Saddle-point Configuration Research Articles

Determinacy in linear rational expectations models

The purpose of this paper is to assess the relevance of rational expectations solutions to the class of linear univariate models where both the number of leads in expectations and the number of lags in predetermined variables are arbitrary. It recommends to rule out all the solutions that would fail to be locally unique, or equivalently, locally determinate. So far, this determinacy criterion has been applied to particular solutions, in general some steady state or periodic cycle. However solutions to linear models with rational expectations typically do not conform to such simple dynamic patterns but express instead the current state of the economic system as a linear difference equation of lagged states. The innovation of this paper is to apply the determinacy criterion to the sets of coefficients of these linear difference equations. Its main result shows that only one set of such coefficients, or the corresponding solution, is locally determinate. This solution is commonly referred to as the fundamental one in the literature. In particular, in the saddle point configuration, it coincides with the saddle stable (pure forward) equilibrium trajectory.

Critical Conditions for Dislocation Nucleation from Surface Steps

AbstractThe critical conditions for dislocation nucleation from the surface steps of various geometries are analyzed based on the Peierls-Nabarro dislocation model. By modeling a surface step as part of a three- dimensional crack surface, the half space problem is transferred into an equivalent three dimensional crack problem in an infinite medium. The profiles of embryonic dislocations, corresponding to the relative displacements between the two adjacent atomic layers along slip planes, are then rigorously solved through the variational boundary integral method. The critical conditions for dislocation nucleation are determined by solving the stress dependent activation energies required to activate embryonic dislocations from their stable to unstable saddle point configurations. For a given slip plane, the effects of step geometry such as the step height and inclined angle on dislocation nucleation are analyzed in detail. The results show that the atomic scale steps may reduce the critical stress required for dislocation nucleation from the surface by several factors. Compared to previous analyses of this type of problem based on continuum elastic dislocation theory, the presented analysis eliminates the uncertain core cutoff parameter by allowing for the existence of an extended dislocation core as the embryonic dislocation evolves. Because of the serious limitation of direct atomic simulation for this type of problem, the presented methodology of incorporating atomic information into continuum approach appears to be particularly noteworthy for providing insights of energetics of the atomic processes involved in dislocation nucleation.

Analysis of dislocation nucleation from a crystal surface based on the Peierls–Nabarro dislocation model

Dislocation nucleation from a stressed crystal surface is analyzed based on the Peierls–Nabarro dislocation model. The variational boundary integral approach is used to obtain the profiles of the embryonic dislocations in various three-dimensional nucleation configurations. The stress-dependent activation energies required to activate dislocations from their stable to unstable saddle point configurations are determined. Compared to previous analyses of this type of problem based on continuum elastic dislocation theory, the present analysis eliminates the uncertain core cutoff parameter by allowing for the existence of an extended dislocation core as the embryonic dislocation evolves. Moreover, atomic information can be incorporated to reveal the dependence of the nucleation process on the profile of the atomic interlayer potential as compared to continuum elastic dislocation theory in which only elastic constants and Burgers vector are relevant. Finally, the presented methodology can also be readily used to study dislocation nucleation from the surface heterogeneities such as cracks, steps, and quantum structures of electronic devices.

Saddle point for oxygen reorientation in the vicinity of a silicon vacancy

A piezospectroscopic analysis of the vacancy-oxygen complex in silicon has enabled us to demonstrate that this defect in the unstable configuration of the saddle point on the reconfiguration trajectory has a trigonal symmetry. This unstable defect configuration may be considered as the precursor for an oxygen diffusion process where the migrating oxygen atom is accompanied by a vacancy. The trigonal saddle point configuration results in a strong electrical polarization of the pair which can aid the jumping to a neighboring unit cell. This scenario is very plausible to explain how vacancies can drag oxygen atoms through the crystal to form larger oxygen aggregates.

UNIQUENESS OF BUBBLE-FREE SOLUTION IN LINEAR RATIONAL EXPECTATIONS MODELS

One usually identifies bubble solutions to linear rational expectations models by extra components (irrelevant lags) arising in addition to market fundamentals. Although there are still many solutions relying on a minimal set of state variables, i.e., relating in equilibrium the current state of the economic system to as many lags as initial conditions, there is a conventional wisdom that the bubble-free (fundamentals) solution should be unique. This paper examines the existence of endogenous stochastic sunspot fluctuations close to solutions relying on a minimal set of state variables, which provides a natural test for identifying bubble and bubble-free solutions. It turns out that only one solution is locally immune to sunspots, independently of the stability properties of the perfect-foresight dynamics. In the standard saddle-point configuration for these dynamics, this solution corresponds to the so-called saddle stable path.

Energetics of Dislocation Nucleation under a Nanoindenter

AbstractDislocation nucleation under an idealized nanoindenter is analyzed based on the Peierls-Nabarro dislocation model. The variational boundary integral method is used to obtain the dislocation profiles that correspond to the shear displacements between the adjacent atomic layers along the slip plane. The critical condition for dislocation nucleation at absolute zero and the activation energies required to thermally activate dislocations from their stable to unstable saddle point configurations are determined. By treating the surface as part of an infinite crack embedded in the infinite medium, a rather straightforward approach is adopted to account for the surface effect. The emission of multiple dislocations from the surface of a film-substrate system is also studied. The size effects of the indenter width and film thickness are characterized by the typical load-displacement relation.

Energetics of nucleation of half dislocation loops at a surface crack

Abstract A previously developed variational boundary integral method in the Peierls-Nabarro framework is extended to analyse nucleation of half dislocation loops at a semicircular surface crack of near-atomic dimensions. The saddle-point configurations of embryonic dislocations and their associated thermal activation energies are determined. The influence of the crack on the energetics of dislocation nucleation is ascertained by calculating these activation energies for cracks of various radii. As the radius of the crack approaches zero, the result naturally converges to that for homogeneous nucleation of a half dislocation loop from the perfect surface. Compared with previous studies of this problem based on continuum elastic dislocation theory, the model presented in this paper provides a more definitive solution because it eliminates the uncertain core cutoff parameter and presumption that the dislocation loop is of semicircular shape.

(1+1)-dimensional baryons from the SU( N) color-flavor transformation

The color-flavor transformation, an identity that connects two integrals, each of which is over one of a dual pair of Lie groups acting in the fermionic Fock space, is extended to the case of the special unitary group. Using this extension, a toy model of lattice QCD is studied: N f species of spinless fermions interacting with strongly coupled SU( N c ) lattice gauge fields in 1+1 dimensions. The color-flavor transformed theory is expressed in terms of gauge singlets, the meson fields, organized into sectors distinguished by the distribution of baryonic flux. A comprehensive analytical and numerical search is made for saddle-point configurations of the meson fields, with various topological charges, in the vacuum and single-baryon sectors. Two definitions of the static baryon on the square lattice, straight and zigzag, are investigated. The masses of the baryonic states are estimated using the saddle-point approximation for large N c .

Vortex correlation functions in Maxwell–Chern–Simons models

Maxwell-Chern-Simons models in the presence of an instanton anti-instanton background are studied. The saddle-point configuration corresponds to the creation and annihilation of a vortex localized around the Dirac string needed to support the nontrivial background. This configuration is generalized to the case in which a nonlocal Maxwell term is allowed in order to fulfill the finite action requirement. Following 't Hooft procedure, we compute the vortex correlation functions and we study the possibility of obtaining spin 1/2 excitations. A possible connection with the bosonization of interacting three-dimensional massive fermionic systems is also discussed.

A hierarchical family of global analytic Born–Oppenheimer potential energy surfaces for the H+H2 reaction ranging in quality from double-zeta to the complete basis set limit

A hierarchical family of analytical Born–Oppenheimer potential energy surfaces has been developed for the H+H2 system. Ab initio calculations of near full configuration interaction (FCI) quality (converged to within ≈1 μEh) were performed for a set of 4067 configurations with the aug-cc-pVDZ, aug-cc-pVTZ, and aug-cc-pVQZ basis sets. The complete basis set (CBS) limit energies were obtained using a highly accurate many-body basis set extrapolation scheme. Surfaces were fitted for the estimated CBS limit, as well as for the aug-cc-pVDZ, aug-cc-pVTZ, and aug-cc-pVQZ basis sets using a robust new functional form. The mean unsigned fitting error for the CBS surface is a mere 0.0023 kcal/mol, and deviations for data not included in the fitting process are of similarly small magnitudes. Highly accurate calculations of the saddle point and van der Waals minimum configurations were performed using basis sets as large as aug-mcc-pV7Z, and these data show excellent agreement with the results of the extrapolated potential surface. The remaining errors from fitting, correlation treatment, and basis set incompleteness for the new CBS-limit surface are lower by over an order of magnitude compared to any prior analytic surface, and are all now significantly smaller than non-Born–Oppenheimer effects. We expect that this new family of potential surfaces will prove useful in studies elucidating the sensitivity of dynamical quantities to the quality of the potential surface.

Kinetic Calculation of L10 Ordered System Based on Phase Field Methodand Cluster Variation Method

ABSTRACTDuring ordering process, anti-site ordering proceeds in atomistic scale and anti-phase domain structure evolves in microstructural scale. In order to describe both the processes, a hybridized calculation of the Phase Field Method(PFM) and Cluster Variation Method(CVM) is attempted. The main objective of the present study is focused on the time evolution of atomic configuration during L10 ordering processes below and above the spinodal ordering temperature and their resultant microstructures. In order to investigate the interplay between atomistic and microstructural processes, we conducted two types of calculations. One is for a homogeneous system without an anti-phase boundary and the other is for an inhomogeneous system in which microstructure is formed by anti-phase domains.For the homogeneous system, the relaxation curve of Long-Range-Order parameter(LRO) indicates a transient appearance of an L12-like atomic configuration below the spinodal ordering temperature. Such an L12–like state corresponds to a saddle point configuration in the CVM free energy surface. When the composition of an alloy is located near L10 + L12 phase field in the phase diagram, the L12–like phase becomes highly ordered state.For the inhomogeneous system, it is implied that the appearance of the L12-like phase affects the resultant microstructure by providing the nucleation sites for the L10 ordered phase.

Energetics of homogeneous nucleation of dislocation loops under a simple shear stress in perfect crystals

We present an analysis of homogeneous nucleation of a dislocation loop under an applied simple shear stress in a perfect crystal. By using a variational boundary integral method together with an interlayer potential in the Peierls-Nabarro (P-N) framework, we have determined the saddle-point configurations of embryonic dislocation loops and their associated activation energies under stress levels up to the ideal shear strength. The results, for typical materials such as Au, Cu, Al, and Si, indicate that energy barriers are far too high to suggest that thermal motion could play a role in nucleation of a dislocation loop in perfect crystals under usual stress levels, markedly below the ideal shear strength. These new findings broaden the earlier understanding of homogeneous nucleation of dislocations in crystalline materials and have significant implications on a number of existing models based on the mechanism of dislocation nucleation in deformation and fracture.

Dislocation nucleation from a penny-shaped crack

We present an analysis of dislocation nucleation from a penny-shaped crack of near-atomic dimensions under a simple shear stress. By using a variational boundary integral method in the Peierls–Nabarro framework, we have determined the saddle-point configurations of embryonic dislocation loops and their associated activation energies under stress levels up to the critical loading of athermal nucleation. The results show that the crack facilitates dislocation nucleation not only by stress concentration but also by supplying elastic strain energy. These new findings broaden the previous understanding of dislocation nucleation in crystalline materials and have significant implications on the development of mechanical models based on dislocation nucleation mechanisms in plastic deformation and fracture.

Analysis of Nucleation of Dislocation Loops from Stressed Surfaces Based on the Peierls-Nabarro Dislocation Model

ABSTRACTNucleation of dislocation loops from stressed crystal surfaces is analyzed based on a variational boundary integral method in the Peierls-Nabarro framework. The stress dependent activation energies required to activate dislocation loops from their stable to unstable saddle point configurations are determined. Compared to previous analyses of this problem based on continuum elastic dislocation theory, the presented analysis provides more definitive solutions because it eliminates the uncertain core cutoff parameter by allowing for the existence of an extended dislocation core as the embryonic dislocation evolves. Moreover, the shape of the dislocation loop is solved by the variational principle instead of assumed to be semicircular as in previous analyses based on continuum elastic dislocation theory. It is noteworthy that the presented methodology can be readily used to study effects of surface inhomogeneities such as cracks and steps on dislocation nucleation.

Temperature dependence of the flow stress of III–V compounds

Abstract An interpretation for the temperature dependence of the plastic flow stress of III–V compounds is presented, invoking only non-dissociated screw dislocations. With the aid of the Stillinger-Weber potential for the interatomic interaction, the Peierls potential of a non-dissociated screw dislocation is deduced in twodimensional space normal to the dislocation line. A saddle-point configuration and the formation energy of a kink pair are calculated in three-dimensional space. The relation obtained between the flow stress T c and temperature T under a constant strain rate describes the experimental T c−T relation of III–V compound well: a strong T dependence at highT c and low T, a stronger T dependence at low Tc and high T, and a plateau-like T c at intermediate T. The hump in the T c−T relation is interpreted as a transition between different paths of the non-dissociated screw dislocation: a planar path and a zigzag path in the (111) plane.

Homogeneous nucleation of dislocation loops under stress in perfect crystals

We present an analysis and results on the homogeneous nucleation of a dislocation loop under stress in a perfect crystal. By using a variational boundary integral method in the Peierls-Nabarro framework, we have determined the saddle-point configurations of embryonic dislocation loops and their associated activation energies under stress levels up to the ideal shear strength. The high-energy barriers under the usual levels of applied shear stresses, differing markedly from the ideal shear strength, confirm the widely held view that thermal motion should play no role in such nucleation. The result provides means for more definitive solutions of fundamental problems involving homogeneous nucleation of dislocation loops and has significant implications for models based on the mechanism of nucleation of dislocations from a perfect crystal.

Non-isotropic surface diffusion of lead on Cu(110): a molecular dynamics study

By using numerical simulation we have studied the diffusion of isolated lead atoms on Cu(110). The calculations rely on a phenomenological many-body potential derived in the framework of the second-moment approximation of the tight-binding method, with parameters fitted on the physical properties of the bulk crystals of copper and lead and to the copper–lead phase diagram. Static calculations, at T=0 K, provide the energy and relaxed atomic positions of the equilibrium and saddle-point configurations of various possible diffusion mechanisms. In spite of the large miscibility gap present in the lead–copper phase diagram, we find that insertion of a lead adatom into the uppermost copper surface layer is a thermodynamically favoured process. Molecular dynamics calculations show that inserted lead atoms diffuse via an exchange mechanism with copper adatoms and via jumps in adatom position along the open [1̄10] direction. These results confirm previously published experimental observations. They also confirm the validity of a statistical model that was developed to account for these observations. The quantity governing the variation of diffusion anisotropy with temperature is the difference E r− E j between the activation energies for the insertion of a lead atom in the copper plane and for its jumps in adatom position. The value of this difference, as determined in the static simulations, compares very well with what can be deduced from experimental observations. The agreement is also very good concerning the value of the main diffusion barrier, which is the energy associated with the de-insertion of a lead atom. Simulations performed in the temperature range 400–700 K show that multiple jumps occur frequently. Their frequency increases with temperature, thus leading to lead diffusion that is more anisotropic and more steeply dependent on temperature than could be expected from the static calculations.

Spontaneous breakdown of Lorentz invariance in IIB matrix model

We study the IIB matrix model, which is conjectured to be a nonperturbative definition of superstring theory, by introducing an integer deformation parameter `nu' which couples to the imaginary part of the effective action induced by fermions. The deformed IIB matrix model continues to be well-defined for arbitrary `nu', and it preserves gauge invariance, Lorentz invariance, and the cluster property. We study the model at `nu' = infinity using a saddle-point analysis, and show that ten-dimensional Lorentz invariance is spontaneously broken at least down to an eight-dimensional one. We argue that it is likely that the remaining eight-dimensional Lorentz invariance is further broken, which can be checked by integrating over the saddle-point configurations using standard Monte Carlo simulation.

Structure of higher order corrections to the Lipatov asymptotic form

High orders of perturbation theory can be calculated by the Lipatov method, whereby they are determined by saddle-point configurations, or instantons, of the corresponding functional integrals. For most field theories, the Lipatov asymptotic form has the functional form ca NΓ(N+b) (N is the order of perturbation theory) and the relative corrections to it are series in powers of 1/N. It is shown that this series diverges factorially and its high-order coefficients can be calculated using a procedure similar to the Lipatov one: the Kth expansion coefficient has the form const[ln(S 1/S 0)]−KΓ(K+(r 1− r 0)/2), where S 0 and S 1 are the values of the action for the first and second instantons of this particular field theory, and r 0 and r 1 are the corresponding number of zeroth-order modes; the instantons satisfy the same equation as in the Lipatov method and are assumed to be renumbered in order of their increasing action. This result is universal and is valid in any field theory for which the Lipatov asymptotic form is as specified above.

Self-Interstitial Configuration in B.C.C. Metals. An Analysis Based on Many-Body Potentials for Fe and Mo

A computer simulation study of the static distortion and stability of self-interstitials in Fe and Mo is presented. Many-body potentials of the type Embedded Atom and Embedded Defect, both developed by us, are employed. We confirm a result found earlier, that relatively short-range potentials predict the (measured) 〈110〉 dumbbell configuration to be of minimum energy, while longer range ones favour 〈111〉 extended structures. This finding is rationalized by considering the resulting different energy distributions within the defect cores. Also, for most potentials studied more than one mechanically stable interstitial is found. Based on the analysis of the saddle point configurations obtained with either interatomic potential, interstitial migration via both short and long jumps are expected in Fe, whereas only short jumps would be relevant in Mo.