Order Terms Research Articles

A personal perspective of the present status and future challenges facing thermal reaction rate theory.

Reaction rate theory has been at the center of physical chemistry for well over one hundred years. The evolution of the theory is not only of historical interest. Reliable and accurate computation of reaction rates remains a challenge to this very day, especially in view of the development of quantum chemistry methods, which predict the relevant force fields. It is still not possible to compute the numerically exact rate on the fly when the system has more than at most a few dozen anharmonic degrees of freedom, so one must consider various approximate methods, not only from the practical point of view of constructing numerical algorithms but also on conceptual and formal levels. In this Perspective, I present some of the recent analytical results concerning leading order terms in an ℏ2m series expansion of the exact rate and their implications on various approximate theories. A second aspect has to do with the crossover temperature between tunneling and thermal activation. Using a uniform semiclassical transmission probability rather than the "primitive" semiclassical theory leads to the conclusion that there is no divergence problem associated with a "crossover temperature." If one defines a semiclassical crossover temperature as the point at which the tunneling energy of the instanton equals the barrier height, then it is a factor of two higher than its previous estimate based on the "primitive" semiclassical approximation. In the low temperature tunneling regime, the uniform semiclassical theory as well as the "primitive" semiclassical theory were based on the classical Euclidean action of a periodic orbit on the inverted potential. The uniform semiclassical theory wrongly predicts that the "half-point," which is the energy at which the transmission probability equals 1/2, for any barrier potential, is always the barrier energy. We describe here how augmenting the Euclidean action with constant terms of order ℏ2 can significantly improve the accuracy of the semiclassical theory and correct this deficiency. This also leads to a deep connection with and improvement of vibrational perturbation theory. The uniform semiclassical theory also enables an extension of the quantum version of Kramers' turnover theory to temperatures below the "crossover temperature." The implications of these recent advances on various approximate methods used to date are discussed at length, leading to the conclusion that reaction rate theory will continue to challenge us both on conceptual and practical levels for years to come.

ℏ2 Corrections to Semiclassical Transmission Coefficients.

The uniform semiclassical expression for the energy-dependent transmission probability through a barrier has been a staple of reaction rate theory for almost 90 years. Yet, when using the classical Euclidean action, the transmission probability is identical to 1/2 when the energy equals the barrier height since the Euclidean action vanishes at this energy. This result is generally incorrect. It also leads to an inaccurate estimate of the leading order term in an ℏ2n expansion of the thermal transmission coefficient. The central result of this paper is that adding an ℏ2 dependent correction to the uniform semiclassical expression, whether as a constant action or as a shift in the energy scale, not only corrects this inaccuracy but also leads to a theory that is more accurate than the previous one for almost any energy. Shifting the energy scale is a generalization of the vibrational perturbation theory 2 (VPT2) and is much more accurate than the "standard" VPT2 theory, especially when the potential is asymmetric. Shifting the action by a constant is a generalization of a result obtained by Yasumori and Fueki (YF) only for the Eckart barrier. The resulting modified VPT2 and YF semiclassical theories are applied to the symmetric and asymmetric Eckart barrier, a Gaussian barrier, and a tanh barrier. The one-dimensional theories are also generalized to many-dimensional systems. Their effect on the thermal instanton theory is discussed.

Applying Lin's method to constructing heteroclinic orbits near the heteroclinic chain

In this paper, we apply Lin's method to study the existence of heteroclinic orbits near the degenerate heteroclinic chain under ‐dimensional periodic perturbations. The heteroclinic chain consists of two degenerate heteroclinic orbits and connected by three hyperbolic saddle points . Assume that the degeneracy of the unperturbed heteroclinic orbit is , the splitting index is . By applying Lin's method, we construct heteroclinic orbits connected and near the unperturbed heteroclinic chain. The existence of these orbits is equivalent to finding zeros of the corresponding bifurcation function. The lower order terms of the bifurcation function is the map from to . Using the contraction mapping principle, we provide a detailed analysis on how zeros can exist based on different cases of splitting indices , and then obtain the existence of the heteroclinic orbits which backward asymptotic to and forward asymptotic to .

Nonlinear analysis of the synchronous reference frame phase-locked loop under unbalanced grid voltage

The synchronous reference frame phase-locked loop (SRF-PLL), also called dqPLL, is an electric circuit commonly used in power electronics to estimate the phase angle of a three-phase AC grid. If the voltage is unbalanced, the PLL is modeled as a periodically forced nonlinear oscillator and is known in practice to converge to a steady oscillation. In the existing literature, the oscillation has been studied via linearization assuming a low level of unbalance. Aiming for stronger nonlocal statements, we present nonlinear analysis. We apply the method of autonomous comparison systems and incremental stability to show that the steady oscillation is unique and attractive in a wider neighborhood. Its lock-in domain is estimated using numerical phase portrait analysis. The oscillation is further approximated up to the terms of the second order in the unbalance factor—it yields an estimation of the time average of the PLL’s phase error which is not visible by linearization only. The results provide stability guarantees and can guide the tuning of SRF-PLL.

Sharp interface limit for a Navier–Stokes/Allen–Cahn system in the case of a vanishing mobility

We consider the sharp interface limit of a Navier–Stokes/Allen Cahn equation in a bounded smooth domain in two space dimensions, in the case of vanishing mobility mε=ε\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$m_\\varepsilon =\\sqrt{\\varepsilon }$$\\end{document}, where the small parameter ε>0\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\varepsilon >0$$\\end{document} related to the thickness of the diffuse interface is sent to zero. For well-prepared initial data and sufficiently small times, we rigorously prove convergence to the classical two-phase Navier–Stokes system with surface tension. The idea of the proof is to use asymptotic expansions to construct an approximate solution and to estimate the difference of the exact and approximate solutions with a spectral estimate for the (at the approximate solution) linearized Allen–Cahn operator. In the calculations we use a fractional order ansatz and new ansatz terms in higher orders leading to a suitable ε\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\varepsilon $$\\end{document}-scaled and coupled model problem. Moreover, we apply the novel idea of introducing ε\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\varepsilon $$\\end{document}-dependent coordinates.

Weyl–invariant scalar–tensor gravities from purely metric theories

We describe a method to generate scalar–tensor theories with Weyl symmetry, starting from arbitrary purely metric higher derivative gravity theories. The method consists in the definition of a conformally-invariant metric g^μν\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\hat{g}_{\\mu \ u }$$\\end{document}, that is a rank (0,2)-tensor constructed out of the metric tensor and the scalar field. This new object has zero conformal weight and is given by ϕ2/Δgμν\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\phi ^{2/\\Delta }g_{\\mu \ u }$$\\end{document}, where (-Δ)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$(-\\Delta )$$\\end{document} is the conformal dimension of the scalar. As gμν\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$g_{\\mu \ u }$$\\end{document} has conformal dimension of 2, the resulting tensor is trivially a conformal invariant. Then, the generated scalar–tensor theory, which we call the Weyl uplift of the original purely metric theory, is obtained by replacing the metric by g^μν\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\hat{g}_{\\mu \ u }$$\\end{document} in the action that defines the original theory. This prescription allowed us to define the Weyl uplift of theories with terms of higher order in the Riemannian curvature. Furthermore, the prescription for scalar–tensor theories coming from terms that have explicit covariant derivatives in the Lagrangian is discussed. The same mechanism can also be used for the derivation of the equations of motion of the scalar–tensor theory from the original field equations in the Einstein frame. Applying this method of Weyl uplift allowed us to reproduce the known result for the conformal scalar coupling to Lovelock gravity and to derive that of Einsteinian cubic gravity. Finally, we show that the cancellation of the volume divergences in the theory given by the conformal scalar coupling to Einstein–Anti-de Sitter gravity is achieved by the Weyl uplift of the original theory augmented by counterterms, which is relevant in the framework of conformalrenormalization.

An improved BRDF hotspot model and its use in VLIDORT for studying the impact of atmospheric scattering on hotspot directional signatures in the atmosphere

Abstract. The term “hotspot” refers to the sharp increase in the reflectance occurring when incident (solar) and reflected (viewing) directions almost coincide in the backscatter direction. The accurate simulation of hotspot directional signatures is important for many remote sensing applications. The RossThick–LiSparse–Reciprocal (RTLSR) bidirectional reflectance distribution function (BRDF) model is widely used in radiative transfer simulations, and the hotspot model mostly used is from Maignan–Bréon, but it typically requires large values of numerical quadrature and Fourier expansion terms in order to represent the hotspot accurately for its use coupled with atmospheric radiative transfer modeling (RTM). In this paper, we have developed a modified version based on the Maignan–Bréon's hotspot BRDF model that converges much faster numerically, making it more practical for use in the RTMs that require Fourier expansion of BRDF to simulate the top-of-atmosphere (TOA) hotspot signatures, such as in the RTM models using the Doubling–Adding or discrete ordinate method. Using the vector linearized discrete ordinate radiative transfer model (VLIDORT), we found that reasonable TOA–hotspot accuracy can be obtained with just 23 Fourier terms for clear atmosphere and 63 Fourier terms for atmosphere with aerosol scattering. In order to study the impact of molecular and aerosol scattering on hotspot signatures, we carried out a number of hotspot signature simulations with VLIDORT. We confirmed that (1) atmospheric molecule scattering and the existence of aerosol tend to smooth out the hotspot signature at the TOA and that (2) the hotspot signature at the TOA in the near-infrared is larger than in the visible, and its impact by surface reflectance is more significant. As the hotspot amplitude at the TOA with aerosol scattering included is smaller than that with molecular scattering only, the amplitude of hotspot signature at the surface is likely underestimated in the previous analysis based on the POLDER measurements, where the atmospheric correction was based on a single-scatter Rayleigh-only calculation. This modified model can calculate the amplitude of the hotspot accurately, and, as it agrees very well with the original RossThick model away from the hotspot region, this model can be simply used in conditions with and without hotspots. However, there are some differences in this modified model compared to the original Maignan–Bréon model for the scattering angles close to the hotspot point; thus, it may not be appropriate for those who need an exact representation of the hotspot angular signature.

Derivation of Miller’s rule for the nonlinear optical susceptibility of a quantum anharmonic oscillator

Miller’s rule is an empirical relation between the nonlinear and linear optical coefficients that applies to a large class of materials but has only been rigorously derived for the classical Lorentz model with a weak anharmonic perturbation. In this work, we extend the proof and present a detailed derivation of Miller’s rule for an equivalent quantum-mechanical anharmonic oscillator. For this purpose, the classical concept of velocity-dependent damping inherent to the Lorentz model is replaced by an adiabatic switch-on of the external electric field, which allows a unified treatment of the classical and quantum-mechanical systems using identical potentials and fields. Although the dynamics of the resulting charge oscillations, and hence the induced polarizations, deviate due to the finite zero-point motion in the quantum-mechanical framework, we find that Miller’s rule is nevertheless identical in both cases up to terms of first order in the anharmonicity. With a view to practical applications, especially in the context of ab initio calculations for the optical response where adiabatically switched-on fields are widely assumed, we demonstrate that a correct treatment of finite broadening parameters is essential to avoid spurious errors that may falsely suggest a violation of Miller’s rule, and we illustrate this point by means of a numerical example.

Implications of the weak gravity conjecture for de Sitter space decay by flux discharge

We examine the implications of the weak gravity conjecture for the mechanisms for discharging cosmological constant via membrane nucleations. Once screening fluxes and membranes which source them enter, and weak gravity bounds are enforced, a generic de Sitter space must be unstable. We show that when all the flux terms which screen and discharge the cosmological constant are dominated by quadratic and higher order terms, the bounds from weak gravity conjecture and naturalness lead toward anthropic outcomes. In contrast, when the flux sectors are dominated by linear flux terms, anthropics may be avoided, and the cosmological constant may naturally decay toward smallest possible values. Published by the American Physical Society 2024

Inter-scale Causality Relations in Wall Turbulence

We investigate the inter-scale energy transfer in the buffer layer between different streamwise and spanwise lengthscales. In a periodic channel, inter-scale transfer of momentum in the periodic directions is only possible through wavenumber interactions in the non-linear advection term. The aim of this study is to map the inter-scale interactions which transfer energy through advection to different lengthscales of interest. The non-linear advection term has nine contributions, which correspond to three different advecting and advected velocity components. Because the flow is anisotropic, each of these contributions can be viewed as portraying a different physical mechanism. Maps of inter-scale transfer are therefore constructed for each of the nine contributions to the advection term in order to understand the relative importance of advecting and advected velocity components in wall turbulence. As a preliminary exploration, two maps of the inter-scale energy transfer in the buffer layer are presented for Reτ = 180, one for transfer to a ‘target’ lengthscale dynamically relevant in the near-wall cycle, λx+=188 , λz+=94 and one for transfer to a ‘target’ lengthscale in the dissipative range, λx+=42 , λz+=21 . In the first case, we observe that the transfer of energy is dominated by transfer into streamwise-velocity (u) by spanwise-elongated structures of spanwise-velocity (w) that advect streamwise-elongated structures of streamwise-velocity (u). This is nothing but the process of streak meandering induced by spanwise-elongated (w) structures. In the second case, the inter-scale transfer can be split into three dominant mechanisms. The first and second mechanisms are observed across many of the nine contributions to the advection term. Velocity structures with size similar or larger than streamwise streaks either advect (mechanism 1) or are advected by (mechanism 2) velocity structures with lengthscales close to the target one. The third mechanism is only significant for streamwise (u) velocity structures advecting streamwise (u) and spanwise (w) velocity, and involves structures with an intermediate scale which is in-between the dissipation scale and the near-wall cycle dynamic lengthscale, but still in the dissipation range. In this mechanism, intermediate-scale streamwise-velocity structures advect intermediate-scale streamwise- and spanwise-velocity structures.

Anharmonic effects on the squeezing of axion perturbations

It is assumed in standard cosmology that the Universe underwent a period of inflation in its earliest phase, providing the seeds for structure formation through vacuum fluctuations of the inflaton scalar field. These fluctuations get stretched by the quasi-exponential expansion of the Universe and become squeezed. The aim of this paper is to deepen the understanding of the squeezing process, considering the effect of self-interactions. Axion-like particles can provide a useful setup to study this effect. Specifically we focus on the consequences that a non-trivial evolution of the background axion field has on the squeezing of the perturbations. We follow the evolution of the axion's fluctuation modes from the horizon exit during inflation to the radiation-dominated epoch. We compute Bogoliubov coefficients and squeezing parameters, which are linked to the axion particle number and isocurvature perturbation. We find that the quantum mechanical particle production and the squeezing of the perturbations are enhanced, if one accounts for anharmonic effects, i.e., the effect of higher order terms in the potential. This effect becomes particularly strong towards the hilltop of the potential.

Bubble wall velocities with an extended fluid Ansatz

We compute the terminal bubble wall velocity during a cosmological phase transition by modelling non-equilibrium effects in the plasma with the so-called “extended fluid Ansatz”. A ϕ 6 operator is included in the Standard Model effective potential to mimic effects of new physics. Hydrodynamical heating of the plasma ahead of the bubble is taken into account. We find that the inclusion of higher order terms in the fluid Ansatz is typically relevant, and may even turn detonation solutions into deflagrations. Our results also corroborate recent findings in the literature that, for a Standard Model particle content in the plasma, only deflagration solutions are viable. However, we also show that this outcome may be altered in a theory with a different particle content.

Normal forms of a class of partial functional differential equations

In this paper, we investigate the normal form a class of partial functional differential equations, which takes account of delays in the diffusion terms as well as a wider scope of nonlinear terms. We first study the associated linear theory, mainly including the spectral properties of infinitesimal generator, formal adjoint and decomposition of phase space. Based on these results, the normal form theory for nonlinear equation is established, which can be used to study the local dynamics near the steady state for such equations. As an application, we consider the Hopf bifurcation problem of a scalar diffusive equation with delay not only involved in the diffusive term but also in reaction terms. The normal form, depending on the original coefficients, up to the third order term is calculated, which allows us to determine the direction of Hopf bifurcation and stability of bifurcated periodic solutions. The results are then applied to study the scalar population model with memory-based diffusion for modeling the movement of highly-developed animals.

Nonlinear random perturbations of PDEs and quasi-linear equations in Hilbert spaces depending on a small parameter

We study a class of quasi-linear parabolic equations defined on a separable Hilbert space, depending on a small parameter in front of the second order term. Through the nonlinear semigroup associated with such equation, we introduce the corresponding SPDE and we study the asymptotic behavior of its solutions, depending on the small parameter. We show that a large deviations principle holds and we give an explicit description of the action functional.

A discrete time-varying grey Fourier model with fractional order terms for electricity consumption forecast

Accurate electricity consumption forecast plays an important role in managing and operating a country’s power system and is a challenging task due to the data exhibiting the mixture characteristics of trend, periodicity and randomness. To accurately describe the dynamic changes of seasonal fluctuations of electricity consumption, a novel discrete time-varying grey Fourier model with fractional order terms is proposed by coupling Fourier functions with fractional order time-varying terms, as the grey action, to fit the amplitude variations hidden in seasonal pattern and improve the fitting and forecasting accuracy. Moreover, the mechanism and properties of the proposed model are discussed. Then the optimal fractional order is determined by the Dingo Optimization Algorithm (DOA) and the optimal Fourier order is selected by the hold-out method to enhance the robustness of the model. Subsequently, the results of designed experiments based on Monte-Carlo method testify the validity of the Fourier order selection method and the proposed model. Finally, this model is applied to predict the monthly residential electricity consumption of China from 2015 to 2022. The results indicate that the proposed model captures the dynamic amplitude variations of the time series and has better prediction performances than other benchmark grey prediction models and non-grey prediction models.

Higher order terms of Mather's β-function for symplectic and outer billiards

We compute explicitly the formal Taylor expansion of Mather's β-function up to seventh order terms for symplectic and outer billiards in a strictly-convex planar domain C. In particular, we specify the third terms of the asymptotic expansions of the distance (in the sense of the symmetric difference metric) between C and its best approximating inscribed or circumscribed polygons with at most n vertices. We use tools from affine differential geometry.

Existence and uniqueness results for an elliptic equation with blowing-up coefficient and singular lower order term

Existence and uniqueness results for an elliptic equation with blowing-up coefficient and singular lower order term

Diagonalizing the Born-Oppenheimer Hamiltonian via Moyal perturbation theory, nonadiabatic corrections, and translational degrees of freedom.

This article describes a method for calculating higher order or nonadiabatic corrections in Born-Oppenheimer theory and its interaction with the translational degrees of freedom. The method uses the Wigner-Weyl correspondence to map nuclear operators into functions on the classical phase space and the Moyal star product to represent operator multiplication on those functions. These are explained in the body of the paper. The result is a power series in κ2, where κ = (m/M)1/4 is the usual Born-Oppenheimer parameter. The lowest order term is the usual Born-Oppenheimer approximation, while higher order terms are nonadiabatic corrections. These are needed in calculations of electronic currents, momenta, and densities. The separation of nuclear and electronic degrees of freedom takes place in the context of the exact symmetries (for an isolated molecule) of translations and rotations, and these, especially translations, are explicitly incorporated into our discussion. This article presents an independent derivation of the Moyal expansion in molecular Born-Oppenheimer theory. We show how electronic currents and momenta can be calculated within the framework of Moyal perturbation theory; we derive the transformation laws of the electronic Hamiltonian, the electronic eigenstates, and the derivative couplings under translations; we discuss in detail the rectilinear motion of the molecular center of mass in the Born-Oppenheimer representation; and we show how the elimination of the translational components of the derivative couplings leads to a unitary transformation that has the effect of exactly separating the translational degrees of freedom.

Nonlinear guided waves in the sandwich nanostructure coupled with piezoelectric actuator: Introducing machine learning approach to measure nonlinear phase velocities

In today's rapidly changing corporate environment, business executives are turning to cutting-edge technology to solve complex problems efficiently. A machine learning approach allows for a more thorough understanding and quantification of decision-making outcomes, leading to increased confidence in business decisions. In this work for the first time, machine learning approach and mathematical simulation are used to model nonlinear guided waves in the sandwich nanostructure coupled with piezoelectric actuator. Using mathematical modeling, the current work studies the nonlinear phase velocity in a sandwich nanoshell composed of a bi-directional functionally graded (BD-FG) core and piezoelectric patch. The modeling uses first-order shear deformation theory with the fairly thick hypothesis to describe the displacement fields of piezoelectric layers. For core layer, a higher order shear deformation theory with terms of higher order in the Taylor series expansion is presented. The nanosystem is accurately modeled using nonlocal strain gradient theory, incorporating nonlocal and length scale characteristics. The solution approach section outlines the multiple scale method for the time domain and the harmonic solution methodology for the displacement domain to address nonlinear equations. To reduce the computational expenses, a machine-learning approach is introduced to address the nonlinear vibration issue in the current study. Validating the conclusions of the current study by comparing them with machine-learning solutions and another published paper can confirm their accuracy. The results section presents the impacts of factors like area of piezoelectric patch, length scale, applied voltage, nonlocality, geometry condition and FG power index on the nonlinear phase velocity of the BD-FG nanoshell coupled with piezoelectric patch.

Isochoric cooling of air in the University Physics Laboratory

In this paper an alternative way of conducting the physics laboratory exercise for determining the adiabatic index of air using the Clement-Desormes method is proposed. The process of isochoric cooling of air has been studied in terms of the dependence of pressure on time, and hence temperature on time, since it is proportional to pressure at a constant volume and mass of air. A theoretical model of the consid- ered process was also made. The experimental results were processed statistically. The coefficient of determination R2 and the F-test statistic were calculated and their val- ues indicate a very good agreement between theory and experiment. The analysis of the residuals, however, implies that the model could be further improved through the inclusion of higher order terms.