Static Approximation Research Articles

Electro-hyperelasticity with phase field for electric breakdown modeling

In this paper, we systematically derive the phase field model for electric breakdown channel propagation. The model includes electric charge, mass, momentum and energy conservation equations as well as Allen–Cahn type equation which governs the phase field evolution. The medium motion is described by the Godunov–Romenski type hyperelastic model. Electromagnetic part of the model consists of Maxwell’s equations in quasi(electro)stationary approximation including electric charge conservation law. Derivation of the model is based on the Coleman–Noll procedure and Gurtin’s microforces and microstress framework.

OnionVQE optimization strategy for ground state preparation on NISQ devices

Abstract The Variational Quantum Eigensolver (VQE) is one of the most promising and widely used algorithms for exploiting the capabilities of current Noisy Intermediate-Scale Quantum (NISQ) devices. However, VQE algorithms suffer from a plethora of issues, such as barren plateaus, local minima, quantum hardware noise, and limited qubit connectivity, thus posing challenges for their successful deployment on hardware and simulators. In this work, we propose a VQE optimization strategy that builds upon recent advances in the literature, and exhibits very shallow circuit depths when applied to the specific system of interest, namely a model Hamiltonian representing a cuprate superconductor. These features make our approach a favorable candidate for generating good ground state approximations on current NISQ devices. Our findings illustrate the potential of VQE algorithmic development for leveraging the full capabilities of NISQ devices.

Application of a Film Model to Mass Transfer and Chemical Reaction at a Plasma-Liquid Interface

Plasma electrodes provide novel ways of conducting electrochemical processes in liquids, in particular because of the ability to generate unique reactive radical species. However, the radicals injected into the liquid and their ensuing reactions are often confined to a narrow region near the interface of the plasma and the liquid. Thus, mass transfer has been found to play an important role in the observed kinetics and a modeling framework that includes both transport and kinetics is required to interpret experimental data. Here, we apply the idea of a film model for interphase mass transfer to plasma-liquid electrochemical processes, whereby transport is described by a stagnant film that is inherently linked to the concentration boundary layer and the mass transfer coefficient. Equations that govern the transport and reaction of radicals and substrates within the film are solved assuming a quasi-steady state approximation. The model is applied to specific case studies from the literature to estimate important parameters that are difficult to measure experimentally, such as the mass transfer coefficient. Our study shows that a film model can elucidate the effect of mass transfer on observed conversion rates and allow the intrinsic kinetics to be unraveled.

EU CIVIL PROCEDURE AFTER THE LISBON TREATY

With the Lisbon Treaty, a new page was opened in the development of EU civil procedure. This is so because тhe Lisbon Treaty has introduced Article 81 (ex-Article 65 EC) as a separate Chapter 3 of Title V on an Area of Freedom, Security, and Justice. With this, civil justice cooperation has gained an autonomous character and become distinct sub-area of the EU policy on Freedom, Security, and Justice.1 More importantly, the approximation of Member States’ civil procedural rules, provided for in Article 81 of the Lisbon Treaty, has gained a central role in the establishment and proper functioning of this Area and the EU in general. Precisely because of this, the subject of analysis of this paper is the new trend in the development of the civil procedure of the EU after the Lisbon Treaty, the advantages that this trend offers versus the old regime in EU and the future directions of development of the civil procedural law of the EU.

Modeling Pitch Angle Dependent Electron Precipitation Using Electron Lifetimes

AbstractElectron precipitation, a crucial link between Earth's magnetosphere and atmosphere, profoundly influences the coupled magnetosphere‐ionosphere‐atmosphere system. Existing models of the ring current often rely on electron lifetimes for characterizing the effects of pitch‐angle scattering, thus limiting accurate predictions of loss cone dynamics. This study introduces a method called steady‐state approximation (steady state approximation) utilizing the steady‐state solution of the pitch‐angle diffusion operator to calculate pitch angle resolved flux within the loss cone. The method enables precise comparisons with low‐earth orbit satellite measurements to validate parameterized electron lifetimes. Applying this approach to reevaluate a prior study, we uncover underestimated electron precipitation during geomagnetic storms, particularly in the pre‐midnight sector. This discrepancy reveals a previously overlooked loss process. Our method enhances the fidelity of global magnetospheric simulations, contributing to improved predictions of ionospheric conductance and atmospheric chemistry dynamics.

Modeling local decoherence of a spin ensemble using a generalized Holstein–Primakoff mapping to a bosonic mode

We show how the decoherence that occurs in an entangling atomic spin–light interface can be simply modeled as the dynamics of a bosonic mode. Although one seeks to control the collective spin of the atomic system in the permutationally invariant (symmetric) subspace, diffuse scattering and optical pumping are local, making an exact description of the many-body state intractable. To overcome this issue we develop a generalized Holstein–Primakoff approximation for collective states which is valid when decoherence is uniform across a large atomic ensemble. In different applications the dynamics is conveniently treated as a Wigner function evolving according to a thermalizing diffusion equation, or by a Fokker–Planck equation for a bosonic mode decaying in a zero-temperature reservoir. We use our formalism to study the combined effect of Hamiltonian evolution, local and collective decoherence, and measurement backaction in preparing nonclassical spin states for application in quantum metrology.

Static Subspace Approximation for Random Phase Approximation Correlation Energies: Implementation and Performance.

Developing theoretical understanding of complex reactions and processes at interfaces requires using methods that go beyond semilocal density functional theory to accurately describe the interactions between solvent, reactants and substrates. Methods based on many-body perturbation theory, such as the random phase approximation (RPA), have previously been limited due to their computational complexity. However, this is now a surmountable barrier due to the advances in computational power available, in particular through modern GPU-based supercomputers. In this work, we describe the implementation of RPA calculations within BerkeleyGW and show its favorable computational performance on large complex systems relevant for catalysis and electrochemistry applications. Our implementation builds off of the static subspace approximation which, by employing a compressed representation of the frequency dependent polarizability, enables the evaluation of the RPA correlation energy with significant acceleration and systematically controllable accuracy. We find that the computational cost of calculating the RPA correlation energy scales only linearly with system size for systems containing up to 50 thousand bands, and is expected to scale quadratically thereafter. We also show excellent strong scaling results across several supercomputers, demonstrating the performance and portability of this implementation.

A robust optimisation approach for the 2D-VO fractional optimal control problems

This paper provides a new numerical scheme with the help of discrete Krawtchouk polynomials (DKPs), the Ritz method, and the two-dimensional (2D) Gauss-Legendre quadrature rule for dealing with two-dimensional variable-order (VO) fractional optimal control problems. For this aim, we construct the derivative operational matrices of DKPs. Also, we impose the conditions into the approximation of state and control functions. Then, we expand the functions in the problem given DKPs and operational matrices. Meanwhile, DKPs and the 2D Gauss-Legendre quadrature rule evaluate the performance index function. According to the numerical algorithm process, the proposed problem is reduced to a system of algebraic equations. The chosen polynomial as basis functions and the Ritz method provide a powerful and flexible scheme. Also, we discuss the error of the VO-fractional derivative of the approximate solution and performance index. At last, we include several illustrative examples to indicate the validity and applicability of the present technique.

Doubly-charged Tcc++ states in the dynamical diquark model

One of the celebrated tools in explaining the hydrogen atom is Born-Oppenheimer approximation. The resemblance of QQq¯q¯ tetraquarks to hydrogen atom within quantum chromodynamics implies usage of Born-Oppenheimer approximation for these multiquark states. In this work, we use dynamical diquark model to calculate mass spectra and sizes of doubly charmed and charged tetraquark states denoted as Tcc++. Our results for mass spectra indicate some bound-state candidates with respect to corresponding two-meson thresholds. Calculation of expectation values of ⟨r2⟩ reflects that doubly charmed and charged tetraquark states are compact. Published by the American Physical Society 2024

Gaussian state approximation of quantum many-body scars

Gaussian state approximation of quantum many-body scars

Hybrid Cluster-Continuum Method for Single-Ion Solvation Free Energy in Acetonitrile Solvent.

A new hybrid discrete-continuum approach named the cluster-continuum static approximation (CCSA) has been proposed for acetonitrile solvent. The continuum part uses the conductor-like polarizable continuum model for electrostatic and a surface area-dependent term for nonelectrostatic solvation. The CCSA includes only one explicit acetonitrile solvent molecule and a damping function, which makes the CCSA method reduce to pure continuum solvation in the case of weaker potential of mean force for solute-solvent interaction. The performance of the model was tested for 22 anions and 22 cations, including challenge species that cannot be adequately described by pure continuum solvation. A comparison was done with the widely used solvent model density (SMD) model. For anions, the CCSA reduces to pure continuum solvation and the method has the same performance as the SMD model, with a standard deviation of the mean signed error (SD-MSE) of 2.7 kcal mol-1 for both models. However, the CCSA method for cations considerably outperforms the SMD model, with an SD-MSE of 3.3 kcal mol-1 for the former and 8.4 kcal mol-1 for the latter. The method can be automated, and the present study suggests that continuum solvation models could be parameterized taking into account the explicit solvation as proposed in this work.

Boosting of enzymatic cascades by intermediates: Theoretical analysis and model-based optimization

A recent paper demonstrated that adding intermediates of an enzymatic cascade reaction at the start of the reaction process can increase the space-time yield significantly. This method was coined boosting by intermediates (BBI) and can potentially increase the productivity of enzymatic cascade reaction systems. BBI remains largely unexplored, and its mechanism, potential, and limitations are not fully understood. In this short communication, the theoretical background of the method is revealed by studying four chemical reaction networks that exhibit boosting through intermediates. The dynamics of these networks are studied via the quasi-steady state approximation and numerical solutions. A novel graphical method that can be used to deduce which intermediates boost linear, redox-neutral enzymatic cascade reaction networks is presented. Further, how such a reaction process can be designed using model-based optimization is demonstrated.

Isomeric effects on the electron impact scattering for selected five membered heterocyclic ring molecules

We present electron scattering cross-sectional data for selected isomers of the five-membered ring molecules, specifically oxazole and isoxazole, thiazole and isothiazole, and imidazole and pyrazole. The ab-initio R-matrix method, incorporating static exchange polarization approximations, is employed for this calculation from energy range 0.1 eV to 20 eV. Three shape resonances are identified and characterized in each system, consisting of two π* and one σ* shape resonances. Notably, thiazole and isothiazole exhibit additional σ* resonance which is which is at a lower energy than other σ* resonance. The calculated resonance positions align well with available experimental as well as theoretical data. We also performed electronic structure calculations to aid in characterization of resonances. Comparison of the scattering cross-sectional data with their respective isomers reveal marginal differences in the magnitude of the elastic cross section particularly below 1 eV. This discrepancy may be attributed to variations in the long-range electronic dipole contribution to the electron-molecule interaction process. Further, to understand dependency of dipole moment on elastic cross section, we performed a fitting procedure for the elastic cross sections at 0.1 eV with square of the dipole moment of all the five membered ring molecules studied here. The fitting formula so obtained was used to estimate the cross section for other three five membered ring molecules: Furan, Thiophene and Pyrrole. Additionally, total ionization cross sections are computed using the binary-encounter-Bethe (BEB) model, demonstrating nearly identical results when compared with their isomers. Our findings offer insights into the influence of structural changes on the electron scattering data, providing guidance for other theoretical investigations.

Doping dependence and multichannel mediators of superconductivity: calculations for a cuprate model

We study two aspects of the superconductivity in a cuprate model system, its doping dependence and the influence of competing pairing mediators. We first include electron–phonon interactions beyond Migdal’s approximation and solve self-consistently, as a function of doping and for an isotropic electron–phonon coupling, the full-bandwidth, anisotropic vertex-corrected Eliashberg equations under a non-interacting state approximation for the vertex correction. Our results show that such pairing interaction supports the experimentally observed dx2−y2 -wave symmetry of the superconducting gap, but only in a narrow doping interval of the hole-doped system. Depending on the coupling strength, we obtain realistic values for the gap magnitude and superconducting critical temperature Tc close to optimal doping, rendering the electron–phonon mechanism an important candidate for mediating superconductivity in this model system. Second, for a doping near optimal hole doping, we study multichannel superconductivity, by including both vertex-corrected electron–phonon interaction and spin and charge fluctuations as pairing mechanisms. We find that both mechanisms cooperate to support an unconventional d-wave symmetry of the order parameter, yet the electron–phonon interaction is mainly responsible for the Cooper pairing and high critical temperature Tc . Spin fluctuations are found to have a suppressing effect on the gap magnitude and critical temperature due to their repulsive interaction at small coupling wave vectors.

Matrix product state approximations to quantum states of low energy variance

We show how to efficiently simulate pure quantum states in one dimensional systems that have both finite energy density and vanishingly small energy fluctuations. We do so by studying the performance of a tensor network algorithm that produces matrix product states whose energy variance decreases as the bond dimension increases. Our results imply that variances as small as ∝1/log⁡N can be achieved with polynomial bond dimension. With this, we prove that there exist states with a very narrow support in the bulk of the spectrum that still have moderate entanglement entropy, in contrast with typical eigenstates that display a volume law. Our main technical tool is the Berry-Esseen theorem for spin systems, a strengthening of the central limit theorem for the energy distribution of product states. We also give a simpler proof of that theorem, together with slight improvements in the error scaling, which should be of independent interest.

Quantum circuits for discrete graphical models

Graphical models are useful tools for describing structured high-dimensional probability distributions. The development of efficient algorithms for generating samples thereof remains an active research topic. In this work, we provide a quantum algorithm that enables the generation of unbiased and independent samples from general discrete graphical models. To this end, we identify a coherent embedding of the graphical model based on a repeat-until-success sampling scheme which clearly identifies whether a drawn sample represents the underlying distribution. Furthermore, we show that the success probability for finding a valid sample can be lower bounded with a quantity that depends on the number of maximal cliques and the model parameter norm. Moreover, we rigorously proof that the quantum embedding conserves the key property of graphical models, i.e., factorization over the cliques of the underlying conditional independence structure. The quantum sampling algorithm also allows for maximum likelihood learning as well as maximum a posteriori state approximation for the graphical model. Finally, the proposed quantum method shows potential to realize interesting problems on near-term quantum processors. In fact, illustrative experiments demonstrate that our method can carry out sampling and parameter learning not only with idealized simulations of quantum computers but also on actual quantum hardware solely supported by simple readout error mitigation.

Beyond homogeneity: Assessing the validity of the Michaelis-Menten rate law in spatially heterogeneous environments.

The Michaelis-Menten (MM) rate law has been a fundamental tool in describing enzyme-catalyzed reactions for over a century. When substrates and enzymes are homogeneously distributed, the validity of the MM rate law can be easily assessed based on relative concentrations: the substrate is in large excess over the enzyme-substrate complex. However, the applicability of this conventional criterion remains unclear when species exhibit spatial heterogeneity, a prevailing scenario in biological systems. Here, we explore the MM rate law's applicability under spatial heterogeneity by using partial differential equations. In this study, molecules diffuse very slowly, allowing them to locally reach quasi-steady states. We find that the conventional criterion for the validity of the MM rate law cannot be readily extended to heterogeneous environments solely through spatial averages of molecular concentrations. That is, even when the conventional criterion for the spatial averages is satisfied, the MM rate law fails to capture the enzyme catalytic rate under spatial heterogeneity. In contrast, a slightly modified form of the MM rate law, based on the total quasi-steady state approximation (tQSSA), is accurate. Specifically, the tQSSA-based modified form, but not the original MM rate law, accurately predicts the drug clearance via cytochrome P450 enzymes and the ultrasensitive phosphorylation in heterogeneous environments. Our findings shed light on how to simplify spatiotemporal models for enzyme-catalyzed reactions in the right context, ensuring accurate conclusions and avoiding misinterpretations in in silico simulations.

An iterative method to solve the time delay optimal control problem with bounded control variable

This paper presents an iterative approach for solving the optimal control problem (OCP) with time-delayed terms and bounded control variables. The projection function is used to construct the iterative method to approximate the control law. Employing the approximation of control law the approximation of state and the co-state variables are obtained. To the best of our knowledge, this paper is the first attempt to solve bounded and delayed OCPs with a projection function. The convergence theorem for the proposed method is also proven. Several numerical examples are added, showing the developed method's effectiveness to solve the time-delayed OCPs with bounded control variables.

Calculation of ATEM Cell Characteristic Impedance Based on Static Field Approximation and MoM

Calculation of ATEM Cell Characteristic Impedance Based on Static Field Approximation and MoM

A dynamical system framework for a mathematical model of atherosclerosis

Atherosclerosis is a chronic inflammatory disease occurs due to plaque accumulation in the inner artery wall. In atherosclerotic plaque formation monocytes and macrophages play a significant role in controlling the disease dynamics. In the present article, the entire biochemical process of atherosclerotic plaque formation is presented in terms of an autonomous system of nonlinear ordinary differential equations involving concentrations of oxidized low-density lipoprotein (LDL), monocytes, macrophages, and foam cells as the key dependent variables. To observe the capacity of monocytes and macrophages the model has been reduced to a two-dimensional temporal model using quasi steady state approximation theory. Linear stability analysis of the two-dimensional ordinary differential equations (ODEs) model has revealed the stability of the equilibrium points in the system. We have considered both one- and two-parameter bifurcation analysis with respect to parameters associated to the rate at which macrophages phagocytose oxidized LDL and the rate at which LDL enters into the intima. The bifurcation diagrams reveal the oscillating nature of the curves representing concentration of monocytes and macrophages with respect to significant model parameters. We are able to find the threshold values at which the plaque accumulation accelerates in an uncontrollable way. Further to observe the impact of diffusion, a spatiotemporal model has been developed. Numerical investigation of the partial differential equations (PDEs) model reveals the existence of travelling wave in the system which ensures the fact that the development of atherosclerotic plaque formation follows reaction–diffusion wave.