Pairwise Summation Research Articles

Pull-in features of nanoswitches in the Casimir regime with account of contact repulsion

The cantilever tip of a nanoswitch in close proximity to the ground plate is considered with account of electrostatic, elastic, van der Waals (Casimir), and also contact repulsive forces. The van der Waals (Casimir) and contact repulsive forces are computed for a Si cantilever and either Au or Ni ground plates using the Lifshitz theory and the method of pairwise summation with account of surface roughness. It is shown that at short separations an impact of the van der Waals (Casimir) force leads to the pull-in and collapse of a cantilever onto the ground plate if the contact repulsion is disregarded. Taking into consideration contact repulsion, the nanoswitch is demonstrated to have the stable cyclic behavior with no pull-in when switching voltage on and off.

FRECKLL: Full and Reduced Exoplanet Chemical Kinetics DistiLLed

We introduce a new Python 1D chemical kinetic code, Full and Reduced Exoplanet Chemical Kinetics distiLLed (FRECKLL), to evolve large chemical networks efficiently. FRECKLL employs “distillation” in computing the reaction rates, which minimizes the error bounds to the minimum allowed by double precision values (ϵ ≤ 10−15). Compared to summation of rates with traditional algorithms like pairwise summation, distillation provides a tenfold reduction in solver time for both full and reduced networks. Both the full and reduced Venot2020 networks are packaged in FRECKLL as well as a TauREx 3.1 plug-in for usage in forward modeling and retrievals of exoplanet atmospheres. We present TauREx retrievals performed on a simulated HD 189733b JWST spectra using the full and reduced Venot2020 chemical networks and demonstrate the viability of total disequilibrium chemistry retrievals and the ability for JWST to detect disequilibrium processes.

Pairwise summation as a method for the additive combination of probabilities in qualitative risk assessments.

Qualitative frameworks are widely employed to tackle urgent animal or public health issues when data are scarce and/or urgent decisions need to be made. In qualitative models, the degree of belief regarding the probabilities of the events occurring along the risk pathway(s) and the outcomes is described in nonnumerical terms, typically using words such as Low, Medium, or High. The main methodological challenge, intrinsic in qualitative models, relates to performing mathematical operations and adherence to the rule of probabilities when probabilities are nonnumerical. Although methods to obtain the qualitative probability from the conditional realization of n events are well-established and consistent with the multiplication rule of probabilities, there is a lack of accepted methods for addressing situations where the probability of an event occurring can increase, and the rule of probability P(AUB)=P(A)+P(B)-P(A∩B) should apply. In this work, we propose a method based on the pairwise summation to fill this methodological gap. Our method was tested on two qualitative models and compared by means of scenario analysis to other approaches found in literature. The qualitative nature of the models prevented formal validation; however, when using the pairwise summation, results consistently appeared more coherent with probability rules. Even if the final qualitative estimate can only represent an approximation of the actual probability of the event occurring, qualitative models have proven to be effective in providing scientific-based evidence to support decision-making. The method proposed in this study contributes to reducing the subjectivity that characterizes qualitative models, improving transparency and reproducibility.

A robust approach for multi-type classification of brain tumor using deep feature fusion.

Brain tumors can be classified into many different types based on their shape, texture, and location. Accurate diagnosis of brain tumor types can help doctors to develop appropriate treatment plans to save patients' lives. Therefore, it is very crucial to improve the accuracy of this classification system for brain tumors to assist doctors in their treatment. We propose a deep feature fusion method based on convolutional neural networks to enhance the accuracy and robustness of brain tumor classification while mitigating the risk of over-fitting. Firstly, the extracted features of three pre-trained models including ResNet101, DenseNet121, and EfficientNetB0 are adjusted to ensure that the shape of extracted features for the three models is the same. Secondly, the three models are fine-tuned to extract features from brain tumor images. Thirdly, pairwise summation of the extracted features is carried out to achieve feature fusion. Finally, classification of brain tumors based on fused features is performed. The public datasets including Figshare (Dataset 1) and Kaggle (Dataset 2) are used to verify the reliability of the proposed method. Experimental results demonstrate that the fusion method of ResNet101 and DenseNet121 features achieves the best performance, which achieves classification accuracy of 99.18 and 97.24% in Figshare dataset and Kaggle dataset, respectively.

Harmonic Surface Mapping Algorithm for Electrostatic Potentials in an Atomistic/Continuum Hybrid Model for Electrolyte Solutions

Simulating charged many-body systems has been a computational demanding task due to the long-range nature of electrostatic interaction. For the multi-scale model of electrolytes which combines the strengths of atomistic/continuum electrolyte representations, a harmonic surface mapping algorithm is developed for fast and accurate evaluation of the electrostatic reaction potentials. Our method reformulates the reaction potential into a sum of image charges for the near-field, and a charge density on an auxiliary spherical surface for the far-field, which can be further discretized into point charges. Fast multipole method is used to accelerate the pairwise Coulomb summation. The accuracy and efficiency of our algorithm, as well as the choice of relevant numerical parameters are demonstrated in detail. As a concrete example, for charges close to the dielectric interface, our method can improve the accuracy by two orders of magnitudes compared to the Kirkwood series expansion method.

Error Analysis and Improving the Accuracy of Winograd Convolution for Deep Neural Networks

Popular deep neural networks (DNNs) spend the majority of their execution time computing convolutions. The Winograd family of algorithms can greatly reduce the number of arithmetic operations required and is used in many DNN software frameworks. However, the performance gain is at the expense of a reduction in floating point (FP) numerical accuracy. In this article, we analyse the worst-case FP error and derive an estimation of the norm and conditioning of the algorithm. We show that the bound grows exponentially with the size of the convolution. Further, the error bound of the modified algorithm is slightly lower but still exponential. We propose several methods for reducing FP error. We propose a canonical evaluation ordering based on Huffman coding that reduces summation error. We study the selection of sampling “points” experimentally and find empirically good points for the most important sizes. We identify the main factors associated with good points. In addition, we explore other methods to reduce FP error, including mixed-precision convolution, and pairwise summation across DNN channels. Using our methods, we can significantly reduce FP error for a given block size, which allows larger block sizes to be used and reduced computation.

A CUDA fast multipole method with highly efficient M2L far field evaluation

Solving an N-body problem, electrostatic or gravitational, is a crucial task and the main computational bottleneck in many scientific applications. Its direct solution is an ubiquitous showcase example for the compute power of graphics processing units (GPUs). However, the naïve pairwise summation has [Formula: see text] computational complexity. The fast multipole method (FMM) can reduce runtime and complexity to [Formula: see text] for any specified precision. Here, we present a CUDA-accelerated, C++ FMM implementation for multi particle systems with [Formula: see text] potential that are found, e.g. in biomolecular simulations. The algorithm involves several operators to exchange information in an octree data structure. We focus on the Multipole-to-Local (M2L) operator, as its runtime is limiting for the overall performance. We propose, implement and benchmark three different M2L parallelization approaches. Approach (1) utilizes Unified Memory to minimize programming and porting efforts. It achieves decent speedups for only little implementation work. Approach (2) employs CUDA Dynamic Parallelism to significantly improve performance for high approximation accuracies. The presorted list-based approach (3) fits periodic boundary conditions particularly well. It exploits FMM operator symmetries to minimize both memory access and the number of complex multiplications. The result is a compute-bound implementation, i.e. performance is limited by arithmetic operations rather than by memory accesses. The complete CUDA parallelized FMM is incorporated within the GROMACS molecular dynamics package as an alternative Coulomb solver.

NUMERICAL AND THEORETICAL STUDY OF THE CASIMIR FORCE CORRECTIONS BETWEEN TWO ROUGH SURFACES

The Casimir force between two rough surfaces grown through statistical growth models is studied, both theoretically and numerically. The models include the random deposition (RD), the random deposition with surface relaxation (RDSR), and the restricted solid-on-solid growth (RSOS). To calculate the Casimir force, the pairwise summation (PWS) method in [Formula: see text] dimensions is used. First, the force between two growing surfaces with RD equations, Edwards–Wilkinson, (EW) and Kardar–Parisi–Zhang (KPZ) equations is theoretically investigated. Then, using the Monte Carlo simulation, the rough surfaces with RD, RDSR, and RSOS models in [Formula: see text] dimensions are simulated in different growth times, and the Casimir force between each pair of rough surfaces in each time step is calculated. The results show that the force between the surfaces increases as the roughness increases.

Upper and Lower Bounds on Infinite Summation of Pair Potential Energy between Atom and Single Graphene Sheet

The infinite pairwise summation of the potential energy between an atom above a graphene sheet and carbon atoms arranged in a hexagonal lattice is considered. Its magnitude depends on the minimum d...

Modification of the Wolf Method and Evaluation for Molecular Simulation of Vapor–Liquid Equilibria

A modification of the Wolf method [ Wolf et al. J. Chem. Phys. 1999 , 110 , 8254 - 8282 ], a spherically truncated pairwise summation to evaluate electrostatic interactions efficiently, is proposed. This method achieves better results for the energy, and the approach is used for determining phase equilibria in the grand canonical ensemble. To assess optimal parameters for the Wolf summation we propose a simple iterative approach using the Ewald summation as a reference. We show that phase equilibrium properties for pure components as well as mixtures can be calculated accurately using the Wolf summation. Our study considers molecular fluids with partial charges but without net charge.

The effect of roughness and correlation on the Casimir torque between two plates

The effect of roughness and correlation on the Casimir torque is studied. The plates are assumed to be perfect conductors. This is a good approximation when the separation between plates is not too small. The pairwise summation (PWS) method is used, which is a good approximation when the correlation length is much larger than the distance between the plates. Torque components both parallel and perpendicular to the plates are obtained. It is seen that the component parallel to the plates is nonvanishing even if the plates are smooth, but there are contributions due to roughness and correlation as well, and the contribution of the correlation is an increasing function of both the roughness exponent and the correlation length. The component perpendicular to the plates, however, is nonvanishing only if the plates are rough and correlated to each other. As the roughness exponent increases, this component increases, reaches a peak and then decreases.

Scaling laws for migrating cloud of low-Reynolds-number particles with Coulomb repulsion

We investigate the evolution of spherical clouds of charged particles that migrate under the action of a uniform external electrostatic field. Hydrodynamic interactions are modelled by Oseen equations and the Coulomb repulsion is calculated through pairwise summation. It is shown that strong long-range Coulomb repulsion can prevent the breakup of the clouds covering a wide range of particle Reynolds number $Re_{p}$ and cloud-to-particle size ratio $R_{0}/r_{p}$. A dimensionless charge parameter $\unicode[STIX]{x1D705}_{q}$ is constructed to quantify the effect of the repulsion, and a critical value $\unicode[STIX]{x1D705}_{q,t}$ is deduced, which successfully captures the transition of a cloud from hydrodynamically controlled regime to repulsion-controlled regime. Our results also reveal that, with sufficiently strong repulsion, the cloud undergoes a universal self-similar expansion. Scaling laws of cloud radius $R_{cl}$ and particle number density $n$ are obtained by solving a continuum convection equation.

Casimir effect for Dirac lattices

We consider polarizable sheets, which recently received some attention, especially in context of the dispersion interaction of thin sheets like graphene. These sheets are modeled by a collection of delta function potentials and resemble "zero range potentials", known in quantum mechanics. We develop a theoretical description and apply the so-called "TGTG"-formula to calculate the interaction of two such lattices. Thereby we make use of the formulation of the scattering of waves off such sheets provided earlier. We consider all limiting cases, providing link to earlier results. Also, we discuss the relation to the pairwise summation method.

Effects of geometrical specification of microengineered surfaces to Casimir-Polder potential

Cold atoms can be trapped and guided using nanofabricated wires on microengineered surfaces, achieving the scales required by quantum information proposals on the consequences of Casimir-Polder (CP) force to its stability and lifetime. The atoms are found to be less attractive to the thin films compared to dielectric half plate, where the attractive potential drops even further when the thickness of the surface is lower than both thermal wavelength and the spacing length. Amiably, for the spherical surface, the CP potential decays even further as radial thickness decreases. We approximate the CP potential using the analytical expression obtained from the pairwise summation (PWS) calculated by taking a sum of single atom with the jumbled atoms inside the surface, treating them as two-bodies system as a whole. The means of PWS are to calculate the possibilities of geometrical effect: planar and curved surface on the CP potential.

Evolution of Clouds of Migrating Micro-particles with Hydrodynamic and Electrostatic Interactions

The evolution of a migrating cloud of particles with electrohydrodynamic interactions is numerically investigated. The hydrodynamic interaction is modelled by Oseen dynamics in the limit of small-but-finite particle Reynolds number. The effects of external field and the particle-particle Coulomb repulsion, calculated through pairwise summation, are quantified by a charge parameter, κ q . With a small or zero κ q , the cloud is seen to flatten into a toroidal configuration and eventually breaks up into two small clouds, indicating that the external electrostatic field has a similar effect with gravity. Increasing the long-range Coulomb repulsion can delay or even totally prevent the breakup. With sufficiently strong repulsion, the cloud undergoes a self-similar expansion. Finally, we show that the breakup of the cloud is strongly related to the initial inhomogeneity of the particle concentration. This chaotic characteristic, however, is dramatically depressed by the long-range Coulomb repulsion.

Interaction Energy between Graphene and a Silicon Substrate Using Pairwise Summation of the Lennard-Jones Potential

Interaction Energy between Graphene and a Silicon Substrate Using Pairwise Summation of the Lennard-Jones Potential

Long-range wetting transparency on top of layered metal-dielectric substrates.

It has been recently shown that scores of physical and chemical phenomena (including spontaneous emission, scattering and Förster energy transfer) can be controlled by nonlocal dielectric environments provided by metamaterials with hyperbolic dispersion and simpler metal/dielectric structures. At this time, we have researched van der Waals interactions and experimentally studied wetting of several metallic, dielectric and composite multilayered substrates. We have found that the wetting angle of water on top of MgF2 is highly sensitive to the thickness of the MgF2 layer and the nature of the underlying substrate that could be positioned as far as ~100 nm beneath the water/MgF2 interface. We refer to this phenomenon as long range wetting transparency. The latter effect cannot be described in terms of the most basic model of dispersion van der Waals-London forces based on pair-wise summation of dipole-dipole interactions across an interface or a gap separating the two media. We infer that the experimentally observed gradual change of the wetting angle with increase of the thickness of the MgF2 layer can possibly be explained by the distance dependence of the Hamaker function (describing the strength of interaction), which originates from retardation of electromagnetic waves at the distances comparable to a wavelength.

Investigation of the range of validity of the pairwise summation method applied to the calculation of the surface roughness correction to the van der Waals force

It has long been recognized that stochastic surface roughness can considerably change the van der Waals (vdW) force between interacting surfaces and particles. However, few analytical expressions for the vdW force between rough surfaces have been presented in the literature. Because they have been derived using perturbative methods or the proximity force approximation the expressions are valid when the roughness correction is small and for a limited range of roughness parameters and surface separation. In this work, a nonperturbative approach, the effective density method (EDM) is proposed to circumvent some of these limitations. The method simplifies the calculations of the roughness correction based on pairwise summation (PWS), and allows us to derive simple expressions for the vdW force and energy between two semispaces covered with stochastic rough surfaces. Because the range of applicability of PWS and, therefore, of our results, are not known a priori, we compare the predictions based on the EDM with those based on the multilayer effective medium model, whose range of validity can be defined more properly and which is valid when the roughness correction is comparatively large. We conclude that the PWS can be used for roughness characterized by a correlation length of the order of its rms amplitude, when this amplitude is of the order of or smaller than a few nanometers, and only for typically insulating materials such as silicon dioxide, silicon nitride, diamond, and certain glasses, polymers and ceramics. The results are relevant for the correct modeling of systems where the vdW force can play a significant role such as micro and nanodevices, for the calculation of the tip-sample force in atomic force microscopy, and in problems involving adhesion.

Lateral Casimir force between self-affine rough surfaces

The effect of self-affine roughness on the lateral Casimir force between two plates is studied using a perturbative expansion method. The PWS (pairwise summation) method is applicable only at lateral correlation lengths much larger than the separation between two plates. The effect of the roughness parameters on the lateral Casimir force is investigated, and it is seen that this effect is significant, enabling one to tailor roughness parameters so that to obtain the desirable Casimir force and increase the yield of micro- or nano-electromechanical devices based on the vacuum fluctuations.

Recent development of atom‐pairwise van der waals corrections for density functional theory: From molecules to solids

Van der Waals (vdW) interactions are important in numerous physical, chemical, and biological systems. However, traditional density functional theory (DFT) within local or semi‐local approximations can hardly treat this interaction. Among various attempts to handle vdW interactions in DFT, semi‐empirical correction methods are known to present the advantages of low additional computational costs and easy implementation in conventional DFT codes. In this review, we summarize the state‐of‐the‐art semi‐empirical vdW correction methods based on pairwise summations within the atoms‐in‐molecules scaling framework, such as the Grimme's D3 methods and variants of the Tkatchenko‐Scheffler method. In addition, we compare the performance of these methods for systems ranging from molecules to solids, which have dispersive to ionic interactions: 128 molecular pairs, 23 molecular crystals, 4 noble gas crystals, 27 two‐dimensional layered materials, and 9 ionic crystals. © 2015 Wiley Periodicals, Inc.