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

Three recently proposed constant temperature molecular dynamics methods by: (i) Nose (Mol. Phys., to be published); (ii) Hoover et al. [Phys. Rev. Lett. 48, 1818 (1982)], and Evans and Morriss [Chem. Phys. 77, 63 (1983)]; and (iii) Haile and Gupta [J. Chem. Phys. 79, 3067 (1983)] are examined analytically via calculating the equilibrium distribution functions and comparing them with that of the canonical ensemble. Except for effects due to momentum and angular momentum conservation, method (1) yields the rigorous canonical distribution in both momentum and coordinate space. Method (2) can be made rigorous in coordinate space, and can be derived from method (1) by imposing a specific constraint. Method (3) is not rigorous and gives a deviation of order N−1/2 from the canonical distribution (N the number of particles). The results for the constant temperature–constant pressure ensemble are similar to the canonical ensemble case.

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A unified formulation of the constant temperature molecular dynamics methods

Glasses are disordered materials that lack the periodicity of crystals but behave mechanically like solids. The most common way of making a glass is by cooling a viscous liquid fast enough to avoid crystallization. Although this route to the vitreous state-supercooling-has been known for millennia, the molecular processes by which liquids acquire amorphous rigidity upon cooling are not fully understood. Here we discuss current theoretical knowledge of the manner in which intermolecular forces give rise to complex behaviour in supercooled liquids and glasses. An intriguing aspect of this behaviour is the apparent connection between dynamics and thermodynamics. The multidimensional potential energy surface as a function of particle coordinates (the energy landscape) offers a convenient viewpoint for the analysis and interpretation of supercooling and glass-formation phenomena. That much of this analysis is at present largely qualitative reflects the fact that precise computations of how viscous liquids sample their landscape have become possible only recently.

Supercooled liquids and the glass transition

The theory of phase-ordering dynamics that is the growth of order through domain coarsening when a system is quenched from the homogeneous phase into a broken-symmetry phase, is reviewed, with the emphasis on recent developments. Interest will focus on the scaling regime that develops at long times after the quench. How can one determine the growth laws that describe the time dependence of characteristic length scales, and what can be said about the form of the associated scaling functions? Particular attention will be paid to systems described by more complicated order parameters than the simple scalars usually considered, for example vector and tensor fields. The latter are needed, for example, to describe phase ordering in nematic liquid crystals, on which there have been a number of recent experiments. The study of topological defects (domain walls, vortices, strings and monopoles) provides a unifying framework for discussing coarsening in these different systems.

Theory of phase-ordering kinetics

Although it has long been recognized that dynamics in supercooled liquids might be spatially heterogeneous, only in the past few years has clear evidence emerged to support this view. As a liquid is cooled far below its melting point, dynamics in some regions of the sample can be orders of magnitude faster than dynamics in other regions only a few nanometers away. In this review, the experimental work that characterizes this heterogeneity is described. In particular, the following questions are addressed: How large are the heterogeneities? How long do they last? How much do dynamics vary between the fastest and slowest regions? Why do these heterogeneities arise? The answers to these questions influence practical applications of glass-forming materials, including polymers, metallic glasses, and pharmaceuticals.

Spatially Heterogeneous Dynamics in Supercooled Liquids

This article presents a calculation of the infinite‐frequency elastic moduli of monatomic fluids. When the intermolecular potential has the standard Lennard‐Jones form, the elastic moduli are shown to be related to the pressure and internal energy of the fluid. Numerical values of the elastic moduli at various densities and temperatures are presented in tabular and graphical form. An identity is derived relating the shear and bulk moduli of any isotropic material in which particles interact by means of two‐body central forces; this is a generalization of the familiar Cauchy identity occurring in the theory of elasticity of solids. The approach is based on analysis of the high‐frequency limit of exact time‐correlation function expressions for shear and bulk viscosity.

High‐Frequency Elastic Moduli of Simple Fluids

The spectral distribution of light scattered by density fluctuations in a dense, monatomic, one-component fluid is calculated from the time dependence of the density fluctuations predicted by the linearized hydrodynamic equations of irreversible thermodynamics. The results of Landau and Placzek are verified and a procedure for deriving correction terms is discussed with the dispersion in the velocity of thermal sound waves obtained as an illustration. Particular attention is paid to the critical region. The properties of carbon dioxide are used to estimate the spectral distribution of critical opalescence. A comparison is made between light-scattering and sound-propagation experiments. Space dispersion near the critical point in the pressure and the thermal conductivity is examined briefly. Finally, some of the experimental problems involved in measuring the spectral distribution of the scattered light are discussed.

Spectral Distribution of Scattered Light in a Simple Fluid

Weak coupling of internal degrees of freedom of molecules to the translational degrees of freedom of a fluid results in additional modes of motion for density fluctuations. These new modes affect the spectral distribution of light scattered by density fluctuations so that the Landau-Placzek ratio is not satisfied. The case of thermal relaxation with a single relaxation time is worked out in detail. Formulas for the spectral distribution of the scattered light, for the ratio of the intensities of the central (Rayleigh) to the Brillouin components and for the phonon velocity are derived and applied to carbon disulfide and carbon tetrachloride. The data for carbon tetrachloride are shown to be inconsistent with the single relaxation time model for thermal relaxation.

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Raymond D. Mountain

Papers

High‐Frequency Elastic Moduli of Simple Fluids

Spectral Distribution of Scattered Light in a Simple Fluid

Thermal Relaxation and Brillouin Scattering in Liquids.

Velocity Verlet algorithm for dissipative-particle-dynamics-based models of suspensions

Ergodic behavior in supercooled liquids and in glasses.