Finite Scaling Research Articles

Multirepresentation Ability Analysis of Wave Matter Using Physics Education Technology

Education is the primary capital of a nation's progress, because with education, the human resources that a nation possesses will become quality. Physics is a branch of natural science, specifically studying natural phenomena and events through several scientific and systematic methods. The research is aimed at developing a tool to evaluate the multirepresentation of wave matter using Physics Education Technology. The methods used in this research are quantitative methods and the research is carried out by providing filling issues that will analyze verbal multirepresentation capabilities, tables and graphs. The research will be carried out at Semarang University using samples of civil engineering students. Research procedures through preliminary stages by drawing up a research plan. Preparing and compiling wave learning modules using the simulation facilities of Physics Education Technology, then compiling evaluation tools that are subject-based with multi-representation verbal approaches, tables and graphs. Validation of learning modules and evaluation tools to determine whether it is worthy of use or not, validation is carried out by 2 experts. Validation of test instruments by experts on material aspects averaged 93%, on construction aspects 94% and on language aspects reached 92%, of which results show that the instruments have excellent validity. In the finite scale test, the highest multirepresentation ability is communicating from mathematics to verbal and its mathematical abilities, while the lowest is in the ability to picture to mathematics and physics.

The use of pairwise comparisons for decision making may lead to grossly inaccurate results

Pairwise comparison matrices (also known as Saaty matrices) in conjunction with a finite discrete scale for their quantification, are considered by many as efficient and effective means for eliciting personal preferences from decision makers. This study demonstrates that under a highly optimistic assumption, called the ultra-accurate decision maker (UADM) assumption, the results obtained from such matrices may be grossly inaccurate. The true ranking of the compared entities may be altered and / or an abnormal rank grouping may occur when processing pairwise comparison matrices. The rates that such effects take place are exceptionally dramatic. Given that this occurs under the deliberately highly optimistic conditions of the UADM assumption, the implication is that under real-life circumstances these rates can be even more dramatic. The results are supported with theoretical and extensive computational investigations. The computational investigations are based on three novel and powerful tests introduced for the first time in this study. This is the first needed step to better understand the potential and limitations of pairwise comparisons before one attempts to remedy this problem.

The stochastic response of fatigue crack growth in scaled components

In fatigue testing the number of cycles and crack length to failure are recognised to be stochastic quantities necessitating repetitive testing to draw out statistically meaningful measures. The origin of this phenomena is uncertainty in the detailed microstructure, where (from a microstructural perspective) there exists no two identical test pieces. The situation is further compounded under scaled testing of polycrystalline specimens due to a form of coarsening with scaled samples containing less grains than the full-size test piece with samples drawn from the same material stock. Although microstructural uncertainly impedes predictability, implying the need for a probabilistic framework, it is nonetheless important to capture deterministic aspects that feature in any stochastic model. The theory advanced in this paper builds on the finite similitude scaling theory and the two-experiment first-order rule that features length as an invariant. Length invariance is shown to be critically important for fatigue as it caters for the many aspects of the microstructure that are not scaled under scaled testing. A diffusional stochastic model is introduced in the paper that is constrained by the first-order finite similitude rule. The approach is shown to enforce necessary determinism despite the uncertainties present and unlike many of the rules that are currently applied in fatigue analysis, it is applicable over a large range of length scales. Popular growth laws for long cracks are examined under the new framework, which transpires to be remarkably straightforward to apply.

The scaling of laser peen forming: A two-experiment finite similitude approach

Laser peen forming (LPF) utilizes laser-induced shock waves to bend and shape metal plates in what is effectively a non-thermal metal-forming process involving no hard tooling. A difficulty with the process, arising from the rapid localized physics involved, is the determination of process conditions for the establishment of desirable process outcomes. The nanosecond physical behaviors induced by the pulsed laser can make simulation impractical, effectively restricting investigations to experiments as the only practical recourse. This paper focuses on the use of scaled experimentation for LPF with the objective of making experimental outcomes more broadly applicable to a wider range of process conditions. Understanding how processes scale can in principle aid in the establishment of process parameters through timely and cost-effective experiments. Scaled experimentation has recently undergone a paradigm shift with the arrival of the finite similitude scaling theory. The theory provides extra degrees of freedom and facilitates the use of unlimited numbers of scaled experiments and allows for anisotropy in plate thickness. It is demonstrated in the work through experimental tests and simulation at two different scales, that geometric and loading similarities can be broken, yet the behavior of LPF can be quantified to good accuracy.

Energy growth in VLVL→ VLVL, VLVLh scattering to probe Higgs cubic and HEFT interactions

We compute the energy scales of perturbative unitarity violation in VLVL → VLVLh processes and compare them to VLVL → hhh process, where VL refers to a longitudinal mode of Z or W boson, and h the Higgs boson. Using these energy scales, we determine which process is more sensitive to potential modifications in the Higgs sector at high-energy colliders. Within the Higgs Effective Field Theory (HEFT), we consider the Higgs cubic coupling and other interactions with and without derivatives. Any HEFT interactions predict the perturbative unitarity violation at a finite scale, and in a generic case, the minimalistic process is 2 → 3 scattering. Our analysis reveals that the energy scales for unitarity violation in VLVL → VLVLh and VLVL → hhh processes are similar across all scenarios considered. If the backgrounds are similar, VLVLh final states are more feasible because VLVLh has higher branching ratios in cleaner decay modes than hhh. We also investigate HEFT derivative interactions derived from various UV models. In these cases, both VLVL → VLVL and VLVL → hh processes exhibit unitarity violating behavior. We demonstrate that the energy scales for unitarity violation in VLVL final states are comparable to or even lower than those in the hh final state.

Conformal structure of singularities in some varying fundamental constants bimetric cosmologies

In this paper, we explore the conformal structure of singularities arising from varying fundamental constants using the method of Penrose diagrams. We employ a specific type of bimetric model featuring two different metrics. One metric describes the causal structure for matter, while the other characterizes the causal structure for gravitational interactions, which is related to variations in fundamental constants such as the gravitational constant and the speed of light. For this reason, we focused on the gravitational metric to calculate the conformal transformation and compose Penrose diagrams for the singularities arising from the varying fundamental constants. We have shown that, in one case, the parameter such as the scale factor, the density and the pressure resemble those of the finite scale factor singularity (FSF). Despite singularity appears in constant conformal time in our case and in the case of FSF the Misner-Sharp horizon looks different. Our another case is similar to sudden future singularity (SFS), but there are differences in the conformal structures. We have also shown that in our cases the behavior of Misner-Sharp horizon strongly depends on initial conditions. The last analytical solution which we introduced is identical to conformal structure of the standard exotic singularity for the matter.

Minimal model for reservoir computing

A minimal model for reservoir computing is studied. We demonstrate that a reservoir computer exists that emulates given coupled maps by constructing a modularised network. We describe a possible mechanism for collapses of the emulation in the reservoir computing by introducing a measure of finite scale deviation. Such transitory behaviour is caused by either (i) an escape from a finite-time stagnation near an unstable chaotic set, or (ii) a critical transition driven by the effective parameter drift. Our approach reveals the essential mechanism for reservoir computing and provides insights into the design of reservoir computer for practical applications.

On the lifespan of solutions and control of high Sobolev norms for the completely resonant NLS on tori

We consider a completely resonant nonlinear Schrödinger equation on the d-dimensional torus, for any d≥1, with polynomial nonlinearity of any degree 2p+1, p≥1, which is gauge and translation invariant. We study the behaviour of high Sobolev Hs-norms of solutions, s≥s1+1>d/2+2, whose initial datum u0∈Hs satisfies an appropriate smallness condition on its lowHs1 and L2-norms respectively. We prove a polynomial upper bound on the possible growth of the Sobolev norm Hs over finite but long time scale that is exponential in the regularity parameter s1. As a byproduct we get stability of the low Hs1-norm over such time interval. A key ingredient in the proof is the introduction of a suitable “modified energy” that provides an a priori upper bound on the growth. This is obtained by combining para-differential techniques and suitable tame estimates.

Effects of wave damping and finite perpendicular scale on three-dimensional Alfvén wave parametric decay in low-beta plasmas

Effects of wave damping and finite perpendicular scale on three-dimensional Alfvén wave parametric decay in low-beta plasmas

Lorentzian Quantum Cosmology from Effective Spin Foams

Effective spin foams provide the most computationally efficient spin foam models yet and are therefore ideally suited for applications, e.g., to quantum cosmology. Here, we provide the first effective spin foam computations of a finite time evolution step in a Lorentzian quantum de Sitter universe. We will consider a setup that computes the no-boundary wave function and a setup describing the transition between two finite scale factors. A key property of spin foams is that they implement discrete spectra for the areas. We therefore study the effects that are induced by the discrete spectra. To perform these computations, we had to identify a technique to deal with highly oscillating and slowly converging or even diverging sums. Here, we illustrate that high-order Shanks transformation works very well and is a promising tool for the evaluation of Lorentzian (gravitational) path integrals and spin foam sums.

Enhancing the description of multi-time-scale geophysical phenomena: Incorporating finite time Scale separation and feedback, illustrated with the case of a 1D variable of interest.

Stochastic approaches play a vital role in weather, climate, and, more in general, geophysics systems, addressing processes and scales beyond the resolution of deterministic models. Similar to equilibrium/non-equilibrium thermodynamics, intricate fast and local dynamics may not always be the primary focus. Practical applications often prioritize observables capturing phenomena at dominant temporal and spatial scales. Developing models for these "large-scale" observables, resulting from averaging fast and local contributions, can be simplified into Low Order Models (LOMs) with reduced degrees of freedom described by ordinary differential equations. Unresolved degrees of freedom are introduced as stochastic components, exhibiting either Markovian or non-Markovian characteristics. The challenge lies in deriving dependable stochastic differential equations representing the statistics of real large-scale, slow features in the climate/ocean system. While paralleling material physics, it is crucial to recognize that direct transfer of tools and outcomes is hindered by the non-Hamiltonian nature of climate/geophysical LOMs and the impracticality of a Markovian treatment of noise due to wide-ranging time scales. A critical examination of the conventional statistical mechanics approach, customized for such LOMs, becomes essential. To this end, we propose utilizing an approach based on the operator cumulant method, which has been recently revisited and generalized, along with the linear response method in a non-Hamiltonian setting. Formal results are then derived, and applications to some typical classes of examples are presented to clarify this approach.

Dispersive decay bound of small data solutions to Kawahara equation in a finite time scale

In this article, we prove that small localized data yield solutions to Kawahara-type equations which have linear dispersive decay on a finite time scale depending on the size of the initial data. We use the similar method used by Ifrim and Tataru to derive the dispersive decay bound of the solutions to the KdV equation, with some steps being simpler. This result is expected to be the first result of the small data global bounds of the fifth-order dispersive equations with quadratic nonlinearity.

Perturbative reheating and thermalization of pure Yang-Mills plasma

We investigate the thermalization of high-energy particles injected from the perturbative decay of inflaton during the pre-thermal phase of reheating in detail. In general, thermalization takes a relatively long time in a low-temperature plasma; therefore, the instantaneous thermalization approximation is not justified, even for the reheating of the Standard Model (SM) sector. We consider a pure Yang-Mills (YM) theory as an approximation of the SM sector or a possible dark sector, considering the Landau-Pomeranchuk-Migdal effect, a quantum interference effect in a finite temperature plasma. We perform the first numerical calculation to solve the time evolution of the system, including the redshift due to the expansion of the Universe, and show the details of the temperature evolution near the maximum and the behavior of the quasi-attractors at later times. The maximal temperature Tmax and time scale tmax are determined quantitatively, such as Tmax ≃ 0.05 × ΓIMPI2/mI32/5mI\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ {\\left({\\Gamma}_I{M}_{\ extrm{PI}}^2/{m}_I^3\\right)}^{2/5}{m}_I $$\\end{document} and tmax ≃ 2 × 103 × ΓIMPI2/mI3−3/5mI−1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ {\\left({\\Gamma}_I{M}_{\ extrm{PI}}^2/{m}_I^3\\right)}^{-3/5}{m}_I^{-1} $$\\end{document} in the SM-like system, where mI and ΓI are the mass and decay rate of inflaton. We also provide a similar formula for pure SU(N) and SO(N) YM theories for general values of N and coupling constant α, including Tmax ∝ α4/5 and tmax ∝ N−2α−16/5 behaviors and their numerical coefficients. The thermalization occurs in a finite time scale, resulting in a lower maximal temperature of the Universe after inflation than that under the instantaneous thermalization approximation.

The separate-universe approach and sudden transitions during inflation

The separate-universe approach gives an intuitive way to understand the evolution of cosmological perturbations in the long-wavelength limit. It uses solutions of the spatially-homogeneous equations of motion to model the evolution of the inhomogeneous universe on large scales. We show that the separate-universe approach fails on a finite range of super-Hubble scales at a sudden transition from slow roll to ultra-slow roll during inflation in the very early universe. Such transitions are a feature of inflation models giving a large enhancement in the primordial power spectrum on small scales, necessary to produce primordial black holes after inflation. We show that the separate-universe approach still works in a piece-wise fashion, before and after the transition, but spatial gradients on finite scales require a discontinuity in the homogeneous solution at the transition. We discuss the implications for the δN formalism and stochastic inflation, which employ the separate-universe approximation.

Ideal gas thermodynamics with an invariant energy scale

A viable approach towards Quantum Gravity is the Doubly Special Relativity (DSR) framework in which an observer-independent finite energy upper bound (or a finite smallest length scale) appears quite naturally. In this work, we have studied the thermodynamic properties of an ideal gas in a specific DSR framework, known as the Magueijo-Smolin (MS) Model. We use the fact that DSR can be considered as nonlinear representation of Lorentz Group. Subsequently, various thermodynamic parameters of ideal gas have been derived in this modified framework to compare the corresponding deviations from the usual scenario due to the presence of the invariant energy (length) scale.

Integral field spectroscopy supports atmospheric optics to reveal the finite outer scale of the turbulence

The spatial coherence wavefront outer scale ( characterizes the size of the largest turbulence eddies in Earth's atmosphere, determining low spatial frequency perturbations in the wavefront of the light captured by ground-based telescopes. Advances in adaptive optics (AO) techniques designed to compensate for atmospheric turbulence emphasize the crucial role of this parameter for the next generation of large telescopes. The motivation of this work is to introduce a novel technique for estimating \ from seeing-limited integral field spectroscopic (IFS) data. This approach is based on the impact of a finite \ on the light collected by the pupil entrance of a ground-based telescope. We take advantage of the homogeneity of IFS observations to generate band filter images spanning a wide wavelength range, enabling the assessment of image quality (IQ) at the telescope's focal plane. Comparing the measured wavelength-dependent IQ variation with predictions derived from a first-order analytical approach based on turbulence statistics simplifications using the von Kármán model provides valuable insights into the prevailing \ parameter during the observations. We applied the proposed technique to observations from the Multi-Unit Spectroscopic Explorer (MUSE) in the wide-field mode obtained at the Paranal Observatory. Our analysis successfully validates the first-order analytical expression, which combines the seeing ($ $) and the \ parameters, to predict the IQ variations with the wavelength in ground-based astronomical data. However, we observed some discrepancies between the measured and predictions of the IQ that are analyzed in terms of uncertainties in the estimated $ $ and dome-induced turbulence contributions. This work constitutes the empirical validation of the analytical expression for estimating IQ at the focal plane of ground-based telescopes under specific $ $ and finite \ conditions. Additionally, we provide a simple methodology to characterize the \ and dome-seeing ($ dome $) as by-products of IFS observations routinely conducted at major ground-based astronomical observatories.

Exploring Lee-Yang and Fisher zeros in the 2D Ising model through multipoint Padé approximants

We present a numerical calculation of the Lee-Yang and Fisher zeros of the 2D Ising model using multipoint Padé approximants. We perform simulations for the 2D Ising model with ferromagnetic couplings both in the absence and in the presence of a magnetic field using a cluster spin-flip algorithm. We show that it is possible to extract genuine signature of Lee-Yang and Fisher zeros of the theory through the poles of magnetization and specific heat, using the multipoint Padé method. We extract the poles of magnetization using Padé approximants and compare their scaling with known results. We verify the circle theorem associated to the well known behavior of Lee-Yang zeros. We present our finite volume scaling analysis of the zeros done at T=Tc for a few lattice sizes, extracting to a good precision the (combination of) critical exponents βδ. The computation at the critical temperature is performed after the latter has been determined via the study of Fisher zeros, thus extracting both βc and the critical exponent ν. Results already exist for extracting the critical exponents for the Ising model in two and three dimensions making use of Fisher and Lee-Yang zeros. In this work, multipoint Padé is shown to be competitive with this respect and thus a powerful tool to study phase transitions. Published by the American Physical Society 2024

Nonlinear electron scattering by electrostatic waves in collisionless shocks

We present a theoretical analysis of electron pitch-angle scattering by ion-acoustic electrostatic fluctuations present in the Earth's bow shock and, presumably, collisionless shocks in general. We numerically simulate electron interaction with a single wave packet to demonstrate the scattering through phase bunching and phase trapping and quantify electron pitch-angle scattering in dependence on the wave amplitude and wave normal angle to the local magnetic field. The iterative mapping technique is used to model pitch-angle scattering of electrons by a large number of wave packets, which have been reported in the Earth's bow shock. Assuming that successive electron scatterings are not correlated, we revealed that the long-term dynamics of electrons is diffusive. The diffusion coefficient depends on the ratio $\varPhi _0/W$ between the wave packet amplitude and electron energy, $D\propto (\varPhi _0/W)^{\nu }$ . A quasi-linear scaling ( $\nu \approx 2$ ) is observed for sufficiently small wave amplitudes, $\varPhi _0\lesssim 10^{-3}W$ , while the diffusion is nonlinear ( $1<\nu <2$ ) above this threshold. We show that pitch-angle diffusion of ${\lesssim }1$ keV electrons in the Earth's bow shock can be nonlinear. The corresponding diffusion coefficient scales with the intensity $E_{w}^{2}$ of the electrostatic fluctuations in a nonlinear fashion, $D\propto E_{w}^{\nu }$ with $\nu <2$ , while its expected values in the Earth's bow shock are $D\sim 0.1\unicode{x2013}100$ $(T_{e}/W)^{\nu -1/2}\,{\rm rad}^{2}\,{\rm s}^{-1}$ . We speculate that in the Earth's quasi-perpendicular bow shock the stochastic shock drift acceleration mechanism with pitch-angle scattering provided by the electrostatic fluctuations can contribute to the acceleration of thermal electrons up to approximately 1 keV. The potential effects of a finite perpendicular coherence scale of the wave packets on the efficiency of electron scattering are discussed.

Emergent Conformal Boundaries from Finite-Entanglement Scaling in Matrix Product States.

The use of finite entanglement scaling with matrix product states (MPS) has become a crucial tool for studying one-dimensional critical lattice theories, especially those with emergent conformal symmetry. We argue that finite entanglement introduces a relevant deformation in the critical theory. As a result, the bipartite entanglement Hamiltonian defined from the MPS can be understood as a boundary conformal field theory with a physical and an entanglement boundary. We are able to exploit the symmetry properties of the MPS to engineer the physical conformal boundary condition. The entanglement boundary, on the other hand, is related to the concrete lattice model and remains invariant under this relevant perturbation. Using critical lattice models described by the Ising, Potts, and free compact boson conformal field theories, we illustrate the influence of the symmetry and the relevant deformation on the conformal boundaries in the entanglement spectrum.

Efficient computational homogenisation of 2D beams of heterogeneous elasticity using the patch scheme

Modern ‘smart’ materials have complex heterogeneous microscale structure, often with unknown macroscale closure but one we need to realise for large scale engineering and science. The multiscale Equation-Free Patch Scheme empowers us to non-intrusively, efficiently, and accurately predict large scale, system level, solutions through computations on only small sparse patches of the given detailed microscale system. Here the microscale system is that of a 2D beam of heterogeneous elasticity, with either fixed–fixed, fixed–free, or periodic boundary conditions. We demonstrate that the described multiscale Patch Scheme simply, efficiently, and stably predicts the beam’s macroscale, with a controllable accuracy, at finite scale separation. Dynamical systems theory supports the scheme. This article points the way for others to use this innovative systematic non-intrusive approach, via a developing toolbox of functions, to model and compute accurately macroscale system-levels of general complex physical and engineering systems.