Particle-particle Particle-mesh Research Articles

Mesoscale Modeling of Polyelectrolyte Brushes with Salt

We report dissipative particle dynamics (DPD) simulations of a polyelectrolyte brush under athermal solvent conditions. The electrostatic interactions are calculated using the particle-particle particle-mesh (PPPM) method with charges distributed over the particles. The polymer beads, counterions, co-ions, and solvent particles are modeled explicitly. The DPD simulations show a dependence of the brush height on the grafting density and the charge fraction that is typical of the nonlinear osmotic brush regime. We report the effect of the addition of salt on the structural properties of the brush. In the case of a polyelectrolyte brush with a high surface coverage, the simulations reproduce the transition between the nonlinear osmotic brush regime where the thickness of the brush is independent of the salt concentration and the salted regime where the brush height decreases weakly with the salt concentration.

Multi-Level Ewald: A Hybrid Multigrid/Fast Fourier Transform Approach to the Electrostatic Particle-Mesh Problem

We present a new method for decomposing the one convolution required by standard Particle-Particle Particle-Mesh (P(3)M) electrostatic methods into a series of convolutions over slab-shaped subregions of the original simulation cell. Most of the convolutions derive data from separate regions of the cell and can thus be computed independently via FFTs, in some cases with a small amount of zero padding so that the results of these sub-problems may be reunited with minimal error. A single convolution over the entire cell is also performed, but using a much coarser mesh than the original problem would have required. This "Multi-Level Ewald" (MLE) method therefore requires moderately more FFT work plus the tasks of interpolating between different sizes of mesh and accumulating the results from neighboring sub-problems, but we show that the added expense can be less than 10% of the total simulation cost. We implement MLE as an approximation to the Smooth Particle Mesh Ewald (SPME) style of P(3)M, and identify a number of tunable parameters in MLE. With reasonable settings pertaining to the degree of overlap between the various sub-problems and the accuracy of interpolation between meshes, the errors obtained by MLE can be smaller than those obtained in molecular simulations with typical SPME settings. We compare simulations of a box of water molecules performed with MLE and SPME, and show that the energy conservation, structural, and dynamical properties of the system are more affected by the accuracy of the SPME calculation itself than by the additional MLE approximation. We anticipate that the MLE method's ability to break a single convolution into many independent sub-problems will be useful for extending the parallel scaling of molecular simulations.

Error Analysis of the Poisson P3M Force Field Scheme for Particle-Based Simulations of Biological Systems

The ability to accurately simulate electrolyte solutions is a strong requirement for the modeling of charge transport in ion channels. In this work, a Particle-Particle–Particle-Mesh (P3M) algorithm is used to model the electrostatic interactions governing the physics of electrolyte solutions. The goal of this study is to define the parameters relevant for this force field, and their respective influence on the accuracy of the simulation. Simulations have been performed for extended ranges of these parameters and the results are compared to theoretical models. Finally, the trade-offs between optimal algorithmic efficiency and accuracy are analyzed.

Early performance data on the Blue Matter molecular simulation framework

Blue Matter is the application framework being developed in conjunction with the scientific portion of the IBM Blue Gene® project. We describe the parallel decomposition currently being used to target the Blue Gene/L machine and discuss the application-based trace tools used to analyze the performance of the application. We also present the results of early performance studies, including a comparison of the performance of the Ewald and the particle-particle particle-mesh (P3ME) methods, compare the measured performance of some key collective operations with the limitations imposed by the hardware, and discuss some future directions for research.

Fast and Accurate P 3 M Cosmological Simulations Using a Hardware Accelerator

We implemented a fast and accurate particle-particle particle-mesh (P3M) method using MDGRAPE-2, a hardware accelerator for arbitrary central forces. The results of cosmological N-body simulations using the MDGRAPE-2 P3M method agree with those using other P3M methods. We measured calculation times required for the MDGRAPE-2 P3M method on the system of a PC and one MDGRAPE-2 board. When we reduce memory requirement, the MDGRAPE-2 P3M method is ~10 and ~20 times faster than the adaptive and original P3M methods on the PC, respectively, under heavy particle clustering. When each P3M method is optimized for performance, the MDGRAPE-2 P3M method is as fast as the adaptive P3M method and ~10 times faster than the original P3M method.

Spatial and Dynamical Biases in Velocity Statistics of Galaxies

We present velocity statistics of galaxies and their biases inferred from the statistics of the underlying dark matter using a cosmological hydrodynamic simulation of galaxy formation in low-density and spatially flat (Ohm(0) = 0.3 and lambda(0) = 0.7) cold dark matter cosmogony. Our simulation is based on a particle-particle-particle-mesh ((PM)-M-3) N-body Poisson solver and smoothed particle hydrodynamics. Galaxies in our simulation are identified as clumps of cold and dense gas particles and classified as young and old galaxies according to their formation epochs. We find that the pairwise velocity dispersion (PVD) of all galaxies is significantly lower than that of the dark matter particles and that the PVD of the young galaxies is lower than that of the old types, and even of all galaxies together, especially at small separations. These results are in reasonable agreement with the recent measurements of PVDs in the Las Campanas redshift survey, the IRAS Point Source Catalogue Redshift Survey, and the Sloan Digital Sky Survey data. We also find that the low PVD of young galaxies is due to the effects of dynamical friction as well as the different spatial distribution, while the difference in the PVD between all galaxies and dark matter can be mostly ascribed to their different spatial distributions. We also consider the mean infall velocity and the POTENT density reconstruction that are often used to measure the cosmological parameters, and investigate the effects of spatial bias and dynamical friction. In our simulation, the mean infall velocity of young galaxies is significantly lower than that of all the galaxies or of the old galaxies, and the dynamical bias becomes important on scales less than 3 h(-1) Mpc. The mass density field reconstructed from the velocity field of young galaxies using the POTENT-style method suffers in accuracy from both the spatial bias and the dynamical friction on the smoothing scale of R-s = 8 h(-1) Mpc. On the other hand, in the case of R-s = 12 h(-1) Mpc, which is typically adopted in the actual POTENT analysis, the density reconstruction based on various tracers of galaxies is reasonably accurate. We also analyze the motions of central galaxies and the velocity dispersion of galaxies within halos and discuss their implications for the motion of cD galaxies and the determination of the mass of galaxy groups.

3-D Parallel Monte Carlo Simulation of Sub-0.1 Micron MOSFETs on a Cluster Based Supercomputer

A full band, three-dimensional, Monte Carlo simulator for deep sub-micron Si MOSFET like devices has been developed, with the goal to obtain optimal performance on a parallel system built from a cluster of commodity computers. A short-range carrier-carrier and carrier-ion model has been implemented within this framework, using Particle-Particle Particle-Mesh (P3M) algorithm. Test simulations include the 90 nm “well-tempered MOSFET” for which measurements are available. Simulation benchmarks have identified several factors limiting the overall performance of the code and suggestions for improvements in these areas are made.

Molecular dynamics studies of electron-hole bilayers

We investigated the properties of strongly coupled classical electron-hole bilayers using a molecular dynamics simulation code based on the PPPM (particle-particle particle-mesh) algorithm and applying the transformation of the two layers into a single layer with modified interaction potentials. The calculations were carried out for layer separations d comparable to the Wigner-Seitz radius a. The analysis of the (intra- and interlayer) pair correlation functions g 11 (r) and g 12 (r) showed that at layer separations d/a ≥ 3 the layers are independent of each other. With decreasing d/a a strongly increasing peak of g 12 develops at r/a = 0, indicating the formation of electron-hole pairs (classical excitons). The intralayer correlation function g 11 is essentially independent of the layer separation for d/a ≥ 1, the interaction of particles between the two layers does not influence the long range order (observed at high r values) in the layers. At d/a < 1 the long range order is destroyed due to the fact that the interaction is due to the dipole field of the electron-hole pairs.

A theoretical evaluation of the ionization potentials for one-electron oxidized states of cytochrome c3

The ionization potentials of one-electron oxidized states of cytochrome c 3 were evaluated by numerical calculations. In the ionization processes from cytochrome c 3 0+ to c 3 1+, four ionization potentials were observed with respect to four hemes involved in cytochrome c 3. Ab initio SCF MO calculations were carried out for the four heme complexes to obtain their ionization potentials under vacuum (IP vacuums). The electrostatic potentials (EPs) which the apo-protein and the aqueous solvent produce at the heme irons were calculated by the particle–particle/particle–mesh (PPPM) algorithm based on the self-consistent boundary (SCB) method. Final theoretical ionization potentials of cytochrome c 3 (IP proteins) were evaluated as the sum of IP vacuums and EPs. The calculated IP proteins successfully reproduced the order of corresponding experimental values. The maximum difference among IP proteins was 130 mV which was approximately twice as large as that of experimental ones. The small differences among IP proteins were given by the delicate cancellations of IP vacuums and EPs. We also discuss the detailed electronic structures of hemes in cytochrome c 3.

An Iterative PPPM Method for Simulating Coulombic Systems on Distributed Memory Parallel Computers

We describe results obtained from a new implementation of Hockney's Particle-Particle Particle-Mesh (PPPM) method for evaluation of Coulomb energies and forces in simulations of charged particles. Rather than taking the usual approach, solving Poisson's equation by means of a Fourier transformation, we use an iterative Poisson solver. In a molecular dynamics (MD) simulation the solution from the previous time-step provides a good starting point for the next solution. This reduces the number of iterations per time-step to acceptable values. The iterative scheme has a complexity O(N), and, in contrast with the Fourier transform based approach, it is easily implemented on a parallel architecture with a minimum of communication overhead. We examine the origin of the errors in the algorithm and find that reasonable accuracies in the Coulomb interaction can best be attained by making the charge density profile as smooth as possible. This involves spreading the particle charges over a large number of grid points. Assigning these charges then becomes the most time consuming part of the algorithm. We show how we can then gain a considerable saving in computing time by employing a diffusion equation as a charge spreading mechanism. The effect of employing the algorithm with an accuracy less than that typically tolerated in an Ewald summation is studied by computing, from an MD simulation of silica, quantities that are sensitive to the long range part of the Coulomb interaction. These results are compared to full Ewald sum reference simulations and found to be within the statistical error.

A load balancing package on distributed memory systems and its application to particle-particle particle-mesh (P3M) methods

We present a tool, Bisect, for balanced decomposition of spatial domains. In addition to applying a nested bisection algorithm to determine the boundaries of each subdomain, Bisect replicates a user specified zone along the boundaries of the subdomain in order to minimize future interactions between subdomains. Results of running the tool on the Cray T3D system using both shared memory operations and MPI communications are reported and discussed. In addition, Bisect is used in a parallel implementation of a particle-particle/particle-mesh (P3M) simulation program on the Cray T3D system. The performance of the P3M program with different load-balancing criteria is evaluated and compared. The results show that the use of the Bisect package balances the load efficiently and minimizes communication on the T3D massively parallel system.

Comments on P 3M, FMM, and the Ewald method for large periodic Coulombic systems

Prompted by the need to simulate large molecular or gravitational systems and the availability of multiprocessor computers, alternatives to the standard Ewald calculation of Coulombic interactions have been developed. The two most popular alternatives, the fast multipole method (FMM) and the particle-particle particle-mesh (P 3M) method are compared here to the Ewald method for a single processor machine. Parallel processor implementations of the P 3M and Ewald methods are compared. The P 3M method is found to be both faster than the FMM and easier to implement efficiently as it relies on commonly available software (FFT subroutines). Both the Ewald and P 3M method are easily implemented on parallel architectures with the P 3M method the clear choice for large systems.

Calculating Electrostatic Interactions Using the Particle−Particle Particle−Mesh Method with Nonperiodic Long-Range Interactions

The particle−particle particle−mesh (PPPM) method is an accurate and computationally efficient method for calculating interactions in molecular simulations. The PPPM method is based on separating the total interaction between particles into the sum of short-range interactions, which are computed by direct particle−particle summation, and long-range interactions, which are calculated by solving Poisson's equation using periodic boundary conditions (PBCs). For nonperiodic systems there is a concern that use of PBCs may introduce an artificial periodicity into the system. In this article we propose, and test, the use of boundary conditions for the long-range interactions which mimic surrounding the explicit system by high dielectric material such as water.

Lattice-sum methods for calculating electrostatic interactions in molecular simulations

When simulating a periodic molecular system, lattice sum methods may be used to evaluate the electrostatic interactions without resorting to an unphysical truncation of the potential. In a previous publication, we compared the computational efficiency of the Ewald and Particle-Particle Particle-Mesh (PPPM) lattice-sum methods. Because the PPPM method discretizes the field equations and utilizes the highly efficient fast Fourier transform algorithm, it requires significantly less computational effort than the Ewald method and scales almost linearly with system size. In this paper, we take a more detailed look into the theory behind the lattice-sum methods to clarify the underlying similarities between the Ewald method and the PPPM method. We also investigate the errors introduced by different approximation used in the discretization of the field equations in the PPPM method.

A Comparison of Particle-Particle, Particle-Mesh and Ewald Methods for Calculating Electrostatic Interactions in Periodic Molecular Systems

Abstract We compare the Particle-Particle Particle-Mesh (PPPM) and Ewald methods for calculating electrostatic interactions in periodic molecular systems. A brief comparison of the theories shows that the methods are very similar differing mainly in the technique which is used to perform the “k-space” or mesh calculation. Because the PPPM utilizes the highly efficient numerical Fast Fourier Transform (FFT) method it requires significantly less computational effort than the Ewald method and scales almost linearly with system size.

Implementation of particle-in-cell stellar dynamics codes on the connection machine-2

The development of massively parallel supercomputers provides a unique opportunity to advance the state of the art inN-body simulations. TheseN-body codes are of great importance for simulations in stellar dynamics and plasma physics. For systems with long-range forces, such as gravity or electromagnetic forces, it is important to increase the number of particles toN ≥ 107 particles. Significantly improved modeling ofN body systems can be expected by increasingN, arising from a more realistic representation of physical transport processes involving particle diffusion and energy and momentum transport. In addition, it will be possible to guarantee that physically significant portions of complex physical systems, such as Lindblad resonances of galaxies or current sheets in magnetospheres, will have an adequate population of particles for a realistic simulation. Particle-mesh (PM) and particle-particle particle-mesh (P3M) algorithms present the best prospects for the simulation of large-scaleN-body systems. As an example we present a two-dimensional PM simulation of a disk galaxy that we have developed on the Connection Machine-2, a massively parallel boolean hypercube supercomputer. The code is scalable to any CM-2 configuration available and, on the largest configuration, simulations withN = 128 M = 227 particles are possible in reasonable run times.