalues (multipliers) are found to converge fast to a universal complex constant. We explain this result on the basis of the renormalization group approach and suggest using it for accurate estimates of the location of the golden-mean critical points in parameter space for a class of nonlinear dissipative systems defined analytically. As an example, we obtain data for the golden-mean critical point in the two-dimensional dissipative invertible map of Zaslavsky. We give a set of graphical illustrations for the scaling properties and emphasize that demonstration of self-similarity on two-dimensional diagrams of Arnold tongues requires the use of a properly chosen curvilinear coordinate system. We discuss a procedure of construction of the appropriate local coordinate system in the parameter plane and present the corresponding data for the circle map and Zaslavsky map.

A Hopf/steady-state mode interaction in rotating convection: bursts and heteroclinic cycles in a square periodic domain

We investigate the transition between oscillatory and steady convection at onset in low Prandtl number rotating convection in a plane layer. This transition is dominated by mode interactions which, at one point in parameter space, can be posed on a square lattice. This allows a rigorous reduction to a finite-dimensional bifurcation problem. We construct the normal form and compute the normal form coefficients for this codimension-2 bifurcation directly from the PDEs for Boussinesq rotating convection with stress-free upper and lower boundaries. The dynamics near the codimension-2 point are investigated fully; they explain behaviour found in numerical simulations of the PDEs at parameter values near the transition boundary. In particular, the normal form exhibits bursting dynamics created by a heteroclinic cycle containing points ‘at infinity’.

Phase synchronization with type-II intermittency in chaotic oscillators

We study the phase synchronization (PS) with type-II intermittency showing +/-2pi irregular phase jumping behavior before the PS transition occurs in a system of two coupled hyperchaotic Rossler oscillators. The behavior is understood as a stochastic hopping of an overdamped particle in a potential which has 2pi-periodic minima. We characterize it as type-II intermittency with external noise through the return map analysis. In epsilon(t)<epsilon<epsilon(c) (where epsilon(t) is the bifurcation point of type-II intermittency and epsilon(c) is the PS transition point in coupling strength parameter space), the average length of the time interval between two successive jumps follows <l> approximately exp(|epsilon(t)-epsilon|(2)), which agrees well with the scaling law obtained from the Fokker-Planck equation.