Perturbation Theory Expansion Research Articles

Note on the bispectrum and one-loop corrections in single-field inflation with primordial black hole formation

Primordial black holes can be formed from the collapse of large-amplitude perturbation on small scales in the early Universe. Such an enhanced spectrum can be realized by introducing a flat region in the potential of single-field inflation, which makes the inflaton go into a temporary ultraslow-roll period. In this paper, we calculate the bispectrum of curvature perturbation in such a scenario. We explicitly confirm that bispectrum satisfies Maldacena’s theorem. At the end of the ultraslow-roll period, the bispectrum is generated by bulk interaction and field redefinition. At the end of inflation, bispectrum is generated only by bulk interaction. We also calculate the one-loop correction to the power spectrum from the bispectrum, called the source method. We find it consistent with the calculation of the one-loop correction from the second-order expansion of in-in perturbation theory. Published by the American Physical Society 2024

Physical and dynamic characteristics of high-amplitude ultrasonic wave propagation in nonlinear and dissipative media

This study presents a comprehensive investigation into the soliton solutions for the Khokhlov–Zabolotskaya–Kuznetsov ([Formula: see text]) problem using the Bernoulli sub-equation function approach, generalized Kudryashov method, and Homotopy perturbation method. The [Formula: see text] equation, a nonlinear wave equation that governs the propagation of high-amplitude ultrasonic waves through nonlinear media, is studied in detail. Perturbation theory and asymptotic expansions are employed to derive approximate solutions for the [Formula: see text] problem. These methodologies offer valuable insights into the behavior of ultrasonic waves in different media, thereby facilitating the optimization of ultrasonic transducer design and enhancing the precision of ultrasonic imaging systems. The analytical and semi-analytical methods utilized for solving the [Formula: see text] equation are computationally efficient and serve as valuable resources for researchers and engineers working in the field of ultrasonics. The outcomes of this study carry significant implications for the understanding of ultrasonic wave behavior in nonlinear and dissipative media, ultimately contributing to the development of more accurate and efficient ultrasonic imaging techniques.

On the connection between perturbation theory and new semiclassical expansion in quantum mechanics

On the connection between perturbation theory and new semiclassical expansion in quantum mechanics

Oscillatory and transient dynamics of a slow–fast predator–prey system with fear and its carry-over effect

In almost every ecological system, growth of various interacting species evolve in different time scales and the implementation of this time scale difference in the corresponding mathematical model exhibits some rich and complex oscillatory dynamics. In this article, we consider a predator–prey model with Beddington–DeAngelis functional response in which the prey reproduction is affected by the predation induced fear and its carry-over effect. Considering the growth of prey species occurs on a faster time scale than that of predator, the proposed system reduces to a ‘slow–fast predator–prey’ system. Using the geometric singular perturbation theory and asymptotic expansion technique, we investigate the system both analytically and numerically, and observe a wide range of rich and complex dynamics such as canard cycles (with or without head) near the singular Hopf-bifurcation threshold and relaxation oscillation cycles. The system experiences a canard explosion through which a rapid transition from small amplitude limit cycle to large amplitude limit cycle occurs in a tiny parametric interval. These types of complex oscillatory dynamics are absent in non slow–fast systems. Furthermore, it is shown that the interplay between fear and its carry-over effect, and the variation of time scale parameter may lead to a regime shift of the oscillatory dynamics. We also study the impact of fear and its carry-over effect on the properties of long transient dynamics. Thus our study provides some valuable biological insights of a slow–fast predator–prey system which will aid in understanding the interplay between fear and its carry-over effect.

Pseudospectrum of horizonless compact objects: A bootstrap instability mechanism

Recent investigations of the pseudospectrum in black hole spacetimes have shown that quasinormal mode frequencies suffer from spectral instabilities. This phenomenon may severely affect gravitational-wave spectroscopy and limit precision tests of general relativity. We extend the pseudospectrum analysis to horizonless exotic compact objects that possess a reflective surface arbitrarily close to the Schwarzschild radius and find that their quasinormal modes also suffer from an overall spectral instability. Even though all the modes themselves decay monotonically, the pseudospectrum contours of equal resonance magnitude around the fundamental mode and the lowest overtones can cross the real axis into the unstable regime of the complex plane, unveiling the existence of nonmodal pseudoresonances. A pseudospectrum analysis further predicts that fluctuations to the system may destabilize the object when next to leading-order effects are considered, as the triggering of pseudoresonant growth can break the order expansion of black hole perturbation theory.

Memory-aware end-to-end learning of channel distortions in optical coherent communications.

We implement a new variant of the end-to-end learning approach for the performance improvement of an optical coherent-detection communication system. The proposed solution enables learning the joint probabilistic and geometric shaping of symbol sequences by using auxiliary channel model based on the perturbation theory and the refined symbol probabilities training procedure. Due to its structure, the auxiliary channel model based on the first order perturbation theory expansions allows us performing an efficient parallelizable model application, while, simultaneously, producing a remarkably accurate channel approximation. The learnt multi-symbol joint probabilistic and geometric shaping demonstrates a considerable bit-wise mutual information gain of 0.47 bits/2D-symbol over the conventional Maxwell-Boltzmann shaping for a single-channel 64 GBd transmission through the 170 km single-mode fiber link.

Keldysh rotation in the large- N expansion and string theory out of equilibrium

We extend our study of the large-$N$ expansion of general nonequilibrium many-body systems with matrix degrees of freedom $M$, and its dual description as a sum over surface topologies in a dual string theory, to the Keldysh-rotated version of the Schwinger-Keldysh formalism. The Keldysh rotation trades the original fields ${M}_{\ifmmode\pm\else\textpm\fi{}}$---defined as the values of $M$ on the forward and backward segments of the closed time contour---for their linear combinations ${M}_{\mathrm{cl}}$ and ${M}_{\mathrm{qu}}$, known as the ``classical'' and ``quantum'' fields. First we develop a novel ``signpost'' notation for nonequilibrium Feynman diagrams in the Keldysh-rotated form, which simplifies the analysis considerably. Before the Keldysh rotation, each world-sheet surface $\mathrm{\ensuremath{\Sigma}}$ in the dual string theory expansion was found to exhibit a triple decomposition into the parts ${\mathrm{\ensuremath{\Sigma}}}^{\ifmmode\pm\else\textpm\fi{}}$ corresponding to the forward and backward segments of the closed time contour, and ${\mathrm{\ensuremath{\Sigma}}}^{\ensuremath{\wedge}}$ which corresponds to the instant in time where the two segments meet. After the Keldysh rotation, we find that the world-sheet surface $\mathrm{\ensuremath{\Sigma}}$ of the dual string theory undergoes a very different natural decomposition: $\mathrm{\ensuremath{\Sigma}}$ consists of a ``classical'' part ${\mathrm{\ensuremath{\Sigma}}}^{\mathrm{cl}}$ and a ``quantum embellishment'' part ${\mathrm{\ensuremath{\Sigma}}}^{\mathrm{qu}}$. We show that both parts of $\mathrm{\ensuremath{\Sigma}}$ carry their own independent genus expansion. The nonequilibrium sum over world-sheet topologies is naturally refined into a sum over the double decomposition of each $\mathrm{\ensuremath{\Sigma}}$ into its classical and quantum part. We apply this picture to the classical limits of the quantum nonequilibrium system (with or without interactions with a thermal bath), and find that in these limits, the dual string perturbation theory expansion reduces to its appropriately defined classical limit.

Theory-guided physics-informed neural networks for boundary layer problems with singular perturbation

Physics-informed neural networks (PINNs) are a recent trend in scientific machine learning research and modeling of differential equations. Despite progress in PINN research, large gradients and highly nonlinear patterns remain challenging to model. Thin boundary layer problems are prominent examples of large gradients that commonly arise in transport problems. In this study, boundary-layer PINN (BL-PINN) is proposed to enable a solution to thin boundary layers by considering them as a singular perturbation problem. Inspired by the classical perturbation theory and asymptotic expansions, BL-PINN is designed to replicate the procedure in singular perturbation theory. Namely, different parallel PINN networks are defined to represent different orders of approximation to the boundary layer problem in the inner and outer regions. In different benchmark problems (forward and inverse), BL-PINN shows superior performance compared to the traditional PINN approach and is able to produce accurate results, whereas the classical PINN approach could not provide meaningful solutions. BL-PINN also demonstrates significantly better results compared to other extensions of PINN such as the extended PINN (XPINN) approach. The natural incorporation of the perturbation parameter in BL-PINN provides the opportunity to evaluate parametric solutions without the need for retraining. BL-PINN demonstrates an example of how classical mathematical theory could be used to guide the design of deep neural networks for solving challenging problems.

Molecular Ordering and Nematic Character of Nonyloxy Benzoic Acid: Solvent and Thermodynamic Potentials Effect

Abstract: Benzoic acid derivative p-n-Nonyloxy Benzoic Acid (NOBA) has been chosen for the present investigation to analyze its properties under the effect of a polar aprotic solvent. Indication of stable phase has been examined by interaction-energy values both in pure form and in a polar aprotic solvent dimethylformamide (DMF, ε=38) for different modes of interaction. Complete neglect differential overlap (CNDO/2) method has been applied for net atomic charges and dipole moments at the atomic center. Modified version of Rayleigh–Schrödinger perturbation theory and multicentered-multipole expansion method have been used to regulate long-range intermolecular interactions, where short-range interactions have been studied by 6-exponential potential function. Details on phase behavior and phase stability of NOBA have been reported using statistical thermodynamic calculations of Helmholtz free energies and entropies in several modes of interaction. Keywords: NOBA, Interaction energy, Helmholtz free energy, Entropy, Stability.

Quantum orders in the frustrated Ising model on the bathroom tile lattice

We determine the zero and finite temperature phase diagram of the fully frustrated quantum Ising model on the bathroom tile (4-8) lattice. The phase diagram exhibits a wealth of 2+1d physics, including 1. classical Coulomb dimer liquids of both square and triangular lattice types; 2. quantum order-by-disorder induced phases breaking $\mathbb{Z}_4$, $\mathbb{Z}_6$, and $\mathbb{Z}_8$ symmetries; 3. finite temperature Kosterlitz-Thouless (KT) phases floating over the $\mathbb{Z}_6$ and $\mathbb{Z}_8$ orders; and, 4. staircases of (in)-commensurate symmetry breaking phases at intermediate coupling. We establish this elaborate phase diagram using a combination of dimer model mapping, perturbation theory, Landau analysis and Stochastic Series Expansion Quantum Monte Carlo (QMC-SSE). Our results provide a baseline for studying frustrated magnetism with D-Wave architecture annealers, where the 4-8 lattice can be embedded naturally without `cloning', reducing the number of competing energy scales. Simulations with the D-Wave 2000Q demonstrate qualitative agreement with the high temperature portion of the phase diagram, but are unable to access the low temperature phases.

The cosmic web connection to the dark matter halo distribution through gravity

ABSTRACT This work investigates the connection between the cosmic web and the halo distribution through the gravitational potential at the field level. We combine three fields of research, cosmic web classification, perturbation theory expansions of the halo bias, and halo (galaxy) mock catalogue making methods. In particular, we use the invariants of the tidal field and the velocity shear tensor as generating functions to reproduce the halo number counts of a reference catalogue from full gravity calculations, populating the dark matter field on a mesh well into the non-linear regime ($3\, h^{-1}\, {\rm Mpc}$ scales). Our results show an unprecedented agreement with the reference power spectrum within 1 per cent up to $k=0.72\, h\, {\rm Mpc}^{-1}$. By analysing the three-point statistics on large scales (configurations of up to $k=0.2\, h\, {\rm Mpc}^{-1}$), we find evidence for non-local bias at the 4.8σ confidence level, being compatible with the reference catalogue. In particular, we find that a detailed description of tidal anisotropic clustering on large scales is crucial to achieve this accuracy at the field level. These findings can be particularly important for the analysis of the next generation of galaxy surveys in mock galaxy production.

Computing one-loop radiative corrections in τ → π(K)ν[γ] and testing new physics

Without a doubt the ratios Rτ/p = Γ(τ → Pντ[γ])/Γ(P → μνμ[γ]) (P = π, K) give a convenient scenario to test the lepton universality, the CKM unitarity and to search for non-standard interactions in τ decays. Moreover, the previous theoretical estimation of these observables is more than twenty-five years old and some assumptions of that estimation are unrealistic. Therefore, an update of Rτ/P was timely. The one-loop radiative corrections have been computed by considering an expansion of Chiral Perturbation Theory including the lightest spin-one resonances and respecting the short-distance behavior dictated by QCD. We have reported δRτ/π = (0.18±0.57)% and δRτ/K = (0.97±0.58)%, where the uncertainties have been induced fundamentally by the estimation of the counterterms. We have tested the lepton universality, the CKM unitarity and have searched for new physics in τ decays. As a by-product, we have also determined the theoretical radiative corrections to the τ → Pντ[γ] decay rates, δτπ = —(0.24 ± 0.56)% and δτK = —(0.15 ± 0.57)%.

The large scale structure bootstrap: perturbation theory and bias expansion from symmetries

We investigate the role played by symmetries in the perturbative expansion of the large-scale structure. In particular, we establish which of the coefficients of the perturbation theory kernels are dictated by symmetries and which not. Up to third order in perturbations, for the dark matter density contrast (and for the dark matter velocity) only three coefficients are not fixed by symmetries and depend on the particular cosmology. For generic biased tracers, where number/mass and momentum conservation cannot be imposed in general, this number rises to seven in agreement with other bias expansions discussed in the literature. A crucial role in our analysis is provided by extended Galilean invariance, which follows from diffeomorphism invariance in the non-relativistic limit. We identify a full hierarchy of extended Galilean invariance constraints, which fix the analytic structure of the perturbation theory kernels as the sums of an increasing number of external momenta vanish.Our approach is especially relevant for non-standard models that respect the same symmetries as ΛCDM and where perturbation theory at higher orders has not been exhaustively explored, such as dark energy and modified gravity scenarios.In this context, our results can be used to systematically extend the bias expansion to higher orders and set up model independent analyses.

Variety of scaling laws for DNA thermal denaturation

We discuss possible mechanisms that may impact the order of the transition between denaturated and bound DNA states and lead to changes in the scaling laws that govern conformational properties of DNA strands. To this end, we re-consider the Poland–Scheraga model and apply a polymer field theory approach to calculate entropic exponents associated with the denaturated loop distribution. We discuss in particular variants of this transition that may occur due to the properties of the solution and may affect the self- and mutual interaction of both single and double strands. We find that the effects studied significantly influence the strength of the first order transition. This is manifested in particular by the changes in the scaling laws that govern DNA loop and strand distribution. As a quantitative measure of these changes we present the values of corresponding scaling exponents. For the d=4−ε case we get corresponding ε4 expansions and evaluate the perturbation theory expansions at space dimension d=3 by means of resummation technique.

Structure and phase behavior of alkoxy benzoic acids in DMSO for thermodynamic applications: Theoretical investigation

This manuscript emphasizes the phase behavior of p-n-Octyloxy Benzoic acid (OOBA) and p-n-Nonyloxy Benzoic acid (NOBA) which are homologues of alkoxy benzoic acid. Fluctuation of the intermolecular interaction energies have been noticed according to rotational and translational co-ordinates for both the compounds in vacuum and under the influence of polar aprotic solvent Dimethyl sulfoxide (DMSO, ε=47). Structures of the molecules have been framed for both the compounds using published crystallographic data. All the charges of atoms and dipole moments of each atom have been estimated on the basis of complete neglect differential overlap (CNDO/2) method. Long-range intermolecular interactions have been considered by modified Rayleigh–Schrodinger perturbation theory and multicentered-multipole expansion method. Short-range interactions have been described by 6-exponential potential function. Theoretical thermodynamic formulations are chosen for the determination of Helmholtz free energies, entropies and internal energies. Existence of liquid crystal phase and its behavior have been discussed by the combined results achieved from relative free energy (ΔA) and relative entropy (ΔS) data. The possibility of addressing the multi-phase phenomenon is discussed from these relative values. The present data is expected to provide a route to analyze behavior of molecules in various solvents, and property prediction for thermodynamic applications.

High-Frequency Radar Cross Section of the Ocean Surface With Arbitrary Roughness Scales: A Generalized Functions Approach

A correction term for electromagnetic scattering from the ocean surface with large roughness scales in the HF band is proposed. A generalized functions' method has been employed to obtain the operator equation of the electric field over the ocean surface. Using a dipole antenna with a pulsed sinusoidal signal at the transmitter and employing Fourier series and perturbation theory expansions of the ocean surface, three expressions were derived for the electric field up to the second order, with one of them being a correction term for the first order for large roughness scales. Through a cross section analysis, it is observed that the additional term would represent waves with both large heights and wavelengths but small slopes. This allows for the analysis of extreme wave events using HF radar, which was limited by previously derived small heights and small slope methods.

Quark flavor phenomenology of the QCD axion

Axion models with generation-dependent Peccei-Quinn charges can lead to flavor-changing neutral currents, thus motivating QCD axion searches at precision flavor experiments. We rigorously derive limits on the most general effective flavor-violating couplings from current measurements and assess their discovery potential. For two-body decays we use available experimental data to derive limits on $q\to q' a$ decay rates for all flavor transitions. Axion contributions to neutral-meson mixing are calculated in a systematic way using chiral perturbation theory and operator product expansion. We also discuss in detail baryonic decays and three-body meson decays, which can lead to the best search strategies for some of the couplings. For instance, a strong limit on the $\Lambda\to n a$ transition can be derived from the supernova SN 1987A. In the near future, dedicated searches for $q\to q' a$ decays at ongoing experiments could potentially test Peccei-Quinn breaking scales up to $10^{12}$ GeV at NA62 or KOTO, and up to $10^{9}$ GeV at Belle II or BES III.

Implementing Chiral Three-Body Forces in Terms of Medium-Dependent Two-Body Forces

Three-nucleon (3N) forces are an indispensable ingredient for accurate few-body and many-body nuclear structure and reaction theory calculations. While the direct implementation of chiral 3N forces can be technically very challenging, a simpler approach is given by employing instead a medium-dependent NN interaction V_med that reflects the physics of three-body forces at the two-body normal-ordered approximation. We review the derivation and construction of V_med from the chiral 3N interaction at next-to-next-to-leading order (N2LO), consisting of a long-range $2\pi$-exchange term, a mid-range $1\pi$-exchange component and a short-range contact term. Several applications of V_med to the equation of state of cold nuclear and neutron matter, the nucleon single-particle potential in nuclear matter, and the nuclear quasiparticle interaction are discussed. We also explore differences in using local vs. nonlocal regulating functions on 3N forces and make direct comparisons to exact results at low order in perturbation theory expansions for the equation of state and single-particle potential. We end with a discussion and numerical calculation of the in-medium NN potential V_med from the next-to-next-to-next-to-leading order (N3LO) chiral 3N force, which consists of a series of long-range and short-range terms.

Interval Analysis of the Eigenvalues of Closed-Loop Control Systems with Uncertain Parameters

Uncertainty caused by a parameter measurement error or a model error causes difficulties for the implementation of the control method. Experts can divide the uncertain system into a definite part and an uncertain part and solve each part using various methods. Two uncertainty problems of the control system arise: problem A for the definite part—how does one find out the optimal number and position of actuators when the actuating force of an actuator is smaller than the control force? Problem B for the uncertain part—how does one evaluate the effect of uncertainty on the eigenvalues of a closed-loop control system? This paper utilizes an interval to express the uncertain parameters and converts the control system into a definite part and an uncertain part using interval theory. The interval state matrix is constructed by physical parameters of the system for the definite part of the control system. For Problem A, the paper finds out the singular value element sensitivity of the modal control matrix and reorders the optimal location of the actuators. Then, the paper calculates the state feedback gain matrix for a single actuator using the receptance method of pole assignment and optimizes the number and position of the actuators using the recursive design method. For Problem B, which concerns the robustness of closed-loop systems, the paper obtains the effects of uncertain parameters on the real and imaginary parts of the eigenvalues of a closed-loop system using the matrix perturbation theory and interval expansion theory. Finally, a numerical example illustrates the recursive design method to optimize the number and location of actuators and it also shows that the change rate of eigenvalues increases with the increase in uncertainty.

Rigorous vibrational Fermi resonance criterion revealed: two different approaches yield the same result

Molecular vibrational resonances are manifest by abnormally small energy differences in denominators of anharmonic constants derived within the second-order vibrational perturbation theory (VPT2), or in abnormally large coefficients Ξ of unitary transformation generators in the canonical Van Vleck perturbation theory (CVPT). A quantitative measure of the vibrational resonance can be derived by assuming the perturbation parameter to be a complex variable , and the numerical analysis of the diverging high order Rayleigh-Schrödinger perturbation theory (RSPT) expansions for closely spaced states. The location of branch points for complex values of within the unit circle serves as the definitive evidence for a resonance. In practice, RSPT series for diverging resonant states can be treated by resummation using quadratic Padé-Hermite approximants. The critical values of can be further found and checked against the condition . A comparative analysis of selected resonances for some molecules (HO, HDO, HS and HCO) revealed a threshold value of that is equivalent to the RSPT parameter . The fundamentally different approach based on the polyad analysis of the resonance vectors spanning the ()-dimensional subspace for the ethylene molecule led to nearly the same value for the resonance criterion.