Kirkwood Superposition Approximation Research Articles

Computing Acceleration to Genome-Wide Association Study Based on CPU/FPGA Heterogeneous System

Genome Wide Association Study (GWAS) reveals the influence of single nucleotide polymorphisms (SNP) and other genetic markers on the complex genetic disease traits, making a significant contribution to the prevention and treatment of genetic diseases. Since GWAS needs to calculate large numbers of pairwise tests, it is highly time-consuming and this limits its widespread promotion in many medical applications. To solve the above problem, many studies deploy GWAS on the CPU/GPU platform. By taking advantages of GPUs' high parallelism, the completion time of GWAS can be significantly decreased. However, GPUs have high energy consumption, which increases GWAS's operating cost. Considering FPGA has low power, high parallelism and programmability, this article aims to use FPGA to design a hardware accelerator dedicated to GWAS for the purpose of optimizing both completion time and energy cost of GWAS. First, the CPU/FPGA heterogenous computing system is built, and the computationally-intensive GWAS tasks such as the contingency table creation (CTC) and Kirkwood Superposition Approximation (KSA) screening are offloaded to run on FPGA. Then, a specific FPGA architectures dedicated to CTC and KSA tasks of GWAS are implemented by using a set of architectural design technologies such as hardware and software co-optimization, systolic array, loop unrolling, pipeline and parallelism. To evaluate the performance of FPGA, we compare its completion time, energy cost and performance per Watt with those of CPU and GPU. The results show the FPGA and GPU can respectively achieve 105--fold and 148--fold speedup compared with CPU. Although FPGA's computing performance is a bit lower than that of GPU, it has higher energy efficiency and its performance per Watt is 1.4 times that of GPU.

Analysis of the finite-size effect of the long-range Ising model under Glauber dynamics

We considered a long-range Ising model under Glauber dynamics and calculated the difference from the mean-field approximation in a finite-size system using perturbation theory. To deal with the Bogoliubov–Born–Green–Kirkwood–Yvon hierarchy, we assumed that certain types of extensive properties have a Gaussian distribution, which turned out to be equivalent to the Kirkwood superposition approximation within the range of first-order perturbation. After several calculations, ordinary differential equations that describe the time development of a two-body correlation were derived. This discussion is the generalization of our previous study which developed a similar consideration on the infinite-range Ising model. The results of the calculation fit those of the numerical simulations for the case in which the decay of the interaction was sufficiently slow; however, they exhibited different behaviors when the decay became rapid.

Explicit solvation thermodynamics in ionic solution: extending grid inhomogeneous solvation theory to solvation free energy of salt\u2013water mixtures

Hydration thermodynamics play a fundamental role in fields ranging from the pharmaceutical industry to environmental research. Numerous methods exist to predict solvation thermodynamics of compounds ranging from small molecules to large biomolecules. Arguably the most precise methods are those based on molecular dynamics (MD) simulations in explicit solvent. One theory that has seen increased use is inhomogeneous solvation theory (IST). However, while many applications require accurate description of salt–water mixtures, no implementation of IST is currently able to estimate solvation properties involving more than one solvent species. Here, we present an extension to grid inhomogeneous solvation theory (GIST) that can take salt contributions into account. At the example of carbazole in 1 M NaCl solution, we compute the solvation energy as well as first and second order entropies. While the effect of the first order ion entropy is small, both the water–water and water–ion entropies contribute strongly. We show that the water–ion entropies are efficiently approximated using the Kirkwood superposition approximation. However, this approach cannot be applied to the water–water entropy. Furthermore, we test the quantitative validity of our method by computing salting-out coefficients and comparing them to experimental data. We find a good correlation to experimental salting-out constants, while the absolute values are overpredicted due to the approximate second order entropy. Since ions are frequently used in MD, either to neutralize the system or as a part of the investigated process, our method greatly extends the applicability of GIST. The use-cases range from biopharmaceuticals, where many assays require high salt concentrations, to environmental research, where solubility in sea water is important to model the fate of organic substances.

Functional-renormalization-group approach to classical liquids with short-range repulsion: A scheme without repulsive reference system.

The renormalization-group approaches for classical liquids in previous works required a repulsive reference such as a hard-core one when applied to systems with short-range repulsion. The need for the reference is circumvented here by using a functional-renormalization-group approach for integrating the hierarchical flow of correlation functions along a path of variable interatomic coupling. We introduce the cavity distribution functions to avoid the appearance of divergent terms and choose a path to reduce the error caused by the decomposition of higher order correlation functions. We demonstrate using exactly solvable one-dimensional models that the resulting scheme yields accurate thermodynamic properties and interatomic distribution at various densities when compared to integral-equation methods such as the hypernetted chain and the Percus-Yevick equation, even in the case where our hierarchical equations are truncated with the Kirkwood superposition approximation, which is valid for low-density cases.

Calculation of pair and triplet correlation functions for a Lennard-Jones fluid at ρ*≤1.41 and T*≤25

On the basis of Monte Carlo simulations, configurations and pair and triplet correlation functions for a Lennard-Jones fluid are reported over a large range of densities and temperatures (0.002 ≤ ρ* ≤ 1.41 and 0.45 ≤ T* ≤ 25, dimensionless). In total, data for 27 615 ρ*–T*-state points including 750 configurations for each state point are used for the calculation of the pair and triplet correlation functions. For the pair configuration function, an approximation over the whole set of state points is provided, which reproduces the pair configuration with high accuracy (3σ < 0.0075). The results for the triplet configuration functions are compared to the Kirkwood superposition approximation. With the exception of low ρ* (<0.3), the application of the Kirkwood superposition approximation seems not to be a proper choice, showing errors >20%. The configurations and pair and triplet correlation functions of all simulations are openly accessible and can be used as a reference for future theoretical developments of Lennard-Jones fluids and, especially, the liquid state.

Spatial population dynamics: Beyond the Kirkwood superposition approximation by advancing to the Fisher–Kopeliovich ansatz

The superior Fisher–Kopeliovich closure is applied to the hierarchy of master equations for spatial moments in population dynamics for the first time. As a consequence, the density, pair and triplet distribution functions of entities are calculated within this closure for a birth–death model with nonlocal dispersal and competition in continuous space. The new results are compared with those obtained by “exact” individual-based simulations as well as by the inferior mean-field and Kirkwood superposition approximations. It is shown that the Fisher–Kopeliovich approach significantly improves the quality of the description in a wide range of varying parameters of the model.

Continuum Approximations to Systems of Correlated Interacting Particles

We consider a system of interacting particles with random initial conditions. Continuum approximations of the system, based on truncations of the BBGKY hierarchy, are described and simulated for various initial distributions and types of interaction. Specifically, we compare the Mean Field Approximation (MFA), the Kirkwood Superposition Approximation (KSA), and a recently developed truncation of the BBGKY hierarchy (the Truncation Approximation - TA). We show that KSA and TA perform more accurately than MFA in capturing approximate distributions (histograms) obtained from Monte Carlo simulations. Furthermore, TA is more numerically stable and less computationally expensive than KSA.

Definition and Time Evolution of Correlations in Classical Statistical Mechanics.

The study of dense gases and liquids requires consideration of the interactions between the particles and the correlations created by these interactions. In this article, the N-variable distribution function which maximizes the Uncertainty (Shannon’s information entropy) and admits as marginals a set of (N−1)-variable distribution functions, is, by definition, free of N-order correlations. This way to define correlations is valid for stochastic systems described by discrete variables or continuous variables, for equilibrium or non-equilibrium states and correlations of the different orders can be defined and measured. This allows building the grand-canonical expressions of the uncertainty valid for either a dilute gas system or a dense gas system. At equilibrium, for both kinds of systems, the uncertainty becomes identical to the expression of the thermodynamic entropy. Two interesting by-products are also provided by the method: (i) The Kirkwood superposition approximation (ii) A series of generalized superposition approximations. A theorem on the temporal evolution of the relevant uncertainty for molecular systems governed by two-body forces is proved and a conjecture closely related to this theorem sheds new light on the origin of the irreversibility of molecular systems. In this respect, the irreplaceable role played by the three-body interactions is highlighted.

Implicit Solvation Using the Superposition Approximation (IS-SPA): An Implicit Treatment of the Nonpolar Component to Solvation for Simulating Molecular Aggregation.

Nonpolar solute-solvent interactions are the driving force for aggregation in important chemical and biological phenomena including protein folding, peptide self-assembly, and oil-water emulsion formation. Currently, the most accurate and computationally efficient description of these processes requires an explicit treatment of all solvent and solute atoms. Previous computationally feasible implicit solvent models, such as solute surface area approaches, are unsuccessful at capturing aggregation features including both structural and energetic trends while more theoretically rigorous approaches, such as Reference Interaction Site Model (RISM), are accurate but extremely computationally demanding. Our approach, denoted Implicit Solvation using the Superposition Approximation (IS-SPA), builds on previous theory utilizing the Kirkwood superposition approximation to approximate the mean force of the solvent from solute parameters. We introduce and verify a parabolic first solvation shell truncation of atomic solvation, fitting water distributions around a molecule, and a Monte Carlo integration of the mean solvent force. These extensions allow this method to be implemented as an efficient nonpolar implicit solvent model for molecular simulation. The approximations in IS-SPA are first explored and justified for the homodimerization of an array of different sized Lennard-Jones spheres. The accuracy and transferability of the approach are demonstrated by its ability to capture the position and relative energies of the desolvation barrier and free energy minimum of alkane homodimers. The model is then shown to reproduce the phase separation and solubility of cyclohexane and water. These promising results, coupled with 2 orders of magnitude speed-up for dilute systems as compared to explicit solvent simulations, demonstrate that IS-SPA is an appealing approach to boost the time- and length-scale of molecular aggregation simulations.

Experimental investigation of triplet correlation approximations for fluid water

Triplet correlations play a central role in our understanding of fluids and their properties. Of particular interest is the relationship between the pair and triplet correlations. Here we use a combination of Fluctuation Solution Theory and experimental pair radial distribution functions to investigate the accuracy of the Kirkwood Superposition Approximation (KSA), as given by integrals over the relevant pair and triplet correlation functions, at a series of state points for pure water using only experimental quantities. The KSA performs poorly, in agreement with a variety of other studies. Several additional approximate relationships between the pair and triplet correlations in fluids are also investigated and generally provide good agreement for the fluid thermodynamics for regions of the phase diagram where the compressibility is small. A simple power law relationship between the pair and triplet fluctuations is particularly successful for state points displaying low to moderately high compressibilities.

Continuum approximations for lattice-free multi-species models of collective cell migration

Cell migration within tissues involves the interaction of many cells from distinct subpopulations. In this work, we present a discrete model of collective cell migration where the motion of individual cells is driven by random forces, short range repulsion forces to mimic crowding, and longer range attraction forces to mimic adhesion. This discrete model can be used to simulate a population of cells that is composed of K ≥ 1 distinct subpopulations. To analyse the discrete model we formulate a hierarchy of moment equations that describe the spatial evolution of the density of agents, pairs of agents, triplets of agents, and so forth. To solve the hierarchy of moment equations we introduce two forms of closure: (i) the mean field approximation, which effectively assumes that the distributions of individual agents are independent; and (ii) a moment dynamics description that is based on the Kirkwood superposition approximation. The moment dynamics description provides an approximate way of incorporating spatial patterns, such as agent clustering, into the continuum description. Comparing the performance of the two continuum descriptions confirms that both perform well when adhesive forces are sufficiently weak. In contrast, the moment dynamics description outperforms the mean field model when adhesive forces are sufficiently large. This is a first attempt to provide an accurate continuum description of a lattice-free, multi-species model of collective cell migration.

Expansion of the Kullback-Leibler Divergence, and a New Class of Information Metrics

Inferring and comparing complex, multivariable probability density functions is fundamental to problems in several fields, including probabilistic learning, network theory, and data analysis. Classification and prediction are the two faces of this class of problem. This study takes an approach that simplifies many aspects of these problems by presenting a structured, series expansion of the Kullback-Leibler divergence—a function central to information theory—and devise a distance metric based on this divergence. Using the Möbius inversion duality between multivariable entropies and multivariable interaction information, we express the divergence as an additive series in the number of interacting variables, which provides a restricted and simplified set of distributions to use as approximation and with which to model data. Truncations of this series yield approximations based on the number of interacting variables. The first few terms of the expansion-truncation are illustrated and shown to lead naturally to familiar approximations, including the well-known Kirkwood superposition approximation. Truncation can also induce a simple relation between the multi-information and the interaction information. A measure of distance between distributions, based on Kullback-Leibler divergence, is then described and shown to be a true metric if properly restricted. The expansion is shown to generate a hierarchy of metrics and connects this work to information geometry formalisms. An example of the application of these metrics to a graph comparison problem is given that shows that the formalism can be applied to a wide range of network problems and provides a general approach for systematic approximations in numbers of interactions or connections, as well as a related quantitative metric.

Two and three-particles spatial correlation in weak coupling plasma and applications

ABSTRACTIn this work, we studied the pair and triplet correlation functions in plasma composed of particles interacting via the Debye screening potential in the conditions of weak coupling parameter (Γ ≤ 1). The pair correlation functions are analysed numerically and compared to the hypernetted chains data, to the Monte Carlo data and to the results of Abramo and Tosi ​​The triplet correlation functions are also analysed numerically and compared to the Kirkwood superposition approximation. At the end of our work, we give some applications to static structure factor and the dielectric constant. The results are well satisfactory.

Dynamical and transport properties in plasmas including three-particle spatial correlations

In this work, we study the two and triplet static correlation functions in plasma when the ions interact via the Debye screened potential and via the Deutsch screened potential. The latter takes into consideration the possible quantum effects at short distances. The ratio of the mean distance between two ions and the thermal De Broglie wavelength r_{i}/lambda_{mathrm{T}} gives the measure of these effects. Our investigation is developed in the conditions of weak coupling parameter (varGamma <1). The pair and the triplet correlation functions are calculated numerically and compared to the correlation functions due to the Kirkwood superposition approximation (KSA). Some applications to the ion velocity auto-correlation function D(t) and the electric field auto-correlation function C(t) at an ion (assumed to be an impurity) and the diffusion coefficient D are calculated for the two kinds of potentials in different plasma conditions. The comparison with other results found in the literature shows a well satisfactory agreement, for the static as well as the dynamic properties.

Diffusion of Particles with Short-Range Interactions

A system of interacting Brownian particles subject to short-range repulsive potentials is considered. A continuum description in the form of a nonlinear diffusion equation is derived systematically in the dilute limit using the method of matched asymptotic expansions. Numerical simulations are performed to compare the results of the model with those of the commonly used mean-field and Kirkwood-superposition approximations, as well as with Monte Carlo simulation of the stochastic particle system, for various interaction potentials. Our approach works best for very repulsive short-range potentials, while the mean-field approximation is suitable for long-range interactions. The Kirkwood-superposition approximation provides an accurate description for both short- and long-range potentials but is considerably more computationally intensive.

Conductivity estimates of spherical-particle suspensions based on triplet structure factors.

In this paper, we present an estimation of the conductivity of composites constituted of identical spheres embedded in a host material. A family of polarization integral equations for the localization problem is constructed and the operator is then minimized to yield an optimal integral equation. As a result, the corresponding Neumann series converges with the fastest rate and can be used to estimate the effective conductivity. By combining this series and integral approximation, one can derive explicit expressions for the overall property using expansions in Fourier domain. For random hard-sphere systems, relations to structure factors and triplet structure factors have been made and Kirkwood superposition approximation is used to evaluate the effective conductivity, taking into account third-order correlations. This presents an original means to account for the statistical information up to third-order correlation when determining the effective properties of composite materials.

Multi-particle critical correlations

We study the role of multi-particle spatial correlations in the appearance of a liquid–vapour critical point. Our analysis is based on the exact infinite hierarchy of equations relating spatial integrals of (k + 1)-particle correlations to the k-particle ones and their derivatives with respect to the density. Critical exponents corresponding to generalised compressibility equations resulting from the hierarchy are shown to grow linearly with the order of correlations. We prove that the critical behaviour requires taking into account correlation functions of arbitrary order. It is however only a necessary condition. Indeed, approximate closures of the hierarchy obtained by expressing higher order correlations in terms of lower order ones (Kirkwood's superposition approximation and its generalisations) turn out to be inconsistent with the critical behaviour.

Simple transform methods of a force curve obtained by surface force apparatus to the density distribution of a liquid near a surface

We propose two simple methods that transform a force curve obtained by a surface force apparatus (SFA) into a density distribution of a liquid near a surface of the SFA probe. The transform methods are derived based on the statistical mechanics of simple liquids, where the liquid is an ensemble of small spheres. The solvent species is limited to only one component and two-body potential between the solvent spheres is arbitrary. However, two-body potential between the SFA probe and the solvent is restricted to rigid potential (i.e., the transform methods are derived within the restriction of the rigid potential). In addition, Kirkwood and linear superposition approximations are applied in order to derive the transform methods. The transform methods are simply tested in both hard-sphere fluid and Lennard-Jones (LJ) fluid with hard core potential. The tests are computationally practiced using a three-dimensional integral equation theory. It is found that the transform method with Kirkwood superposition approximation (transform method 1) generally reproduces the more precise solvation structure than that with linear superposition approximation (transform method 2). In the test of the hard sphere solvent, it is found that the reproducibility becomes better as the number density of the solvent is lower. Furthermore, it is found in the test of the LJ fluid that the reproducibility becomes better as the two-body potential between the SFA probe and the solvent approaches rigid potential. This is because, the transform methods are derived within the model of the rigid potential. It is verified that the transform methods are useful for obtaining of a rough image of the solvation structure. (However, if evaporation or solidification, a phase transition in a local space sandwiched between the two surfaces, occurs while the experiment, the transform methods should not be used.)

Triplet correlation functions in liquid water.

Triplet correlations have been shown to play a crucial role in the transformation of simple liquids to anomalous tetrahedral fluids [M. Singh, D. Dhabal, A. H. Nguyen, V. Molinero, and C. Chakravarty, Phys. Rev. Lett. 112, 147801 (2014)]. Here we examine triplet correlation functions for water, arguably the most important tetrahedral liquid, under ambient conditions, using configurational ensembles derived from molecular dynamics (MD) simulations and reverse Monte Carlo (RMC) datasets fitted to experimental scattering data. Four different RMC data sets with widely varying hydrogen-bond topologies fitted to neutron and x-ray scattering data are considered [K. T. Wikfeldt, M. Leetmaa, M. P. Ljungberg, A. Nilsson, and L. G. M. Pettersson, J. Phys. Chem. B 113, 6246 (2009)]. Molecular dynamics simulations are performed for two rigid-body effective pair potentials (SPC/E and TIP4P/2005) and the monatomic water (mW) model. Triplet correlation functions are compared with other structural measures for tetrahedrality, such as the O-O-O angular distribution function and the local tetrahedral order distributions. In contrast to the pair correlation functions, which are identical for all the RMC ensembles, the O-O-O triplet correlation function can discriminate between ensembles with different degrees of tetrahedral network formation with the maximally symmetric, tetrahedral SYM dataset displaying distinct signatures of tetrahedrality similar to those obtained from atomistic simulations of the SPC/E model. Triplet correlations from the RMC datasets conform closely to the Kirkwood superposition approximation, while those from MD simulations show deviations within the first two neighbour shells. The possibilities for experimental estimation of triplet correlations of water and other tetrahedral liquids are discussed.

Quantum Binary and Triplet Distribution Functions of Plasma by using Green's Function

AbstractThe aim of this paper is to calculate the binary and triplet distribution functions for one and two component Plasmas in terms of Green's function technique. We obtained the triplet distribution function in two form; one of them is based on the Kirkwood Superposition Approximation (KSA) which is consisting of the assumption that the potential in a set of three particles is the sum of the three pair potentials, this is equivalent to assuming that the triplet distribution function is the product of the three binary distribution functions, but the other form is calculated from BBGKY hierarchy. (© 2014 WILEY‐VCH Verlag GmbH &amp; Co. KGaA, Weinheim)