Semi-analytical Approximation Research Articles

Dynamics of the effective mass and the anomalous velocity in two-dimensional lattices

The semiclassical description of the dynamics of wave packets in periodic potentials and subject to an applied force relies on the concepts of effective mass and anomalous transport. This picture is valid if the force changes slowly in time and space, so that the particle described by the wave packet has time to respond according to the properties of the lattice. We analyze the dynamical corrections to this picture when a uniform force is suddenly applied, identifying separate corrections to the usual group and anomalous velocities. We find approximate semianalytical expressions for generalized "dynamical" group and anomalous velocities, and the associated accelerations. We use a two-dimensional optical lattice with finite Berry curvature to illustrate the semianalytical approximation in a regime where the dynamical corrections are significant, suggesting the possibility of experiments to detect them; we compare the results with a full numerical solution, showing excellent agreement for weak forces.

Integral representations combining ladders and crossed-ladders

We use the worldline formalism to derive integral representations for three classes of amplitudes in scalar field theory: (i) the scalar propagator exchanging N momenta with a scalar background field (ii) the "half-ladder" with N rungs in x - space (iii) the four-point ladder with N rungs in x - space as well as in (off-shell) momentum space. In each case we give a compact expression combining the N! Feynman diagrams contributing to the amplitude. As our main application, we reconsider the well-known case of two massive scalars interacting through the exchange of a massless scalar. Applying asymptotic estimates and a saddle-point approximation to the N-rung ladder plus crossed ladder diagrams, we derive a semi-analytic approximation formula for the lowest bound state mass in this model.

Transient volume of evaporating sessile droplets: 2/3, 1/1, or another power law?

The transient shape and volume of evaporating sessile droplets are critical to our understanding and prediction of deposits left over on the solid surface after droplet evaporation. The 2/3 power law of scaling, (V/Vo)(β) = 1 - t/tf with β = 2/3, has been widely used. The 1/1 power law of scaling with β = 1 was also obtained for vanishingly small contact angles. Here we show that β significantly deviates from 2/3 and 1 when the droplet base is pinned: β depends on both initial and transient contact angles. The 1/1 power law presents the upper limit of β = 1, while β = 2/3 is the lower limit if contact angles are smaller than 148°. Unexpectedly, β can be smaller than 2/3 if contact angles are larger than 148°. We also present a semianalytical approximation for β as a function of the initial contact angle.

Hybrid Equity-Interest Rate Model with Stochastic Interest Rate and Volatility

This paper presents an Equity-IR hybrid model that fits in the class of affine diffusion processes. In the absence of cash dividend payment, the moment generating function can be easily and rapidly computed. This allows for an efficient calibration of the model based on Vanilla European Options prices. In practice, cash dividend payments are common and are modeled via the piece-wise and escrowed approaches. We show how to compute Vanilla European Options prices efficiently in this case. The semi-analytical approximation pricing formulas obtained in the piece-wise approach are one of our primary results.In order to assess the semi-analytical pricing method, we compare the computed prices to those obtained via Monte Carlo simulation. Three discretization schemes are used: the two first ones, which have been introduced by Andersen, are the Quadratic Exponential (QE) and its extension with martingale correction QE-M. The last scheme is a new extension of the QE scheme denoted QE-I, that we introduced in order to allow the use of larger simulation time steps by comparison to QE and QE-M.

DYNAMICAL FRICTION IN CUSPY GALAXIES

In this paper we treat the problem of the dynamical friction decay of a massive object moving in an elliptical galaxy with a cuspidal inner distribution of the mass density. We present results obtained by both self-consistent, direct summation, N-body simulations, as well as by a new semi-analytical treatment of dynamical friction valid in such cuspy central regions of galaxies. A comparison of these results indicates that the proposed semi-analytical approximation is the only reliable in cuspy galactic central regions, where the standard Chandrasekhar's local approximation fails, and, also, gives estimates of decay times that are correct at 1% respect to those given by N-body simulations. The efficiency of dynamical friction in cuspy galaxies is found definitively higher than in core galaxies, especially on more radially elongated satellite orbits. As another relevant result, we find a proportionality of the dynamical friction decay time to the -0.67 power of the satellite mass, M, shallower than the standardly adopted 1/M dependence.

Semi-analytical Approximation for Solving High-order Sturm-Liouville Problems

In this paper, an algorithm for solving high-order non-singular Sturm-Liouville eigenvalue problems is proposed. A modified form of Adomian decomposition method is implemented to provide a semianalytical solution in the form of a rapidly convergent series. Convergent analysis and error estimate based on the Banach fixed-point is discussed. Five high-order Sturm-Liouville problems are solved numerically. Numerical results demonstrate reliability and efficiency of the proposed scheme.

Tension Analysis of Submarine Cables during Laying Operations

This paper presents a semi-analytical approximation for a two-dimensional (2D) tension analysis of submarine cables during laying operations. In the analysis, based on geometric compatibility relations and equilibrium equations, a set of non-linear differential equations are obtained. The present model considers effects of ocean currents, cable ship motion, pay-out rate, water depth and material properties on submarine cable behavior in water, which are crucial during laying operations. As shown in numerical examples, with consideration of currents and cable ship motion, the cable tension appears to be smaller and cable configuration curve tends to be fatter than the conventional catenary theory.

Absorption of planar massless scalar waves by Kerr black holes

We consider planar massless scalar waves impinging upon a Kerr black hole, for general angles of incidence. We compute the absorption cross section via the partial wave approach, and present a gallery of results. In the low-frequency regime, we show that the cross section approaches the horizon area; in the high-frequency regime, we show that the cross section approaches the geodesic capture cross section. In the aligned case, we extend the complex angular momentum method to obtain a `sinc' approximation, which relates the regular high-frequency oscillations in the cross section to the properties of the polar null orbit. In the non-aligned case, we show, via a semi-analytic approximation, that the reduction in symmetry generates a richer, less regular absorption cross section. We separate the absorption cross section into corotating and counterrotating contributions, showing that the absorption is larger for counterrotating waves, as expected.

Graphene with time-dependent spin-orbit coupling: truncated Magnus expansion approach

We analyze the role of ac-driven Rashba spin-orbit coupling in monolayer graphene including a spin-dependent mass term. Using the Magnus expansion as a semi-analytical approximation scheme a full account of the quasienergie spectrum of spin states is given. We discuss the subtleties arising in correctly applying the Magnus expansion technique in order to determine the quasienergy spectrum. Comparison to the exact numerical solution gives appropriate boundaries to the validity of the Magnus expansion solution.

Structural properties of fluids interacting via piece-wise constant potentials with a hard core

The structural properties of fluids whose molecules interact via potentials with a hard core plus two piece-wise constant sections of different widths and heights are presented. These follow from the more general development previously introduced for potentials with a hard core plus n piece-wise constant sections [A. Santos, S. B. Yuste, and M. Lopez de Haro, Condens. Matter Phys. 15, 23602 (2012)] in which use was made of a semi-analytic rational-function approximation method. The results of illustrative cases comprising eight different combinations of wells and shoulders are compared both with simulation data and with those that follow from the numerical solution of the Percus-Yevick and hypernetted-chain integral equations. It is found that the rational-function approximation generally predicts a more accurate radial distribution function than the Percus-Yevick theory and is comparable or even superior to the hypernetted-chain theory. This superiority over both integral equation theories is lost, however, at high densities, especially as the widths of the wells and/or the barriers increase.

Advective Transport in Heterogeneous Formations: The Impact of Spatial Anisotropy on the Breakthrough Curve

Water flow and solute transport take place in formations of spatially variable conductivity K. The logconductivity Y = ln K is modeled as a random stationary space function, of normal univariate pdf (of mean In KG and variance \({\sigma_{Y}^{2}}\)) and of axisymmetric autocorrelation of integral scales Ih,Iv (anisotropy ratio f = Iv/Ih 1 }\) , which is of interest for many aquifers and is more difficult to solve either numerically or by approximations. We approach the three dimensional problem by modeling the structure as an ensemble of densely packed oblate spheroids of semi-major and semi-minor axis R and f R, respectively, and independent lognormal K, submerged in a matrix of uniform conductivity Kef, the effective conductivity of the ensemble. The detailed numerical simulations of transport show that the BTC is insensitive to the value of the anisotropy ratio f, i.e., μ (t, x) Ih/U depends only on \({\sigma _{Y}^{2}}\) (except for small differences in the tail). This important result implies that transport, as quantified by BTCs or spatial longitudinal mass distributions, can be modeled accurately by the much simpler solutions developed in the past for isotropic media, like e.g., the semi-analytical self-consistent approximation.

A semi-analytical closure approximation for pipe flows of two Herschel–Bulkley fluids with a stratified interface

We propose a semi-analytical closure approximation to the stratified flow of two Herschel–Bulkley fluids flowing along a circular pipe. The closure proposed appears to approximate the volumetric fluxes within 10–15% over a broad range of rheological parameters and interface heights. Accuracy is diminished at more extreme viscosity ratios when the less viscous fluid is in a thin layer. We demonstrate that the closure is able to give a good prediction of displacement front velocities in the viscous regime, comparing well with experimental values, and can be used for approximating other design features of a displacement process, e.g. the maximal static wall layer (in the case of a yield stress fluid) and the minimal flow rate to prevent backflow.

Semi-analytical approaches to assess the crack driving force in periodically heterogeneous elastic materials

When a crack propagates in a heterogeneous elastic material, its crack driving force depends strongly on the distribution of the local stiffness near the crack tip. In materials with periodic spatial variations of the Young’s modulus, shielding and anti-shielding effects appear, i.e. the crack driving force is reduced or enhanced, compared to a homogeneous material. The effect is of great practical relevance, since it may lead to a strong increase of the fracture resistance. The concept of configurational forces (CCF) offers an established procedure for calculating the crack driving force. A very general relation for the periodic variation of Young’s modulus is applied, allowing the description of both harmonically varying and layered microstructures. Numerical results are presented. Two semi-analytical approximation concepts, based on either the CCF or the moduli perturbation concept, are introduced and discussed. Comparisons are provided and recommendations given.

Three-dimensional simulations of internal wave breaking and the fate of planets around solar-type stars

We study the fate of internal gravity waves approaching the centre of a non-rotating solar-type star, by performing 3D numerical simulations using a Boussinesq-type model. These waves are excited at the top of the radiation zone by the tidal forcing of a short-period planet on a circular, coplanar orbit. This extends previous work done in 2D by Barker & Ogilvie. We first derive a linear wave solution, which is not exact in 3D; however, the reflection of ingoing waves from the centre is close to perfect for moderate amplitude waves. Waves with sufficient amplitude to cause isentropic overturning break, and deposit their angular momentum near the centre. This forms a critical layer, at which the angular velocity of the flow matches the orbital angular frequency of the planet. This efficiently absorbs ingoing waves, and spins up the star from the inside out, while the planet spirals into the star. We also perform numerical integrations to determine the linearised adiabatic tidal response throughout the star, for solar-type stellar models with masses in the range 0.5 < m_star/M_sun < 1.1, throughout their main sequence lifetimes. The aim is to study the launching region for these waves at the top of the radiation zone in more detail, and to determine the accuracy of a semi-analytic approximation for the tidal torque on the star, that was derived under the assumption that all ingoing wave angular momentum is absorbed in a critical layer. The main conclusions are that this nonlinear mechanism of tidal dissipation could provide an explanation for the survival of all short-period extrasolar planets observed around FGK stars, while it predicts the destruction of more massive planets. This work provides further support for the model outlined in a previous paper by Barker & Ogilvie, and makes predictions that will be tested by ongoing observational studies, such as WASP and Kepler.

Integrated optical frequency-resolved Mach-Zehnder interferometers for label-free affinity sensing

Integrated Optical Frequency-Resolved Mach-Zehnder Interferometry (IO FR-MZI) is introduced as an alternative, cost-efficient operation principle for integrated optical label-free affinity sensors that can combine high sensitivity with high versatility in terms of potential applications and experimental configurations. A detailed theoretical analysis of the method is presented followed by a semi-analytical approximation and numerical calculations in order to quantify the sensitivity and limits of detection of the FR-MZI over Single Wavelength MZI. The obtained results substantiate that IO FR-MZI- based sensors constitute a generic technological platform of high sensitivity that can be implemented into a plethora of detection schemes. For an optimized optical design well below 1mm in length the limit of detection can be as low as 0.025A in terms of adlayer effective thickness allowing for truly miniaturized integrated optical sensors fabricated with high yield with standard microfabrication techniques.

Thermodynamics of blue phases in electric fields

We present extensive numerical studies to determine the phase diagrams of cubic and hexagonal blue phases in an electric field. We confirm the earlier prediction that hexagonal phases, both two and three dimensional, are stabilized by a field, but we significantly refine the phase boundaries, which were previously estimated by means of a semianalytical approximation. In particular, our simulations show that the blue phase I-blue phase II transition at fixed chirality is largely unaffected by electric field, as observed experimentally.

Semi-analytic calculation of the gravitational wave signal from the electroweak phase transition for general quartic scalar effective potentials

Upcoming gravitational wave (GW) detectors might detect a stochastic background of GWs potentially arising from many possible sources, including bubble collisions from a strongly first-order electroweak phase transition. We investigate whether it is possible to connect, via a semi-analytical approximation to the tunneling rate of scalar fields with quartic potentials, the GW signal through detonations with the parameters entering the potential that drives the electroweak phase transition. To this end, we consider a finite temperature effective potential similar in form to the Higgs potential in the Standard Model (SM). In the context of a semi-analytic approximation to the three dimensional Euclidean action, we derive a general approximate form for the tunneling temperature and the relevant GW parameters. We explore the GW signal across the parameter space describing the potential which drives the phase transition. We comment on the potential detectability of a GW signal with future experiments, and physical relevance of the associated potential parameters in the context of theories which have effective potentials similar in form to that of the SM. In particular we consider singlet, triplet, higher dimensional operators, and top-flavor extensions to the Higgs sector of the SM. We find that the addition of a temperature independent cubic term in the potential, arising from a gauge singlet for instance, can greatly enhance the GW power. The other parameters have milder, but potentially noticeable, effects.

The Sommerfeld enhancement for dark matter with an excited state

We present an analysis of the Sommerfeld enhancement to dark matter annihilation in the presence of an excited state, where the interaction inducing the enhancement is purely off-diagonal, such as in models of exciting or inelastic dark matter. We derive a simple and accurate semi-analytic approximation for the s-wave enhancement, which is valid provided the mass splitting between the ground and excited states is not too large, and discuss the cutoff of the enhancement for large mass splittings. We reproduce previously derived results in the appropriate limits, and demonstrate excellent agreement with numerical calculations of the enhancement. We show that the presence of an excited state leads to generically larger values of the Sommerfeld enhancement, larger resonances, and shifting of the resonances to lower mediator masses. Furthermore, in the presence of a mass splitting the enhancement is no longer a monotonic function of velocity: the enhancement where the kinetic energy is close to that required to excite the higher state can be up to twice as large as the enhancement at zero velocity.

A semi-analytic approximation of charge induction in monolithic pixelated CdZnTe radiation detectors

A semi-analytic approximation to the weighting potential within monolithic pixelated CdZnTe radiation detectors is presented. The approximation is based on solving the multi-dimensional Laplace equation that results upon replacing rectangular pixels with equal-area circular pixels. Further, we utilize the simplicity of the resulting approximate weighting potential to extend the well-known Hecht equation, describing charge induction in a parallel plate detector, to that approximating the multi-dimensional charge induction within a pixelated detector. These newly found expressions for the weighting potential and charge induction in a pixelated detector are compared throughout to full 3D electrostatic and monte carlo simulations using eV dsim ( eV Microelectronics Device SIMulator). The semi-analytic expressions derived in this paper can be evaluated quickly, and can therefore be used to efficiently reduce the size and dimensionality of the parameter space on which a detailed 3D numerical analysis is needed for pixelated detector design in a wide range of applications.

Semi-Analytical Treatment of Wall Heat Transfer Coupled to a Numerical Simulation Model of Fire

This analysis focuses on the importance of accurate description of walls' thermal behavior with particular relevance to fire safety cases. To accurately predict thermal loads on the walls in a complex fire situation, time dependent phenomena of thermal convection, diffusion, radiation, and heat transfer across the wall have to be taken into account. The heat transfer across the walls is often neglected as it requires fine grid resolution to approximate large temperature gradients. The analysis shows that this can lead to substantial errors and proposes a transient semi-analytical approximation of thermal wall behavior. Unsteady heat diffusion equation is solved by semi-analytical approximation of a temperature profile in a wall. The temperature profile is approximated in each time step and for each wall grid node separately using the temperature at the wall internal side and Biot number at the external side for boundary conditions. Wall heat transfer coefficient is calculated as a temperature profile derivative and then used in the coupled numerical model of the fire. The semi-analytical approach does not rely on a discretization procedure and the accuracy of its solutions is independent of grid spacing. Therefore, such semi-analytical technique is computationally less expensive and especially attractive for long transient calculations.