Finite Size Research Articles

Estimation of the Largest Lyapunov Exponent for a Model of Cross-Shaped Waves in a Rectangular Channel of Finite Size

Estimation of the Largest Lyapunov Exponent for a Model of Cross-Shaped Waves in a Rectangular Channel of Finite Size

Condensate and superfluid fraction of homogeneous Bose gases in a self-consistent Popov approximation

We study the condensate and superfluid fraction of a homogeneous gas of weakly interacting bosons in three spatial dimensions by adopting a self-consistent Popov approximation, comparing this approach with other theoretical schemes. Differently from the superfluid fraction, we find that at finite temperature the condensate fraction is a non-monotonic function of the interaction strength, presenting a global maximum at a characteristic value of the gas parameter, which grows as the temperature increases. This non-monotonic behavior has not yet been observed, but could be tested with the available experimental setups of ultracold bosonic atoms confined in a box potential. We clearly identify the region of parameter space that is of experimental interest to look for this behavior and provide explicit expressions for the relevant observables. Finite size effects are also discussed within a semiclassical approximation.

Modeling drag coefficients of spheroidal particles in rarefied flow conditions

Transport of particles in flows is often modeled in a combined Eulerian–Lagrangian framework. The flow is evaluated on an Eulerian grid, while particles are modeled as Lagrangian points whose positions and velocities are evolved in time, resulting in particle trajectories embedded in the time-dependent flow field. The method essentially resolves the flow field in complex geometries in detail but uses a closure model for the particle dynamics aimed at including the essential particle–fluid interactions at the cost of detailed small-scale physics. Rarefaction effects are usually included through the phenomenological Cunningham correction on the drag force experienced by the particles. In this Lagrangian point-particle approach, any explicit reference to the finite size and the shape of the particles, and their local orientation in the flow field, is typically ignored. In this work we aim to address this gap by deriving, from fully-resolved Direct Simulation Monte Carlo (DSMC) studies, heuristic or closure models for the drag force acting on prolate and oblate spheroidal particles with different aspect ratios, and a fixed orientation, in uniform ambient rarefied flows covering the transition regime between the continuum and free-molecular limits. These closure models predict the drag in the transition regime for all considered parameter settings (validated with DSMC data). The continuum limit is enforced a priori and we retrieve the free-molecular limit with reasonable accuracy (based on comparisons with literature data). We also include in the models the capability to predict effects related to basic gas-surface interactions via the tangential momentum accommodation coefficient. We furthermore assess the validity of the proposed closure model for particle dynamics in proximity to solid walls. This investigation extends our previous work, which focused on small aspect ratio spheroids with exclusively diffusive gas-surface interactions [see Livi et al. (2022)]. The derived models are obtained for isothermal, subsonic flows relevant for particle contamination control in semiconductor manufacturing.

Theoretical Investigation of Transport Properties of X-doped (X = Cl, P, N) Graphene Quantum Dots Layer Adsorbed with Ammonia Molecule Using Non-Equilibrium Green’s Function

This study investigates the various transport properties, including Density of States (DOS), Transmission Coefficient, Current-Voltage Characteristics (I-V), and HOMO-LUMO energy levels, of X-doped (X = Cl, P, N) graphene quantum dot (GQD) layers with adsorbed ammonia molecules using density functional theory and non-equilibrium Green's function techniques. Our findings revealed the profound influence of different doping and adsorption scenarios on the transport properties of GQD layers. Specifically, chlorine doping leads to notably stronger resonance peaks, resulting in enhanced conductivity of the GQD layer. This effect is less pronounced for P-doped and N-doped GQDs, which exhibit minimal changes in conductance capability. Importantly, we observed a prominent interaction between ammonia molecules and chlorine-doped GQD layers. The HOMO and LUMO energy levels exhibit double degeneracy, significantly impacting edge passivation and the finite dimensions of GQDs, consequently reducing the energy gap. As a result, chlorine-doped GQD layers with adsorbed ammonia molecules exhibit increased energy levels and stronger current characteristics compared to P-GQDs and N-GQDs. HOMO and LUMO are doubly degenerated which has a clear effect on edge passivation as well as the finite size of GQDs and reduces energy gap. Therefore, more energy level and stronger current is available for chlorine doped graphene quantum dots layer with adsorbed ammonia molecule than P-GQDs and N-GQDs. Our theoretical finding highlighted the significant features of edge passivation, doping effect and adsorption of ammonia molecule onto graphene quantum dots layer. Our theoretical analysis highlighted key features encompassing edge passivation, doping effects, and ammonia molecule adsorption on GQD layers. These insights not only develop our fundamental understanding of these systems but also hold promise for novel applications that harness their distinct electronic properties.

On the equivalence between n-state spin and vertex models on the square lattice

In this paper we investigate a correspondence among spin and vertex models with the same number of local states on the square lattice with toroidal boundary conditions. We argue that the partition functions of an arbitrary n-state spin model and of a certain specific n-state vertex model coincide for finite lattice sizes. The equivalent vertex model has n3 non-null Boltzmann weights and their relationship with the edge weights of the spin model is explicitly presented. In particular, the Ising model in a magnetic field is mapped to an eight-vertex model whose weights configurations combine both even and odd number of incoming and outcoming arrows at a vertex. We have studied the Yang-Baxter algebra for such mixed eight-vertex model when the weights are invariant under arrows reversing. We find that while the Lax operator lie on the same elliptic curve of the even eight-vertex model the respective R-matrix can not be presented in terms of the difference of two rapidities. We also argue that the spin-vertex equivalence may be used to imbed an integrable spin model in the realm of the quantum inverse scattering framework. As an example, we show how to determine the R-matrix of the 27-vertex model equivalent to a three-state spin model devised by Fateev and Zamolodchikov.

Designing edge states from fractional polarization insulators

We theoretically investigated disconnected dispersive edge states in an anisotropic honeycomb lattice without chiral symmetry. When both mirror and chiral symmetries are present, this system is defined by a topological quantity known as fractional polarization (FP) term and exhibits a bulk band gap, classifying it as an FP insulator. While the FP insulator accommodates robust, flat topological edge states (TES), it also offers the potential to engineer these edge states by deliberately disrupting a critical symmetry that safeguards the underlying topology. These symmetry-breaking terms allow the edge states to become dispersive and generate differing configurations along the open boundaries. Furthermore, disconnected helical-like and chiral-like edge states analogous to TES seen in quantum spin and anomalous hall effect are achieved by the finite size effect, not possible from the symmetry-breaking terms alone. The demonstration of manipulating these edge states from a FP insulator can open up new avenues in constructing devices that utilize topological domain walls.

A rank-based adaptive independence test for high-dimensional data

There are lots of methods for complete independence test for high-dimensional random vector. However, it is difficult for practitioners to choose a powerful test because the true alternative hypothesis is unknown. Combining the L 2-type with the L ∞ -type test statistic, we propose a rank-based test method. From a technical point of view, the proposed test is distribution-free and consequently the corresponding critical values or p-values can be obtained by Monte Carlo methods. Compared with permutation or bootstrap test methods, the proposed test method saves calculation cost. Simulation results show that the resulting method has excellent performance with finite sample size. We also provide a real data application to demonstrate the practicality and effectiveness of the proposed test method.

Characteristic times for radiation belt drift phase mixing

Impulsive radial transport events occurring in the radiation belts leave lasting marks in the form of drift echoes, that is, energy-dependent drift phase structures in the radiation belts that evolve at the drift frequency. Drift echoes are known to be transient structures that dissipate due to phase mixing. The objective of this paper is to discuss how much time it takes for drift echoes to dissipate, and what drives this phase-mixing process. While any uncertainty or perturbation in the variables controlling trapped particles’ drift frequency contributes to phase mixing, we highlight two main drivers: the observational uncertainty associated with the finite size of the instrument energy channels, and the natural field fluctuations driving perturbations in trapped particles’ drift frequency. It is the combination of both instrumental and natural sources of phase mixing that determines the observed dissipation and lifetime of drift echoes. This means that the observed magnitude and lifetime of a drift echo are always underestimations of the natural magnitude and lifetime of the structure. This calls into question the applicability of the standard, drift-averaged formulation of radial diffusion. The three key points of the study are the following: First, the time it takes for particles initially localized in local time to phase-mix is measured in hours in the Earth’s radiation belts. Second, phase mixing at the drift scale is primarily due to uncertainties in measured kinetic energy and field perturbations. Third, our analysis can be utilized to set an energy resolution requirement for future particle instruments.

The study of the influence of the geometry of a lateral electric field resonator on its resonant characteristics

An experimental study of the dependence of the electrical impedance of a lateral electric field resonator on its thickness and the size of the gap between the electrodes was carried out. The resonator was made of PZT-19 piezoceramics in the form of a rectangular parallelepiped with the shear dimensions of 18 × 20 mm2. Two rectangular electrodes with a gap that varied in the range from 4 to 14 mm were applied on one side of the resonator. For each gap width, the frequency dependences of the real and imaginary parts of the electrical impedance were measured using an impedance analyzer. It has been found that increasing the gap width leads to an increase in the resonant frequency and to an increase in the maximum value of the real part of the impedance. Three series of such experiments were carried out for three values of the resonator thickness: 3.02, 2.38 and 1.9 mm. The resonant characteristics of the resonator were also theoretically analyzed by finite element analysis using two models. One resonator model was based on a two-dimensional finite element method. In this case, the vibration modes that existed due to the finite size of the plate in the direction parallel to the gap between the electrodes were not taken into account. The second model of the resonator used a three-dimensional finite element method, which correctly took into account all vibration modes existing in the resonator. Comparison of theory with experiment has shown that the three-dimensional model provides a better agreement between theoretical and experimental results.

Symmetry-preserving quadratic Lindbladian and dissipation driven topological transitions in Gaussian states

The dynamical evolution of an open quantum system can be governed by the Lindblad equation of the density matrix. In this paper, we propose to characterize the density matrix topology by the topological invariant of its modular Hamiltonian. Since the topological classification of such Hamiltonians depends on their symmetry classes, a primary issue we address is determining the requirement for the Lindbladian operators, under which the modular Hamiltonian can preserve its symmetry class during the dynamical evolution. We solve this problem for the fermionic Gaussian state and for the modular Hamiltonian being a quadratic operator of a set of fermionic operators. When these conditions are satisfied, along with a nontrivial topological classification of the symmetry class of the modular Hamiltonian, a topological transition can occur as time evolves. We present two examples of dissipation-driven topological transitions where the modular Hamiltonian lies in the AIII class with U(1) symmetry and the DIII class without U(1) symmetry. By a finite size scaling, we show that this density matrix topology transition occurs at a finite time. We also present the physical signature of this transition.

Analysis of the ion collection model in the ExB probe

An ExB probe is a pass-band velocity filter for an ion beam. The ExB probe measurement is related to the ion velocity distribution function (IVDF); however, the finite pass-band filter window size leads to differences in the true IVDF and measured ExB probe spectrum. We derive for the first time an analytical ExB probe transmittancy matrix and use it to examine differences in the IVDF and the probe spectrum. The probe spectra are compared to a synthetically defined test IVDF to study how probe geometry and ion species affect the differences. It is found that the difference in the probe spectrum and the true IVDF is dependent on the ion species, and the deviation is larger for heavier ions. The peak velocity of the spectrum was shifted by up to 13%, and the velocity spread was broadened by up to 276%. The relative ion species fraction is calculated from the probe spectra by two different methods and compared to the true fraction. Using the approach of this work, direct integration of the spectrum resulted in a 2% difference, while a more common approach from literature overestimated one ion species by 184%.

Contrasting Views of the Electric Double Layer in Electrochemical CO2 Reduction: Continuum Models vs Molecular Dynamics.

In the field of electrochemical CO2 reduction, both continuum models and molecular dynamics (MD) models have been used to understand the electric double layer (EDL). MD often focuses on the region within a few nm of the electrode, while continuum models can span up to the device level (cm). Still, both methods model the EDL, and for a cohesive picture of the CO2 electrolysis system, the two methods should agree in the regions where they overlap length scales. To this end, we make a direct comparison between state-of-the-art continuum models and classical MD simulations under the conditions of CO2 reduction on a Ag electrode. For continuum modeling, this includes the Poisson-Nernst-Planck formulation with steric (finite ion size) effects, and in MD the electrode is modeled with the constant potential method. The comparison yields numerous differences between the two modeling methods. MD shows cations forming two adsorbed layers, including a fully hydrated outer layer and a partial hydration layer closer to the electrode surface. The strength of the inner adsorbed layer increases with cation size (Li+ < Na+ < K+ < Cs+) and with more negative applied potentials. Continuum models that include steric effects predict CO2 to be mostly excluded within 1 nm of the cathode due to tightly packed cations, yet we find little evidence to support these predictions from the MD results. In fact, MD shows that the concentration of CO2 increases within a few Å of the cathode surface due to interactions with the Ag electrode, a factor not included in continuum models. The EDL capacitance is computed from the MD results, showing values in the range of 7-9 μF cm-2, irrespective of the electrolyte concentration, cation identity, or applied potential. The direct comparison between the two modeling methods is meant to show the areas of agreement and disagreement between the two views of the EDL, so as to improve and better align these models.

Classical and quantum computing of shear viscosity for (2+1)D SU(2) gauge theory

We perform a nonperturbative calculation of the shear viscosity for (2+1)-dimensional SU(2) gauge theory by using the lattice Hamiltonian formulation. The retarded Green’s function of the stress-energy tensor is calculated from real time evolution via exact diagonalization of the lattice Hamiltonian with a local Hilbert space truncation, and the shear viscosity is obtained via the Kubo formula. When taking the continuum limit, we account for the renormalization group flow of the coupling but no additional operator renormalization. We find the ratio of the shear viscosity and the entropy density ηs is consistent with a well-known holographic result 14π at several temperatures on a 4×4 honeycomb lattice with the local electric representation truncated at jmax=12. We also find the ratio of the spectral function and frequency ρxy(ω)ω exhibits a peak structure when the frequency is small. Both the exact diagonalization method and simple matrix product state classical simulation method beyond jmax=12 on bigger lattices require exponentially growing resources. So we develop a quantum computing method to calculate the retarded Green’s function and analyze various systematics of the calculation including jmax truncation and finite size effects, Trotter errors and the thermal state preparation efficiency. Our thermal state preparation method still requires resources that grow exponentially with the lattice size, but with a very small prefactor at high temperature. We test our quantum circuit on both the Quantinuum emulator and the IBM simulator for a small lattice and obtain results consistent with the classical computing ones. Published by the American Physical Society 2024

Distribution-Free Prediction Intervals Under Covariate Shift, With an Application to Causal Inference

Owing to its appealing distribution-free feature, conformal inference has become a popular tool for constructing prediction intervals with a desired coverage rate. In scenarios involving covariate shift, where the shift function needs to be estimated from data, many existing methods resort to data-splitting techniques. However, these approaches often lead to wider intervals and less reliable coverage rates, especially when dealing with finite sample sizes. To address these challenges, we propose methods based on a pivotal quantity derived under a parametric working model and employ a resampling-based framework to approximate its distribution. The resampling-based approach can produce prediction intervals with a desired coverage rate without splitting the data and can be easily applied to causal inference settings where a shift in the covariate distribution can occur between treatment and control arms. Additionally, the proposed approaches enjoy a double robustness property and are adaptable to different prediction tasks. Our extensive numerical experiments demonstrate that, compared to existing methods, the proposed novel approaches can produce substantially shorter conformal prediction intervals with lower variability in the interval lengths while maintaining promising coverage rates and advantages in versatile usage. Supplementary materials for this article are available online, including a standardized description of the materials available for reproducing the work.

A study of layered holographic superconductor

We have investigated a holographic model of a multi-layered superconductor in (2+1)-dimensions using the AdS/CFT correspondence. This correspondence allows us to study strongly interacting condensed matter systems through a weakly interacting gravitational theory. Our study focused on the effects of a finite system size on the superconductor’s properties. We observed significant variations in the critical temperature and critical magnetic field depending on the number of layers and the spacing between them. Notably, our results capture the transition from a two-dimensional (2D) to a three-dimensional (3D) behavior in the limit of a large number of closely spaced layers. This transition signifies a change in how the superconductivity operates within real material.

A radiometric model for demonstration of exoplanets detection by the transit method

This work describes an exact radiometric model for experimental demonstrators of the detection of exoplanets by the transit method. This model generalises the calculation of the depth of occultation from the standard transit method to the case of a finite size demonstrator apparatus. Results show that, for demonstrator apparatuses of moderately small sizes, a significant accuracy improvement in the determination of the size of a planet model can be achieved using the proposed method in comparison to using the formula from the standard transit method. The radiance distribution of the star model is found to be of crucial importance, as deviations from a Lambertian radiance distribution can lead to significantly different results.

Resonator embedded photonic crystal surface emitting lasers

The finite size of 2D photonic crystals results in them being a lossy resonator, with the normally emitting modes of conventional photonic crystal surface emitting lasers (PCSELs) differing in photon lifetime via their different radiative rates, and the different in-plane losses of higher order spatial modes. As a consequence, the fundamental spatial mode (lowest in-plane loss) with lowest out-of-plane scattering is the primary lasing mode. For electrically driven PCSELs, as current is increased, incomplete gain clamping results in additional spatial (and spectral) modes leading to a reduction in beam quality. A number of approaches have been discussed to enhance the area (power) scalability of epitaxy regrown PCSELs through careful design of the photonic crystal atom1–3. None of these approaches tackle the inflexibility in being unable to independently modify the photon lifetime of the different modes at the Γ2 point. As a method to introduce design flexibility, resonator embedded photonic crystal surface emitting lasers (REPCSELs) are introduced. This device, combining comparatively low coupling strength photonic crystal structures along with perimeter mirrors, allow a Fabry–Pérot resonance effect to be realised that provides wavelength selective modification of the photon lifetime. We show that surface emission of different surface emitting modes may be selectively enhanced, effectively changing the character of the modes at the Γ2 point. This is a consequence of the selective modification of in-plane loss for particular modes, and is dependent upon the alignment of the photonic crystal (PhC) band-structure and distributed Bragg reflectors’ (DBRs) reflectance spectrum. These findings offer new avenues in surface emitting laser diode engineering. The use of DBRs to reduce the lateral size of a PCSEL opens the route to small, low threshold current (Ith), high output efficiency epitaxy regrown PCSELs for high-speed communication and power sensitive sensing applications.

Free-field method for inverse characterization of finite porous acoustic materials using feed forward neural networks.

This paper presents a free-field method for inverse estimation of acoustic porous material parameters from sound pressure measurements above small rectangular samples. The finite sample effect, the spherical propagation of the sound field, and a potential lateral material reaction are considered. Using an extensive series of systematically varied finite element simulations, neural network models are developed to replace computationally expensive simulations as a forward model for the calculation of the complex sound pressure above small samples in the inverse optimization. The method is experimentally validated using various porous material samples. The results show that the influence of the finite sample size is successfully removed and thus, the acoustic properties of the materials can be estimated from the determined porous parameters with high accuracy, even based on a single sound pressure measurement over small samples with pronounced edge diffraction. The poroacoustic parameters hence derived can be used directly, e.g., in simulation applications, or to calculate complex surface impedances or absorption coefficients.

Regularization errors introduced by the one-fluid formulation in the solution of two-phase elliptic problems

This manuscript discusses the structure of the errors in the solution of elliptic problems introduced by the regularization of the fluid properties discontinuity in a small region of finite size. By using a multiscale approach, the problem of the error calculation is splitted into an outer problem, where the regularization region is replaced by a sharp interface, and an inner local one dimensional problem that eventually imposes effective jump conditions across the interface for the outer problem. Except in some particular cases, the use of regularization techniques introduces first order errors in the solution imposing an error flux jump that is proportional to the tangential Laplacian of the averaged solution and an error jump that is proportional to the normal flux, both multiplied by a prefactor that depends on the averaging rule used. In general, the optimal averaging procedure is shown to depend on the structure of the problem at hand. The errors introduced by standard arithmetic and harmonic averages are obtained for various analytical and numerical examples which are used to discuss the nature of the errors introduced and the importance of first order errors in the solution. The influence of the ratio between the regularization thickness and the grid size is also investigated in numerical implementations of the one-fluid model.

A finite-time quantum Otto engine with tunnel coupled one-dimensional Bose gases

We undertake a theoretical study of a finite-time quantum Otto engine cycle driven by inter-particle interactions in a weakly interacting one-dimensional (1D) Bose gas in the quasicondensate regime. Utilizing a c-field approach, we simulate the entire Otto cycle, i.e. the two work strokes and the two equilibration strokes. More specifically, the interaction-induced work strokes are modelled by treating the working fluid as an isolated quantum many-body system undergoing unitary evolution. The equilibration strokes, on the other hand, are modelled by treating the working fluid as an open quantum system tunnel-coupled to another quasicondensate which acts as either the hot or cold reservoir, albeit of finite size. We find that, unlike a uniform 1D Bose gas, a harmonically trapped quasicondensate cannot operate purely as a heat engine; instead, the engine operation is enabled by additional chemical work performed on the working fluid, facilitated by the inflow of particles from the hot reservoir. The microscopic treatment of dynamics during equilibration strokes enables us to evaluate the characteristic operational time scales of this Otto thermochemical engine, crucial for characterizing its power output, without any ad hoc assumptions about typical thermalization timescales. We analyse the performance and quantify the figures of merit of the proposed Otto thermochemical engine, finding that it offers a favourable trade-off between efficiency and power output, particularly when the interaction-induced work strokes are implemented via a sudden quench. We further demonstrate that in the sudden quench regime, the engine operates with an efficiency close to the near-adiabatic (near maximum efficiency) limit, while concurrently achieving maximum power output.