Complex Initial Conditions Research Articles

Interference in Bohmian mechanics with complex action

In recent years, intensive effort has gone into developing numerical tools for exact quantum mechanical calculations that are based on Bohmian mechanics. As part of this effort we have recently developed as alternative formulation of Bohmian mechanics in which the quantum action S is taken to be complex [Y. Goldfarb et al., J. Chem. Phys. 125, 231103 (2006)]. In the alternative formulation there is a significant reduction in the magnitude of the quantum force as compared with the conventional Bohmian formulation, at the price of propagating complex trajectories. In this paper we show that Bohmian mechanics with complex action is able to overcome the main computational limitation of conventional Bohmian methods-the propagation of wave functions once nodes set in. In the vicinity of nodes, the quantum force in conventional Bohmian formulations exhibits rapid oscillations that present a severe numerical challenge. We show that within complex Bohmian mechanics, multiple complex initial conditions can lead to the same real final position, allowing for the accurate description of nodes as a sum of the contribution from two or more crossing trajectories. The idea is illustrated on the reflection amplitude from a one-dimensional Eckart barrier. We believe that trajectory crossing, although in contradiction to the conventional Bohmian trajectory interpretation, provides an important new tool for dealing with the nodal problem in Bohmian methods.

Dynamical singularities for complex initial conditions and the motion at a real separatrix

This work investigates singularities occurring at finite real times in the classical dynamics of one-dimensional double-well systems with complex initial conditions. The objective is to understand the relationship between these singularities and the behavior of the systems for real initial conditions. An analytical treatment establishes that the dynamics of a quartic double well system possesses a doubly infinite sequence of singularities. These are associated with initial conditions that converge to those for the real separatrix as the singularity time becomes infinite. This confluence of singularities is shown to lead to the unstable behavior that characterizes the real motion at the separatrix. Numerical calculations confirm the existence of a large number of singularities converging to the separatrix for this and two additional double-well systems. The approach of singularities to the real axis is of particular interest since such behavior has been related to the formation of chaos in nonintegrable systems. The properties of the singular trajectories which cause this convergence to the separatrix are identified. The hyperbolic fixed point corresponding to the potential energy maximum, responsible for the characteristic motion at a separatrix, also plays a critical role in the formation of the complex singularities by delaying trajectories and then deflecting them into asymptotic regions of space from where they are directly repelled to infinity in a finite time.

A modified cellular automata algorithm for the simulation of boundary advancement in deposition topography simulation

This paper presents a two-dimensional (2D) modified cellular automata (CA) algorithm for accurate boundary movement of deposition process simulation. The algorithm incorporates deposition rate calculation methods based on a cell structure. Isotropic and anisotropic deposition processes were successfully simulated by the algorithm, and the simulation profiles show good agreement with available experimental results. The deposition process with defects on the substrate was also implemented using this algorithm. The 2D modified CA algorithm demonstrates accuracy and capability of handling complex boundary and initial conditions. It is useful for the deposition process simulation of microelectromechanical systems and integrated circuits.

Relationship Between Singularities in Classical Mechanics for Complex Initial Conditions and for Complex Time

A relationship is established between the functional forms of two kinds of singularities in dynamical variables that arise in complexified versions classical mechanics: singularities that are treated as a functions of complex initial conditions for real time and those that are treated as a functions of complex time for real initial conditions. The analysis is verified by numerical calculations. The results imply that Kowaleskaya–Painleve condition for integrability can be phrased in terms of singularities with respect to initial conditions.

Estimation of time since death by heat-flow Finite-Element model. Part I: method, model, calibration and validation

The determination of the time since death which often represents the presumed time of an offence plays an important role in medico-legal practice. In the early postmortem phase analyses of postmortem cooling provide the most accurate estimates. Empirical models of postmortem cooling are methodically restricted to standard conditions while heat flow models can in principle be applied to any complex cooling situations. The main problem having so far prevented heat flow models from being used in practice was the difficulty of solving the heat transfer equation for complex geometrical, initial and boundary conditions. This problem is now overcome by using the Finite-Element-Method as a numerical procedure. The study presents a three-dimensional Finite-Element-Model of the human body containing various tissue compartments with different thermal tissue properties. The initial temperature field is modelled inhomogeneously with a temperature gradient between body core and shell. Heat loss by conduction, convection and radiation as well as heat gain by supravital activity or irradiation from external sources can be simulated. One model parameter, the decrease rate of the supravital energy production, was calibrated and the model successfully validated using the experimentally verified empirical model by Marshall and Hoare.

Impact of graben-parallel faults on hydrothermal convection––Rhine Graben case study

Detailed investigations of the temperature distribution in the Rhine Graben indicate a regular pattern of graben-parallel thermal anomalies following major north–south striking faults. These faults, which might imply north–south convection pattern due to higher permeability, have been ignored in conventional models based on 2D east–west striking sections. The present study investigates small-scale temperature anomalies in the Rhine Graben, focusing on its north–south striking major faults and their important permeability structures. Analytical solutions for convective flow in vertical faults were applied to a clearly observable anomalous temperature pattern. Through these calculations the fault geometry, minimal fault permeability and time to convective onset were derived from observed distances between anomalies. Since analytical solutions are limited to simple model geometries, further improvement was achieved through numerical model simulations, which allow the assumption of more complex initial and boundary conditions. Using the finite volume code TOUGH2 we investigated in detail the convective fluid flow pattern along the Γ-fault in the Landau region through a 3D model. There, regions of alternatively higher and lower temperatures follow each other at an average distance of 1800 m. Focusing on the predominant north–south permeability structure, our model consists of a vertical north–south striking fault and surrounding matrix with fault geometries based on analytical predictions. The observed temperature could be explained by a numerical model indicating fluid velocities in the fault of ∼10 −9 m s −1 and an anomaly whose minimal age is 77,000 years. The study proves that the observed small-scale graben-parallel temperature anomalies can be explained only by convection systems within north–south striking fault zones. The regional flow regime is therefore significantly influenced by the convective fluid flux in major north–south striking faults. Investigations of temperature anomalies in the Rhine Graben must take into account this north–south flow. Since the presented results correlate with findings from other geothermal anomalies in the Rhine Graben such as the one at Soultz, we suspect this to be a general feature of the major north–south striking faults there. Moreover, since in most grabenlike structures thermal anomalies are associated with the Graben master faults, the present study may provide common clues for investigating other graben-parallel temperature anomalies.

Planetesimal clusters in a Keplerian disk

It was recently demonstrated by numerical simulations that a turbulent flow in a rotating system is capable of eciently concentrating passively advected particles - having a density larger than the fluid - inside anti-cyclonic vortices. This process has important consequences on the distribution of solid particles in protoplanetary disks, since dust surface densities 1-2 orders of magnitude higher than the background are rapidly reached in vortex cores. However, until now, the role of self-gravitation of captured solids has been neglected. In this work we study the action of mutual gravitational interactions - after the gas has dissipated - over the dynamics of planetesimals inside clusters similar to those created in vortex cores. A comparison is made between the behavior of idealized clusters of planetesimals characterized by ad-hoc velocity profiles, and more complex initial conditions such as those obtained in previous hydrodynamical simulations. We show here that, within the explored interval of parameters, mutual scattering of particles can quickly disperse the cluster. Our results are demonstrated to be not dependent on the resolution employed. It can be concluded that if large planetesimals were formed inside vortex cores, they would be ejected by mutual perturbations.

Three-Dimensional Modeling of Microsphere Contact/Impact with Smooth, Flat Surfaces

A three-dimensional (3D) simulation model of a microsphere in contact with a nominally-smooth, flat surface is established. The model is based on Hertzian contact stresses, an idealized ring force distribution of adhesion, nonlinear damping, and friction. Although originally developed for the study of oblique impact, the model also is shown to describe the motion of a microsphere in sustained contact with a flat surface, including nonlinear normal oscillatory motion in the presence of sliding and rolling. Nonlinear, normal oscillatory motion is illustrated using conventional phaseplane techniques. Methods for the determination of damping coefficients associated with normal motion from impact experiments are discussed. The significance and modeling of rolling resistance and implications of asymmetric contact stress distributions are presented. Simulations show that energy exchange between translation and rotation can play an important role during oblique impact. The effects of complex initial conditions on the rebound and capture of a microsphere are significant as are the effects of contact friction. Results show that the inability to measure and the failure to account for rotational velocities in experimental measurements can limit the intepretation of the results.

Extension of a combined analytical/numerical initial value problem solver for unsteady periodic flow

AbstractHere we describe analytical and numerical modifications that extend the Differential Reduced Ejector/ mixer Analysis (DREA), a combined analytical/numerical, multiple species ejector/mixing code developed for preliminary design applications, to apply to periodic unsteady flow. An unsteady periodic flow modelling capability opens a range of pertinent simulation problems including pulse detonation engines (PDE), internal combustion engine ICE applications, mixing enhancement and more fundamental fluid dynamic unsteadiness, e.g. fan instability/vortex shedding problems. Although mapping between steady and periodic forms for a scalar equation is a classical problem in applied mathematics, we will show that extension to systems of equations and, moreover, problems with complex initial conditions are more challenging. Additionally, the inherent large gradient initial condition singularities that are characteristic of mixing flows and that have greatly influenced the DREA code formulation, place considerable limitations on the use of numerical solution methods. Fortunately, using the combined analytical–numerical form of the DREA formulation, a successful formulation is developed and described. Comparison of this method with experimental measurements for jet flows with excitation shows reasonable agreement with the simulation. Other flow fields are presented to demonstrate the capabilities of the model. As such, we demonstrate that unsteady periodic effects can be included within the simple, efficient, coarse grid DREA implementation that has been the original intent of the DREA development effort, namely, to provide a viable tool where more complex and expensive models are inappropriate. Copyright © 2002 John Wiley & Sons, Ltd.

Generalized Filinov transformation of the semiclassical initial value representation

An efficient method is proposed for the practical solution of the “sign” problem for integrals involved in the semiclassical initial value representation. It is based on a generalization of the conventional Filinov filtering procedure which has the (approximate) effect of incorporating complex initial conditions into the phase space average; it does this by including an explicit oscillatory term in the filtering function that partially cancels the oscillatory part of the original integrand. A systematic procedure is also described for making an optimal choice of the “smoothing parameters,” thus removing this arbitrariness in the overall approach. Tests on systems with chaotic dynamics demonstrate the accuracy and efficiency of the method.

A semianalytical method for solving equations describing the transport of contaminants in two-dimensional homogeneous porous media

The two-dimensional convection-dispersion equation with most complex boundary and initial conditions does not have an exact analytical solution and is therefore solved numerically. However, the solutions obtained with several traditional finite difference or finite element techniques typically exhibit spurious oscillation or numerical dispersion when convection is dominant. The semianalytical model proposed in this paper avoids such oscillations and numerical dispersion when convection dominates. The semianalytical model is the solution of ( n + m − 1)-order ordinary differential equations that are derived from the two-dimensional convection dispersion equation. The semianalytical model describes the contaminant concentration as a function of time and space, the space is reflected in parameters n and m. The results calculated with the semianalytical model are in good agreement with those obtained from the exact solution of a corresponding two-dimensional convection-dispersion equation, but the semianalytical model greatly simplifies the computation, making solutions for complex boundary and initial conditions possible.

Quantum creation of a universe with Ω≠1: Singular and nonsingular instantons

We propose two new classes of instantons which describe the tunneling and/or quantum creation of closed and open universes. The instantons leading to an open universe can be considered as generalizations of the Coleman-De-Luccia solution. They are non-singular, unlike the instantons recently studied by Hawking and Turok, whose prescription has the problem that the singularity is located on the hypersurface connecting to the Lorentzian region, which makes it difficult to remove. We argue that such singularities are harmless if they are located purely in the Euclidean region. We thus obtain new singular instantons leading to a closed universe; unlike the usual regular instantons used for this purpose, they do not require complex initial conditions. The singularity gives a boundary contribution to the action which is small for the instantons leading to sufficient inflation, but changes the sign of the action for small $\phi$ corresponding to short periods of inflation.

Developments in the Physics of Infiltration

AbstractDevelopments in the physics of infiltration, particularly the many theoretical contributions of John Philip, have been based on solutions of Richards' equation for flow in unsaturated soils. Solutions for constant surface soil‐water pressure and constant surface flux into soils initially at a uniform water content have given the physical basis for infiltration that generally takes place in conditions described by more complex initial and boundary conditions. The Green and Ampt step function and Gardner's exponential function models for the relationship between hydraulic conductivity and soil‐water pressure head are often employed to facilitate solutions. However, Richards' equation makes assumptions in the physics of the flow that do not take account of many complicating factors that may vitiate theories based on it. These factors include the influence of the air phase on the soil‐water movement, the effects of soil heterogeneity, soil swelling, soil aggregation and soil instability, deviations from Darcy's law, thermal effects and the hysteresis in soil‐water relationships. These factors need to be addressed in order to provide a better physical understanding of infiltration behavior in the field, first by understanding simple systems, as with much of John Philip's work, and then by progressing to more complex situations.

Breakdown of the slow manifold in the Shallow-Water equations

Abstract Numerical solutions are obtained by implicit multigrid solvers for initial-value problems in the rotating Shallow-Water Equations (SWE) with spatially complex initial conditions. Companion solutions are also obtained with the Shallow-Water Balance Equations (SWBE), both to determine the initial conditions for the SWE and to provide a comparison solution that lies entirely on the slow, advective manifold. We make use of a control parameter (here the Rossby number, R) to regulate the degree of slowness and balance. While there are measurable discrepancies between the evolving SWE and SWBE solutions for all R, there is a distinct, spatially local breakdown both of the slow manifold in the SWE solution and in the closeness of correspondence between the SWE and SWBE solutions. This critical value for breakdown is only slightly smaller than the R values at which, first, the SWE evolution becomes singular (i.e., the fluid depth vanishes), or second, a consistent initial condition for the SWBE cannot be ...

Behavior of Cantilever Retaining Walls

Current methods for retaining‐wall analysis are based on the classical theoretical formulations of Rankine or Coulomb, with the assumption that sufficient lateral yield will occur to mobilize fully active conditions behind the wall. This leads to relatively simple equations for active pressure distributions. The effects of the deformation parameters of the backfill and foundation subsoil, construction sequence, wall flexibility and other complex initial conditions are not taken into consideration. In this paper the finite element method is employed to investigate the behavior of concrete cantilever retaining walls. The analyses reveal that the wall and soil movements are of considerable importance in predicting the lateral earth pressures. The results indicate that the lateral pressures acting on the wall stem are generally close to the active condition only in the top two‐thirds of the wall stem. Based on this study, a simplified design procedure is proposed for estimating the lateral earth pressures in cantilever retaining walls with loosely placed backfill.

Are the equilibrium concentrations for a chemical reaction always uniquely determined by the initial concentrations?

A question not often addressed is whether the equilibrium concentrations for a given set of initial concentrations are always unique, or for some complex reaction and initial conditions is it possible for more than one set of chemically meaningful equilibrium concentrations to be satisfied?

An improved stability result for resonances

We prove stability of shape resonances for the sequence of Schrödinger equations ( − d 2 / d x 2 + U ( x ) + W n ( x ) ) ψ ( x ) = E ψ ( x ) , 0 ⩽ x > ∞ ( - {d^2}/d{x^2} + U(x) + {W_n}(x))\psi (x) = E\psi (x),0 \leqslant x\, > \infty , in the limit n → ∞ n \to \infty where the barrier potentials W n ( x ) {W_n}(x) are integrable, nonnegative, supported in the interval [ 1 , a ] ( 1 > a > ∞ ) [1,a]\;(1 > a > \infty ) , and approach infinity pointwise a.e. for x ∈ [ 1 , a ] x \in [1,a] as n → ∞ n \to \infty . In the course of our investigation we prove that for suitable complex initial conditions the solution to the Riccati equation S ′ ( x ) = 1 − ( W n ( x ) − E ) [ S ( x ) ] 2 S’(x) = 1 - ({W_n}(x) - E){[S(x)]^2} goes to 0 0 as n → ∞ n \to \infty uniformly on compact subsets of [ 1 , a ] [1,a] . Our approach is via ordinary differential equations using outgoing wave boundary conditions to define resonances. Our stability result extends a similar result of Ashbaugh and Harrell, who use an argument based on asymptotics and the implicit function theorem to study the above problem with λ V ( x ) \lambda V(x) replacing W n ( x ) {W_n}(x) . Our approach is to use the Riccati equation analysis mentioned above and an application of Hurwitz’s Theorem from complex variable theory.

Software for a method of finite approximations for the numerical solution of differential equations

solution of the given nonlinear differential equation, is obtained by means of software of a fairly simple structure. Problems with complex initial, boundary or mixed conditions, involving combinations of function and derivative values, can be dealt with by means of our program. Encouraging results have been obtained in the solution of equations with regions of rapid variation, oscilla- tory behaviour and in the presence of stiffness. As well as in the treatment of nonlinear boundary value problems where the solution is not unique. Examples are given in the last section of the paper. Applications to eigenvalue problems and to partial differential equations are reported elsewhere. Finite approximations method based on tim Tau method: different approaches in the design of numerical algorithms

The Determination of Stresses and Temperatures in Cooling Bodies by Finite Elements

This paper presents a finite element method for solving non-linear heat conduction and thermal stress problems. The nonlinearity is caused by the variation of thermo-physical and mechanical properties of the material with respect to temperature. The distribution of temperature and stress in a cooling body can be followed from the onset of solidification. The method is capable of handling complex geometries and initial non-linear boundary conditions.

Backward Formalism to Derive the Size of Secondary Ejected Droplets Produced by Crown Splashing of Drops Impinging on a Solid Wall

Crown splashing, produced by high speed impact of a droplet on a rough or wet wall, is physically very complicated. It is impossible to determine the size of secoundary ejected droplets by literally solving the full set of nonlinear Navier-Stokes equations supplemented by complex initial and boundary conditions. In order to get useful impact laws and, most importantly, to propose a genearl concept of deriving useful results without going through the complex mathematical details, we propose a backward formalism in which we determine the size of the secoundary ejected droplets by tracking the past event whenever it is required and just what is required. This procedure allows us to discard those complex details of negligible importance. Such a formalism, conceptually very simple and possibly meaningful for other complex problems, leads to a reasonably correct formula for the most probable diameter of secondary ejected droplets, compared to know experimental data.