Periodic Box Research Articles

Energy spectrum of two-particle scattering in a periodic box

We aim to compute the discrete energy spectrum for two-body scattering in a three-dimensional box under periodic boundary conditions. The spectrum in the center of mass is obtained by solving the Schödinger equation in a test potential using the Fourier basis. The focus is on how to project the spectrum into the various irreducible representations of the symmetry groups of the box. Four examples are given to show how the infinite-volume spectrum (including both bound and scattering states) is resolved in cubic or elongated boxes, and in systems with integer or half-integer total spin. Such a demonstration is a crucial step in relating the discrete spectrum in the box to the infinite-volume scattering phaseshifts via the Lüscher method.

Ld/Mm+ Simulation of Some Aristolochic and Humic Acids Species Coupled in Periodic Box with Water.

This study is one of the dynamics molecular docking that presents the interactions between a molecular model of the mixture of humic acid structure and 18 aristolochic acid structures, from the PubChem database in a water box that simulates the environment reactions. The major objective was to identify what happens in this procedure (LD/MM+) with the coupled species. LD/MM+ SIMULATION (Langevin dynamics simulation). The R-Squared statistic indicates that the model, as fitted by MLR, explains 90.9437% of the variability in volume. The interactions of these acids, the types of forces, and the way that these molecules can get closer to each other, in terms of total energy density, while identifying the specificities vis-àvis of water-aromaticity or water-reactivity behaviors were observed.

Point defect evolution under irradiation: Finite size effects and spatio-temporal correlations

The evolution of point defect concentrations under irradiation is controlled by their diffusion properties, and by their formation and elimination mechanisms. The latter includes the mutual recombination of vacancies and interstitials, and the elimination of point defects at sinks. Two models are traditionally used to predict the defect concentration evolution, the standard rate theory (SRT) and the atomistic kinetic Monte Carlo (AKMC). We show in this work a large discrepancy in the defect concentrations between both methods when the average number of defects in the AKMC simulation box is close or lower than one. The reason is that AKMC naturally captures strong space and time correlations between vacancies and interstitials generated by the finite size of the periodic simulation box. These correlations strongly affect the recombination rate and the point defect concentrations. SRT fails to predict such correlations, and the corresponding solution deviates from the more accurate solution given by the AKMC simulation under similar conditions. These finite size correlation effects are strong when the elimination of point defects occurs by recombination only, but can still be significant in the presence of sinks. In order to account for the spacio-temporal correlation in a continuum framework, we introduce a Correlated Pair Theory (CPT). This theory fully takes into account the correlations between vacancy and interstitial pairs and predicts point defect concentrations in good agreement with AKMC simulations. Inversely, if the goal is for the AKMC to reproduce defect concentrations in bulk using small simulation boxes, the naturally occurring correlations need to be corrected. We show here that the CPT can be used to modify the elimination rates in the AKMC simulations, so as to yield point defect concentrations in agreement with SRT.

Finite difference scheme for two-dimensional periodic nonlinear Schrödinger equations

A nonlinear Schr\"odinger equation (NLS) on a periodic box can be discretized as a discrete nonlinear Schr\"odinger equation (DNLS) on a periodic cubic lattice, which is a system of finitely many ordinary differential equations. We show that in two spatial dimensions, solutions to the DNLS converge strongly in $L^2$ to those of the NLS as the grid size $h>0$ approaches zero. As a result, the effectiveness of the finite difference method (FDM) is justified for the two-dimensional periodic NLS.

Scaling exponents saturate in three-dimensional isotropic turbulence

From a database of direct numerical simulations of homogeneous and isotropic turbulence, generated in periodic boxes of various sizes, we extract the spherically symmetric part of moments of velocity increments and first verify the following (somewhat contested) results: the $4/5$-ths law holds in an intermediate range of scales and that the second order exponent over the same range of scales is {\it{anomalous}}, departing from the self-similar value of $2/3$ and approaching a constant of $0.72$ at high Reynolds numbers. We compare with some typical theories the dependence of longitudinal exponents as well as their derivatives with respect to the moment order $n$, and estimate the most probable value of the H\"older exponent. We demonstrate that the transverse scaling exponents saturate for large $n$, and trace this trend to the presence of large localized jumps in the signal. The saturation value of about $2$ at the highest Reynolds number suggests, when interpreted in the spirit of fractals, the presence of vortex sheets rather than more complex singularities. In general, the scaling concept in hydrodynamic turbulence appears to be more complex than even the multifractal description.

A fully discrete boundary integral method based on embedding into a periodic box

This paper shows how numerical methods on a regular grid in a box can be used to generate numerical schemes for problems in general smooth domains contained in the box with no need for a domain specific discretization (other than a list of points on it). The focus is mainly on spectral discretizations due to their ability to accurately resolve the interaction of finite order distributions (generalized functions) and smooth functions. Mimicking the analytical structure of the relevant (pseudodifferential) operators leads to viable and accurate numerical representations and algorithms. An important byproduct of the structural insights gained in the process is the introduction of smooth kernels (at the discrete level) to replace classical singular kernels which are typically used in the (numerical) representations of the solution. The new kernel representations yield enhanced numerical resolution.

Phase diagram of the large N Gross-Neveu model in a finite periodic box

We analyze the Gross-Neveu model in the limit of large number of flavors of massless fermions. We study the phase diagram in a two and three dimensional periodic box at a fixed thermal to spatial aspect ratio, $\frac{\beta}{\ell}$, with a flavor independent chemical potential. We assume the bilinear condensate, when one exists, has a specific momentum in the spatial direction(s). The main known features of the phase diagram in the $\ell\to\infty$ limit of the two dimensional model are also seen on a finite $\ell\times\beta$ torus -- a phase with a homogeneous (zero momentum) condensate; a phase with an inhomogeneous (non-zero momentum) condensate and a phase with no condensate. We observe that the inhomogeneous phase contains several sub-phases characterized by a specific spatial momentum. Unlike the two dimensional model, we do not find evidence for a phase with a inhomogeneous condensate in the three dimensional model.

The Landau Equation with the Specular Reflection Boundary Condition

The existence and stability of the Landau equation (1936) in a general bounded domain with a physical boundary condition is a long-outstanding open problem. This work proves the global stability of the Landau equation with the Coulombic potential in a general smooth bounded domain with the specular reflection boundary condition for initial perturbations of the Maxwellian equilibrium states. The highlight of this work also comes from the low-regularity assumptions made for the initial distribution. This work generalizes the recent global stability result for the Landau equation in a periodic box (Kim et al. in Peking Math J, 2020). Our methods consist of the generalization of the wellposedness theory for the Fokker–Planck equation (Hwang et al. SIAM J Math Anal 50(2):2194–2232, 2018; Hwang et al. Arch Ration Mech Anal 214(1):183–233, 2014) and the extension of the boundary value problem to a whole space problem, as well as the use of a recent extension of De Giorgi–Nash–Moser theory for the kinetic Fokker–Planck equations (Golse et al. Ann Sc Norm Super Pisa Cl Sci 19(1):253–295, 2019) and the Morrey estimates (Bramanti et al. J Math Anal Appl 200(2):332–354, 1996) to further control the velocity derivatives, which ensures the uniqueness. Our methods provide a new understanding of the grazing collisions in the Landau theory for an initial-boundary value problem.

On the Correlation Energy of Interacting Fermionic Systems in the Mean-Field Regime

We consider a system of $$N\gg 1$$ interacting fermionic particles in three dimensions, confined in a periodic box of volume 1, in the mean-field scaling. We assume that the interaction potential is bounded and small enough. We prove upper and lower bounds for the correlation energy, which are optimal in their N-dependence. Moreover, we compute the correlation energy at leading order in the interaction potential, recovering the prediction of second order perturbation theory. The proof is based on the combination of methods recently introduced for the study of fermionic many-body quantum dynamics together with a rigorous version of second-order perturbation theory, developed in the context of non-relativistic QED.

Efficient Estimation of Binding Kinetics using Scaled Non-Bonded Interactions and Harmonic Restraints

Understanding the thermodynamics and kinetics of protein-ligand interactions is crucial for computational drug discovery. In this project, we simulate a small host-guest system where the ligand is a single, uncharged Lennard-Jones (LJ) particle, and the binding site is an 11-particle icosahedral pocket made from the same atom type; the system is simulated in a (3.00859 nm)3 periodic box filled with 860 TIP3P waters. We first tuned the LJ parameter ε to construct a system that realistically mimics a protein-ligand binding system.

An $$L^2$$ to $$L^\infty $$ Framework for the Landau Equation

Consider the Landau equation with Coulomb potential in a periodic box. We develop a new $$L^{2}\ \text{to}\ L^{\infty }$$ framework to construct global unique solutions near Maxwellian with small $$L^{\infty }$$ norm. The first step is to establish global $$L^{2}$$ estimates with strong velocity weight and time decay, under the assumption of $$L^{\infty }$$ bound, which is further controlled by such $$L^{2}$$ estimates via De Giorgi’s method (Golse et al. in Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) 19(1), 253–295 (2019), Imbert and Mouhot in arXiv:1505.04608 (2015)). The second step is to employ estimates in $$S_{p}$$ spaces to control velocity derivatives to ensure uniqueness, which is based on Holder estimates via De Giorgi’s method (Golse et al. in Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) 19(1), 253–295 (2019), Golse and Vasseur in arXiv:1506.01908 (2015), Imbert and Mouhot in arXiv:1505.04608 (2015)).

The LOCV asymmetric nucleonic matter in the periodic boxes

The study of the finite asymmetric nucleonic matter (ASM) in the periodic boxes (PBs) could play a significant role in the investigation of the finite nuclei as well as the crust of neutron stars. So, in the present contribution, the lowest order constrained variational (LOCV) method is applied to finite ASM in PBs (PBLOCV). In the PBLOCV ASM computations, different sets of finite nucleonic systems are considered, at various proton to neutron ratios ($$\beta $$), using the Bethe and the central part of the AV$$'$$8 potentials. The asymmetric effects are imposed in the statistical correlation functions, in which their PB versions are anisotropic. It is shown that the mentioned anisotropic behavior is the origin of the anisotropy of the PBLOCV ASM nucleon–nucleon distribution functions. As expected, it is found that, for a large number of nucleons and mesh segments, the PBLOCV ASM energies tend to those of LOCV. In other words, for a large PB, the PBLOCV smallness parameters reasonably agree with those of LOCV. The finite size effects (FSEs), which can be employed in quantum Monet Carlo calculations, are illustrated for PBLOCV ASM energies, especially for small periodic boxes, at high $$\beta $$ values. It is demonstrated that considering 420 (504) nucleons at $$\beta =0.034(0.474)$$, the corresponding PBLOCV energy is notably consistent with those of 1030 nucleons and LOCV. Meanwhile, the PBLOCV ASM data for 14 (16) nucleons at $$\beta =0.000 (0.143)$$ lie far from those of LOCV, for which the FSEs are remarkably evident.

Large Spatio-temporal Fluctuation and Energy Cascade Dynamics in von Kármán Turbulence

Kolmogorov’s local equilibrium hypothesis claims that the statistics of small-scale motion of turbulence are independent of its large-scale spatio-temporal fluctuation. This is the basis of the “K41” picture of the universality in small-scale turbulence. However, the validity of this hypothesis and its relationship with the energy cascade dynamics are not fully understood. In the present study, von Kármán flow is examined by experiments and numerical simulations. By PIV experiments, we obtain results consistent with DNS in a periodic box, i.e. spatially intermittent small-scale clusters of vortices and a very long temporal fluctuation.

Numerical forcing scheme to generate passive scalar mixing on the centerline of turbulent round jets in a triply periodic box

Numerical studies of passive scalars in three-dimensional (3D) periodic box turbulence have often used arbitrary scalar forcing schemes to sustain the variance. These existing methods represent certain flow configurations, but they have not been derived using specific velocity and scalar profiles. In this work, a forcing technique is devised to generate centerline scalar mixing of round jets in a triply periodic box. It is derived from the scalar transport equation using a Reynolds-like decomposition of the scalar field. The equation is closed by applying the known mean velocity and scalar profiles of axisymmetric jets. The result is a combination of a mean gradient term and a linear scalar term. Direct numerical simulations at different Reλ have been performed with these source terms for unity Schmidt numbers. Scalar flux values and scaling exponents of energy spectra from simulations are comparable to experimental values. In addition, a dimensional analysis shows that the normalized scalar statistics, such as variance, flux, and dissipation rate, should only be a function of Reynolds number; indeed, such quantities computed from our simulations approach constant values as the Reynolds number increases. The effects of velocity forcing on scalar fields are also investigated; changing velocity forcing terms may result in unstable scalar fields even under the same scalar forcing. It may indicate that an appropriate relation between the velocity and scalar forcing schemes can help producing a proper scalar mixing environment.

Atomistic treatment of periodic gold nanowire array nanofasteners under shear loading

Comprehensive molecular dynamics simulations are conducted to unravel the mechanics and mechanisms associated with the strength and fracture behavior of a highly ordered gold nanowire (Au-NW) array of a pair of nanofasteners (nanoconnectors) under externally applied shear strain. Large-scale atomic/molecular massively parallel simulator (LAMMPS and embedded atom method were adopted to model the atomic interactions of a number of neighboring nanofasteners. This was affected via the use of a periodic simulation box around a pair of highly ordered nanotube arrays to minimize the cost of the computations. Energy minimization using a conjugate gradient algorithm was first performed and followed by atomic relaxation to achieve an equilibrated configuration under the canonical ensemble of constant temperature and volume. The relaxed equilibrated configuration of the nanofastener was then subjected to an externally applied shear strain at a rate of per nanosecond under the canonical ensemble. Our results reveal the importance of the morphology and the overlap depth of the mating nanowire arrays upon the mechanical and fracture behavior of the nanofastener under shear loading. Our work also disclosed the phenomenon of multiple contacts of some displaced nanowires with their neighbors even after their fracture leading to multiple cold-welds with added redundancy to the nanofastener. Finally, in this research, we identified the locations of dislocation emissions and the resulting fracture processes that govern the mechanical integrity and ultimately the functionality of the Au-NW connector. The proposed highly ordered alignment, as conceived numerically herein, can yield a peak stress two to three times higher than that corresponding to a random alignment reported in a previous study. The Au-NW connector also exhibited resistance to fracture, even in cases where small overlap depth is considered in joint bonding. The nanoconnector was also tested at high temperatures (up to 450 K). Our results show that the rising temperature only leads to a minor reduction in the load transmitted by the nanoconnector.

Manifestly finite derivation of the quantum kink mass

In 1974 Dashen, Hasslacher and Neveu calculated the leading quantum correction to the mass of the kink in the scalar ϕ4 theory in 1+1 dimensions. The derivation relies on the identification of the perturbations about the kink as solutions of the Pöschl-Teller (PT) theory. They regularize the theory by placing it in a periodic box, although the kink is not itself periodic. They also require an ad hoc identification of plane wave and PT states which is difficult to interpret in the decompactified limit. We rederive the mass using the kink operator to recast this problem in terms of the PT Hamiltonian which we explicitly diagonalize using its exact eigenstates. We normal order from the beginning, rendering our theory finite so that no compactification is necessary. In our final expression for the kink mass, the form of the PT potential disappears, suggesting that our mass formula applies to other quantum solitons.

A Theoretical Study of the Molecular Coupled Structures of Aristolochic Acids and Humic Acid, Potential Environmental Contaminants.

An understanding of the fate of organic compounds originating from plants in soil is crucial for determining their persistence and concentrations in the environment. Aristolochic acids are believed to be the causal agents that induce Balkan endemic nephropathy by food contamination through soil adsorption of humic acids, major components of soil. Aristolochic acids are active chemicals in Aristolochia plant species found in endemic villages. In this article, molecular structure interactions between 18 structures of aristolochic acids with an inserted humic acid structure were studied. These structures were optimized in vacuo and by periodic box simulation with water solvate using the computational molecular mechanics MM+ method with HyperChem software. The QSPR models were used for correlation of the relationship between the hydrophobicity values of 18 AA structures coupled with a HA structure by MM+ and QSAR+ properties. Computational hydrophobicity values were considered dependent variables and were related to the structural features obtained by molecular and quantum mechanics calculations by multiple linear regression approaches. The obtained model was validated, and the results indicated differing hydrophobicity between the MM+ and QSAR+ properties.

Tests of acoustic scale shifts in halo-based mock galaxy catalogues

ABSTRACT We utilize mock catalogues from high-accuracy cosmological N-body simulations to quantify shifts in the recovery of the acoustic scale that could potentially result from galaxy clustering bias. The relationship between galaxies and dark matter haloes presents a complicated source of systematic errors in modern redshift surveys, particularly when aiming to make cosmological measurements to sub-per cent precision. Apart from a scalar, linear bias parameter accounting for the density contrast ratio between matter tracers and the true matter distribution, other types of galaxy bias, such as assembly and velocity biases, may also significantly alter clustering signals from small to large scales. We create mocks based on generalized halo occupation populations of 36 periodic boxes from the abacus cosmosrelease, and test various biased models along with an unbiased base case in a total volume of $48 \, h^{-3} \, {\rm Gpc}^{3}$. Two reconstruction methods are applied to galaxy samples and the apparent acoustic scale is derived by fitting the two-point correlation function multipoles. With respect to the baseline, we find a 0.3 per cent shift in the line-of-sight acoustic scale for one variation in the satellite galaxy population, and we find a 0.7 per cent shift for an extreme level of velocity bias of the central galaxies. All other bias models are consistent with zero shift at the 0.2 per cent level after reconstruction. We note that the bias models explored are relatively large variations, producing sizeable and likely distinguishable changes in small-scale clustering, the modelling of which would further calibrate the baryon acoustic oscillations standard ruler.

Incompressible Euler limit of the Boltzmann equation in the whole space and a periodic box

The purpose of this article is to establish the incompressible Euler limit for the Boltzmann equation in the whole space and a periodic box in the case where local smooth solutions of the incompressible Euler system are given.

Non-linear evolution of instabilities between dust and sound waves

ABSTRACT We study the non-linear evolution of the acoustic ‘resonant drag instability’ (RDI) using numerical simulations. The acoustic RDI is excited in a dust–gas mixture when dust grains stream through gas, interacting with sound waves to cause a linear instability. We study this process in a periodic box by accelerating neutral dust with an external driving force. The instability grows as predicted by linear theory, eventually breaking into turbulence and saturating. As in linear theory, the non-linear behaviour is characterized by three regimes – high, intermediate, and low wavenumbers – the boundary between which is determined by the dust–gas coupling strength and the dust-to-gas mass ratio. The high and intermediate wavenumber regimes behave similarly to one another, with large dust-to-gas ratio fluctuations while the gas remains largely incompressible. The saturated state is highly anisotropic: dust is concentrated in filaments, jets, or plumes along the direction of acceleration, with turbulent vortex-like structures rapidly forming and dissipating in the perpendicular directions. The low-wavenumber regime exhibits large fluctuations in gas and dust density, but the dust and gas remain more strongly coupled in coherent ‘fronts’ perpendicular to the acceleration. These behaviours are qualitatively different from those of dust ‘passively’ driven by external hydrodynamic turbulence, with no back-reaction force from dust on to gas. The virulent nature of these instabilities has interesting implications for dust-driven winds in a variety of astrophysical systems, including around cool stars, in dusty torii around active-galactic-nuclei, and in and around giant molecular clouds.