Three parallel algorithms for classical molecular dynamics are presented. The first assigns each processor a fixed subset of atoms; the second assigns each a fixed subset of inter-atomic forces to compute; the third assigns each a fixed spatial region. The algorithms are suitable for molecular dynamics models which can be difficult to parallelize efficiently—those with short-range forces where the neighbors of each atom change rapidly. They can be implemented on any distributed-memory parallel machine which allows for message-passing of data between independently executing processors. The algorithms are tested on a standard Lennard-Jones benchmark problem for system sizes ranging from 500 to 100,000,000 atoms on several parallel supercomputers--the nCUBE 2, Intel iPSC/860 and Paragon, and Cray T3D. Comparing the results to the fastest reported vectorized Cray Y-MP and C90 algorithm shows that the current generation of parallel machines is competitive with conventional vector supercomputers even for small problems. For large problems, the spatial algorithm achieves parallel efficiencies of 90% and a 1840-node Intel Paragon performs up to 165 faster than a single Cray C9O processor. Trade-offs between the three algorithms and guidelines for adapting them to more complex molecular dynamics simulations are also discussed.

Fast parallel algorithms for short-range molecular dynamics

N G van Kampen 1981 Amsterdam: North-Holland xiv + 419 pp price Dfl 180 This is a book which, at a lower price, could be expected to become an essential part of the library of every physical scientist concerned with problems involving fluctuations and stochastic processes, as well as those who just enjoy a beautifully written book. It provides an extensive graduate-level introduction which is clear, cautious, interesting and readable.

https://iopscience.iop.org/article/10.1088/0031-9112/34/4/040/pdf

Stochastic Processes in Physics and Chemistry

Since the subject of traffic dynamics has captured the interest of physicists, many surprising effects have been revealed and explained. Some of the questions now understood are the following: Why are vehicles sometimes stopped by ``phantom traffic jams'' even though drivers all like to drive fast? What are the mechanisms behind stop-and-go traffic? Why are there several different kinds of congestion, and how are they related? Why do most traffic jams occur considerably before the road capacity is reached? Can a temporary reduction in the volume of traffic cause a lasting traffic jam? Under which conditions can speed limits speed up traffic? Why do pedestrians moving in opposite directions normally organize into lanes, while similar systems ``freeze by heating''? All of these questions have been answered by applying and extending methods from statistical physics and nonlinear dynamics to self-driven many-particle systems. This article considers the empirical data and then reviews the main approaches to modeling pedestrian and vehicle traffic. These include microscopic (particle-based), mesoscopic (gas-kinetic), and macroscopic (fluid-dynamic) models. Attention is also paid to the formulation of a micro-macro link, to aspects of universality, and to other unifying concepts, such as a general modeling framework for self-driven many-particle systems, including spin systems. While the primary focus is upon vehicle and pedestrian traffic, applications to biological or socio-economic systems such as bacterial colonies, flocks of birds, panics, and stock market dynamics are touched upon as well.

Traffic and related self-driven many-particle systems

We survey the newly developed Hilbert spectral analysis method and its applications to Stokes waves, nonlinear wave evolution processes, the spectral form of the random wave field, and turbulence. Our emphasis is on the inadequacy of presently available methods in nonlinear and nonstationary data analysis. Hilbert spectral analysis is here proposed as an alternative. This new method provides not only a more precise definition of particular events in time-frequency space than wavelet analysis, but also more physically meaningful interpretations of the underlying dynamic processes.

/pdf/a-new-view-of-nonlinear-water-waves-the-hilbert-spectrum-42def0zzxb.pdf

A new view of nonlinear water waves: the Hilbert spectrum

Nanocrystalline solids, in which the grain size is in the nanometre range, often have technologically interesting properties such as increased hardness and ductility. Nanocrystalline metals can be produced in several ways, among the most common of which are high-pressure compaction of nanometre-sized clusters and high-energy ball-milling1,2,3,4. The result is a polycrystalline metal with the grains randomly orientated. The hardness and yield stress ofthe material typically increase with decreasing grain size, a phenomenon known as the Hall–Petch effect5,6. Here we present computer simulations of the deformation of nanocrystalline copper, which show a softening with grain size (a reverse Hall–Petch effect3,7) for the smallest sizes. Most of the plastic deformation is due to a large number of small ‘sliding’ events of atomic planes at the grain boundaries, with only a minor part being caused by dislocation activity in the grains; the softening that we see at small grain sizes is therefore due to the larger fraction of atoms at grain boundaries. This softening will ultimately impose a limit on how strong nanocrystalline metals may become.

Softening of nanocrystalline metals at very small grain sizes

The general expressions, valid for any potential, for the calculation of elastic constants through computer simulation are given. At zero temperature, the elastic constants are found to be the sum of a generalization of the Born term and a term accounting for internal relaxations that arise when a system with more than one atom in the primitive unit cell is strained. The fluctuation formulae used in finite temperature simulations are found to be straightforward generalizations of those used for pair potentials. The connection between the finite‐temperature and zero‐temperature methods is also made.

Generalized expressions for the calculation of elastic constants by computer simulation

The local stress tensor commonly used in statistical mechanics is cast in a form useful for molecular dynamics (MD) simulations. It is then used to derive fluctuation formulas for the local elastic constants of a system. The formulas are used in an MD measurement of the local elastic constants of an ideal crystal which are found to be in good agreement with the bulk elastic constants of the same system.

Stress and elastic constants in anisotropic solids: Molecular dynamics techniques

We present classical density functional theory calculations of the free-energy landscape for fluids below their triple point as a function of density and crystallinity. We find that, both for a model globular protein and for a simple atomic fluid modeled with a Lennard-Jones interaction, it is free-energetically easier to crystallize by passing through a metastable dense fluid in accord with the Ostwald rule of stages but in contrast to the alternative of ordering and densifying at once as assumed in the classical picture of crystallization.

Theoretical evidence for a dense fluid precursor to crystallization.

The high-temperature behavior of a high-angle twist grain boundary, a free surface, and planar arrays of voids of various sizes, all on the (001) plane in copper, are studied through molecular-dynamics simulation using an embedded-atom-method potential. Independently, we determine the thermodynamic melting point, {ital T}{sub {ital m}} of this potential through an analysis of the free energies of a perfect crystal and the liquid phase. It is found that an ideal crystal consisting of nearly 1000 atoms may be superheated over 200 K above {ital T}{sub {ital m}} while the introduction of any of the defects listed above nucleates melting at any temperature above {ital T}{sub {ital m}}. We conclude that nucleation of the liquid phase at extrinsic defects is the most rapid, and therefore the dominant, mechanism of melting.

Molecular-dynamics study of lattice-defect-nucleated melting in metals using an embedded-atom-method potential.

A fuzzy decision tree is constructed by allowing the possibility of partial membership of a point in the nodes that make up the tree structure. This extension of its expressive capabilities transforms the decision tree into a powerful functional approximant that incorporates features of connectionist methods, while remaining easily interpretable. Fuzzification is achieved by superimposing a fuzzy structure over the skeleton of a CART decision tree. A training rule for fuzzy trees, similar to backpropagation in neural networks, is designed. This rule corresponds to a global optimization algorithm that fixes the parameters of the fuzzy splits. The method developed for the automatic generation of fuzzy decision trees is applied to both classification and regression problems. In regression problems, it is seen that the continuity constraint imposed by the function representation of the fuzzy tree leads to substantial improvements in the quality of the regression and limits the tendency to overfitting. In classification, fuzzification provides a means of uncovering the structure of the probability distribution for the classification errors in attribute space. This allows the identification of regions for which the error rate of the tree is significantly lower than the average error rate, sometimes even below the Bayes misclassification rate.

James F. Lutsko

Papers

Generalized expressions for the calculation of elastic constants by computer simulation

Stress and elastic constants in anisotropic solids: Molecular dynamics techniques

Theoretical evidence for a dense fluid precursor to crystallization.

Molecular-dynamics study of lattice-defect-nucleated melting in metals using an embedded-atom-method potential.

Globally optimal fuzzy decision trees for classification and regression