Weak Approximation Research Articles

Force-extension and longitudinal response of wormlike chains with single cross-link

Certain important biopolymers, such as actin filaments, are known to have cross-links at their interfaces, which significantly influence their mechanical properties. To explore these effects, the force-extension and longitudinal response of wormlike chains (WLCs) with a single cross-link under tension in two-dimension are examined using both analytical methods and Brownian dynamics simulations. The cross-link is modeled as a spring in the analytical method, and mode analysis is used to calculate the path integrals associated with the partition function. These theoretical results are then validated through Brownian dynamics simulations. Final results indicate that the simulation results are consistent with the theoretical predictions, particularly for situations involving large tensile force and short chain, which are prerequisites for the application of the weak bending approximation.

Exact solutions for analog Hawking effect in dielectric media

In the framework of the analog Hawking radiation for dielectric media, we analyze a toy model and also the 2D reduction of the Hopfield model for a specific monotone and realistic profile for the refractive index. We are able to provide exact solutions, which do not require any weak dispersion approximation. The theory of Fuchsian ordinary differential equations is the basic tool for recovering exact solutions, which are rigorously identified and involve the so-called generalized hypergeometric functions F34(α1,α2,α3,α4;β1,β2,β3;z). A complete set of connection formulas are available, both for the subcritical case and for the transcritical one, and also the Stokes phenomenon occurring in the problem is fully discussed. From the physical point of view, we focus on the problem of thermality. Under suitable conditions, the Hawking temperature is deduced, and we show that it is in full agreement with the expression deduced in other frameworks under various approximations. Published by the American Physical Society 2024

Anisotropic error analysis of weak Galerkin finite element method for singularly perturbed biharmonic problems

We consider the weak Galerkin finite element approximation of the singularly perturbed biharmonic elliptic problem on a unit square domain with clamped boundary conditions. Shishkin mesh is used for domain discretization as the solution exhibits boundary layers near the domain boundary. Error estimates in the equivalent H2− norm have been established and the uniform convergence of the proposed method has been proved. Numerical examples are presented corroborating our theoretical findings.

Deflection of charged signals in a dipole magnetic field in Kerr background

This paper investigates charged particle deflection in a Kerr spacetime background with a dipole magnetic field, focusing on the equatorial plane and employing the weak field approximation. We employ the Jacobi–Randers metric to unify the treatment of the gravitational and electromagnetic effects on charged particles. Furthermore, we utilize the Gauss–Bonnet theorem to calculate the deflection angle through curvature integrals. The difference between the prograde and retrograde deflection angles is linked to the non-reversibility of metrics and geodesics in Finsler geometry, revealing that this difference can be considered a Finslerian effect. We analyze the impact of both gravitomagnetic field and dipole magnetic field on particle motion and deflection using the Jacobi–Randers magnetic field. The model considered in this paper exhibits interesting features in the second-order approximation of (M/b), i.e., the ratio between the spacetime mass and impact parameters. When qμ=2MaE\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$q\\mu =2MaE$$\\end{document}, where q,E\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$q,\\,E$$\\end{document} are the charge and asymptotic energy of the charged particle and μ,a\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mu ,\\,a$$\\end{document} are the dipole magnetic moment and spacetime spin, the Jacobi–Randers metric possesses reversible geodesics, leading to equal prograde and retrograde deflection angles. In this case, the gravitomagnetic field and dipole magnetic field cancel each other out, distinguishing it from scenarios involving only the gravitomagnetic field or the dipole magnetic field. We also explore the magnetic field’s impact on gravitational lensing of charged particles.

A weak Galerkin finite element method for fourth-order parabolic singularly perturbed problems on layer adapted Shishkin mesh

In this paper, we propose a weak Galerkin finite element approximation for a class of fourth-order singularly perturbed parabolic problems. The problem exhibits boundary layers and so we have considered layer adapted triangulations, in particular Shishkin triangular mesh in the spatial domain. For temporal discretization, we utilize the Crank-Nicolson scheme on a uniform mesh. Stability and error estimates along with the uniform convergence of the method has been proved. Numerical examples are included which verifies our analysis.

Transport-of-intensity differential phase contrast imaging: defying weak object approximation and matched illumination condition

Transport-of-intensity differential phase contrast imaging: defying weak object approximation and matched illumination condition

CoOMBE: A suite of open-source programs for the integration of the optical Bloch equations and Maxwell-Bloch equations

The programs described in this article and distributed with it aim (1) at integrating the optical Bloch equations governing the time evolution of the density matrix representing the quantum state of an atomic system driven by laser or microwave fields, and (2) at integrating the 1D Maxwell-Bloch equations for one or two laser fields co-propagating in an atomic vapour. The rotating wave approximation is assumed. These programs can also be used for more general quantum dynamical systems governed by the Lindblad master equation. They are written in Fortran 90; however, their use does not require any knowledge of Fortran programming. Methods for solving the optical Bloch equations in the rate equations limit, for calculating the steady-state density matrix and for formulating the optical Bloch equations in the weak probe approximation are also described. Program summaryProgram Title: CoOMBECPC Library link to program files:https://doi.org/10.17632/5wsg9d52dk.1Developers' repository link:https://github.com/durham-qlm/CoOMBELicensing provisions: GPLv3Programming language: Fortran 90Nature of problem: The present programs can be used for the following operations: (1) Integrating the optical-Bloch equations within the rotating wave approximation for a multi-state atomic system. At the choice of the user, the calculation will return either the time-dependent density matrix at given times or the density matrix in the long time limit if the system evolves into a steady state in that limit. The calculation can be done with or without averaging over the thermal velocity distribution of the atoms. The number of atomic states which can be included in the calculation is limited only by the CPU time available and possibly by memory requirements. An arbitrarily large number of laser or microwave fields can be included in the calculation if these fields are all CW. This number is currently limited to one or two for fields that are not all CW. The calculation can be done in the weak probe approximation, or in the rate equations approximation, or without assuming either of these two approximations. Calculating refractive indexes, absorption coefficients and complex susceptibilities is also possible. (2) Integrating the 1D Maxwell-Bloch equations in the slowly varying envelope approximation for one or two fields co-propagating in a single-species atomic vapour. Although geared towards the case of atoms interacting with laser fields, this code can also be used for more general quantum systems with similar equations of motion (e.g., molecular systems, spin systems, etc.).Solution method: The Lindblad master equation is expressed as a system of homogeneous first order linear differential equations, which are transformed as required and solved to obtain the density matrix representing the state of the atomic system. A variety of methods are offered to this end. The same approach is also used in the calculation of the polarisation of the medium when integrating the Maxwell-Bloch equations. The latter are integrated over space using predictor-corrector methods. The library includes a general driving program making it possible to use these codes without additional program development. The distribution also includes examples of the use of a container for running these programs without a pre-installed Fortran compiler.

Retardation theory of eleven galaxies

Abstract The missing mass problem has been with us since the 1970s, as Newtonian gravity using baryonic mass cannot account for various observations. We investigate the viability of retardation theory, an alternative to the Dark Matter paradigm (DM) which does not seek to modify the General Principal of Relativity but to improve solutions within it by exploring its weak field approximation to solve the said problem in a galactic context. This approach have yielded satisfactory results, with respect to galactic rotation curves, the Tully-Fisher relation and missing mass derived from gravitational lensing. Recently it was able to introduce a necessary correction to the virial theorem explaining mass excess in clusters of galaxies. The current work presents eleven rotation curves calculated using Retardation Theory. The calculated rotation curves are compared with observed rotation curves. Values for the change in mass flux to mass ratio are extracted from the fitting process as a free fitting parameter. Those quantities are interpreted here and in previous works using galactic processes. Retardation Theory was able to successfully reproduce rotation curves and a preliminary correlation with star birthrate index is seen, suggesting a possible link between galactic winds and observed rotation curves. Retardation Theory shows promising results within current observations. More research is needed to elucidate the suggested mechanism and the processes which contribute to it. Galactic mass outflows carried by galactic winds may affect rotation curves.

Magnetic field diagnostics of prominences with the Mg II k line 3D Stokes inversions versus traditional methods

The Mg II k resonance line is commonly used for diagnosing the solar chromosphere. We theoretically investigated its intensity and polarization in solar prominences, taking 3D radiative transfer and Hanle and Zeeman effects into account. We used an optically thick 3D model representative of a solar prominence and applied several inversion methods to the synthetic Stokes profiles, clarifying their pros and cons for inferring prominence magnetic fields. We conclude that the self-consistent 3D inversion with radiative transfer is necessary to determine the magnetic field vector, although its geometry cannot be inferred with full fidelity. We also demonstrate that more traditional methods, such as those based on the weak field approximation or the constant-property slab assumption, can offer useful information under certain conditions.

Efficient option pricing in the rough Heston model using weak simulation schemes

We provide an efficient and accurate simulation scheme for the rough Heston model in the standard (H>0) as well as the hyper-rough regime ( -1/2 $ ]]> H > − 1 / 2 ). The scheme is based on low-dimensional Markovian approximations of the rough Heston process derived in [Bayer and Breneis, arXiv:2309.07023], and provides a weak approximation to the rough Heston process. Numerical experiments show that the new scheme exhibits second-order weak convergence, while the computational cost increases linearly with respect to the number of time steps. In comparison, existing schemes based on discretization of the underlying stochastic Volterra integrals such as Gatheral's HQE scheme show a quadratic dependence of the computational cost. Extensive numerical tests for standard and path-dependent European options and Bermudan options show the method's accuracy and efficiency.

The motion of twisted particles in a stellar gravitational field

Abstract In this work, we explore the motion of a twisted particle possessing intrinsic orbital angular momentum (OAM) as it traverses a weak stellar gravitational field, which we approximate using a polytropic model. We disregard the spin characteristic of the twisted particle, modeling it as a massless complex twisted scalar wave packet to simplify its interaction with gravitational fields. Building on this simplification, we determine the trajectory of this twisted particle by using the center of its energy density and investigate the gravitational birefringence induced by its OAM. In a weak field approximation, we find the gravitational birefringence-OAM relationship parallels that with spin, as described by the Mathisson-Papapetrou-Dixon equations. This indicates that the gravitational birefringence induced by OAM can potentially exceed that induced by spin by several orders of magnitude, significantly enhancing its detectability. To broaden our analysis, we introduce a nonminimal coupling term, $\lambda R|\phi|^2$, into the Lagrangian, resulting in the modified expression $\mathcal{L}=-\frac{1}{2}\nabla _\rho\phi\nabla^\rho\phi^*-\frac{1}{2}\lambda R|\phi|^2$. This adjustment is necessitated by the quantization of the scalar field in curved spacetime. We then explore the effects of this term on the motion of the twisted particle. Our findings show that the trajectory of the twisted particle under nonminimal coupling ($\lambda\neq 0$) differs from that in the minimal coupling scenario ($\lambda=0$). Specifically, for a positive nonminimal coupling constant $\lambda$, the trajectory of the twisted particle is expected to deviate away from the stellar center, compared to the minimal coupling scenario.

A coupled-mode analysis of mode localization on weakly-coupled silicon micromechanical resonators with damping

Mode localization on weakly-coupled silicon resonators in the areas of microelectromechanical systems (MEMS) has been widely studied for highly sensitive sensors of physical and chemical parameter monitoring. In terms of its operation principle, i.e., energy conservation, a system for the weakly-coupled silicon resonator is required to have no damping. In fact, however, various types of energy dissipation mechanisms inevitably lead to the damping even under high vacuum packaging, and for mass sensing scenarios it always operates in high damping environments. The effect of damping on the amplitude and frequency of the weakly-coupled silicon resonator is generally explored by numerical methods. Based on a weak coupling approximation, this article has developed a coupled-mode model for analytically analyzing the effects. A pair of electrostatically-coupled silicon resonator were experimentally fabricated and operated through capacitive transduction. The theoretical and experimental results show a quantitative consistency. The results presented here suggest that it may guide the design of sensors based on mode localization in the presence of damping.

Magnetic catalysis and diamagnetism from pion fluctuations

In the framework of the Nambu–Jona-Lasinio model beyond mean field approximation, the effects of pion fluctuations on (inverse) magnetic catalysis and magnetic susceptibility are studied. The negative magnetic susceptibility at low temperature is observed when contributions from both neutral and charged pions are taken into account. In the weak field approximation, it is observed that at finite temperature, the magnetic inhibition effect in the chiral limit, resulting from the difference between the transverse and longitudinal velocities of neutral pions, converts to weak magnetic catalysis when considering a nonzero current quark mass. Moreover, the magnetic catalysis is amplified by the charged pions. Therefore, no inverse magnetic catalysis is observed when considering pion fluctuations. Published by the American Physical Society 2024

Understanding Advanced Attenuated Total Reflection Correction: The Low Absorbance Assumption.

We present an attenuated total reflection (ATR) correction scheme capable of rectifying ATR spectra while considering the polarization state for arbitrary angles of incidence, provided that this angle exceeds the critical angle for the entire ATR spectrum. Due to its reliance on the weak absorption approximation, it cannot achieve perfect correction of the ATR spectra. However, comprehending its functionality may offer valuable insights into the concept behind the weak absorption approximation. Depending on the specific polarization state of an instrument accessory combination, this correction scheme may outperform the proprietary advanced ATR correction authored by ThermoFisher while being as user-friendly, but in contrast to the latter completely transparent with regard to its functionality.

A multilevel Monte Carlo algorithm for stochastic differential equations driven by countably dimensional Wiener process and Poisson random measure

In this paper, we investigate properties of standard and multilevel Monte Carlo methods for weak approximation of solutions of stochastic differential equations (SDEs) driven by infinite-dimensional Wiener process and Poisson random measure with Lipschitz payoff function. The error of the truncated dimension randomized numerical scheme, which depends on two parameters i.e., grid density n∈N and truncation dimension parameter M∈N, is of the order n−1/2+δ(M) such that δ(⋅) is positive and decreasing to 0. We derive a complexity model and provide proof for the complexity upper bound of the multilevel Monte Carlo method which depends on two increasing sequences of parameters for both n and M. The complexity is measured in terms of upper bound for mean-squared error and is compared with the complexity of the standard Monte Carlo algorithm. The results from numerical experiments as well as Python and CUDA C implementation details are also reported.

Simultaneous Spectropolarimetric Observations in the Hα and Ca ii 8662 Å Lines of an Active Region

We present spectropolarimetric observations of an active region recorded simultaneously in the Hα and the Ca ii 8662 Å lines. The sunspot exhibits multiple structures, including a light bridge and a region where the Ca ii 8662 Å line core is in emission. Correspondingly, the Hα line core image displays brightening in the emission region, with the spectral profiles showing elevated line cores. The stratification of the line-of-sight magnetic field is inferred through nonlocal thermodynamic equilibrium multiline inversions of the Ca ii 8662 Å line and the weak field approximation (WFA) over the Hα line. The field strength inferred from the Hα line core is consistently smaller than that inferred from inversions at logτ500 = −4.5. However, the study finds no correlation between the WFA over the core of the Hα line and that inferred from inversions at logτ500 = −4.5. In regions exhibiting emission features, the morphology of the magnetic field at logτ500 = −4.5 resembles that at logτ500 = −1, with slightly higher or comparable field strengths. The magnetic field morphology inferred from the core of the Hα line is also similar to that inferred from the full spectral range of the Hα line in the emission region. The field strength inferred in the light bridge at logτ500 = −1 is smaller than the surrounding umbral regions and comparable at logτ500 = −4.5. Similarly, the field strength inferred in the light bridge from the WFA over the Hα line appears lower compared to the surrounding umbral regions.

Shadows, rings and optical appearance of a magnetically charged regular black hole illuminated by various accretion disks

Abstract The Event Horizon Telescope (EHT) imaging of the supermassive black holes at the centers of Messier 87 galaxy (M87) and the Milky Way galaxy (Sgr A) marks a significant step in observing the photon rings and central brightness depression that define the optical appearance of black holes with an accretion disk scenario. Inspired by this, we take into account a static and spherically symmetric magnetically charged regular black hole (MCRBH) metric characterized by its mass and an additional parameter q, which arises from the coupling of Einstein gravity and nonlinear electrodynamics (NLED) in the weak field approximation. This parameterized model offers a robust foundation for testing the coupling of Einstein gravity and NLED in the weak-field approximation, using the EHT observational results. In this study, we investigate the geodesic motion of particles around the solution, followed by a discussion of its fundamental geometrical characteristics such as scalar invariants. Using null geodesics, we examine how the model parameter influences the behavior of the photon sphere radius and the associated shadow silhouette. We seek constraints on q by applying the EHT results for supermassive black holes M87* and Sgr A*. Furthermore, it is observed that the geodesics of time-like particles are susceptible to variations in q, which can have an impact on the traits of the innermost stable circular orbit and the marginally bounded orbit. Our primary objective is to probe how the free parameter q affects various aspects of the accretion disk surrounding the MCRBH using the thin-disk approximation. Next, we discuss the physical characteristics of the thin accretion disk as well as the observed shadows and rings of the MCRBH, along with its luminosity, across various accretion models. Ultimately, variations in accretion models and the parameter q yield distinct shadow images and optical appearances of the MCRBH.

Weak noise approximation for the Kolmogorov forward equation for a leaky integrate-and-fire neuron subject to stochastic stimulation

We develop a weak noise approximation for the Kolmogorov forward equation governing the dynamics of a leaky integrate-and-fire neuron subject to white noise. Although being very simple, our approximation provides accurate results as far the magnitude of noise-induced fluctuations Δ remains much smaller than the distance A between the mean potential (center of mass) and the excitation threshold. The error for the firing rate is <3% if A/Δ>3 for the stationary stimuli and if A/Δ>5 for time-varying stimuli.

Enhanced Approaches to Gravitational Lensing: A Regular Charged Black Hole in Weak Field Limit

Gravitational lensing, an integral aspect of general relativity, profoundly impacts our understanding of black holes (BH’s). The presence of a centered singularity resulting from gravitational collapse is a fundamental characteristic of a BH Nonetheless, the exploitation of non-linear electrodynamics (NLED) divulges the emphatic concept of non-singularity. Our analysis primarily focuses on calculating the deflection angle in the weak field approximation, deliberately excluding the original spacetime configuration. The photon’s effective geometry is typically linked to NLED. Instead, a photon traverses an effective null geodesic that aligns with the geometry of a regular metric. The findings reveal an additional term in the deflection angle compared to the Reissner-Nordström (RN) metric, evidenced by a shifted positional displacement of the tertiary image associated with a regular BH This study succinctly encapsulates these findings, positioning them within the extensive terminology ambient to BH’s singularity and GL, thus edifying the scientific memoir with precise astrophysical insights.

Massless Dirac perturbations of black holes in f(Q) gravity: quasinormal modes and a weak deflection angle

This article considers a static and spherical black hole (BH) in f(Q) gravity. f(Q) gravity is the extension of symmetric teleparallel general relativity, where both curvature and torsion are vanishing and gravity is described by nonmetricity. In this study, we investigate the possible implications of quasinormal mode (QNM) modified Hawking spectra and deflection angles generated by the model. The Wentzel–Kramers–Brillouin method is used to solve the equations of motion for massless Dirac perturbation fields and explore the impact of the nonmetricity parameter (Q 0). Based on the QNM computation, we can ensure that the BH is stable against massless Dirac perturbations and as Q 0 increases the oscillatory frequency of the mode decreases. We then discuss the weak deflection angle in the weak field limit approximation. We compute the deflection angle up to the fourth order of approximation and show how the nonmetricity parameter affects it. We find that the Q 0 parameter reduces the deflection angle.