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

The thermodynamic properties of 154 mineral end-members, 13 silicate liquid end-members and 22 aqueous fluid species are presented in a revised and updated data set. The use of a temperature-dependent thermal expansion and bulk modulus, and the use of high-pressure equations of state for solids and fluids, allows calculation of mineral–fluid equilibria to 100 kbar pressure or higher. A pressure-dependent Landau model for order–disorder permits extension of disordering transitions to high pressures, and, in particular, allows the alpha–beta quartz transition to be handled more satisfactorily. Several melt end-members have been included to enable calculation of simple phase equilibria and as a first stage in developing melt mixing models in NCKFMASH. The simple aqueous species density model has been extended to enable speciation calculations and mineral solubility determination involving minerals and aqueous species at high temperatures and pressures. The data set has also been improved by incorporation of many new phase equilibrium constraints, calorimetric studies and new measurements of molar volume, thermal expansion and compressibility. This has led to a significant improvement in the level of agreement with the available experimental phase equilibria, and to greater flexibility in calculation of complex mineral equilibria. It is also shown that there is very good agreement between the data set and the most recent available calorimetric data.

An internally consistent thermodynamic data set for phases of petrological interest

A revised regular solution-type thermodynamic model for twelve-component silicate liquids in the system SiO2-TiO2-Al2O3-Fe2O3-Cr2O3-FeO-MgO-CaO-Na2O-K2O-P2O5-H2O is calibrated. The model is referenced to previously published standard state thermodynamic properties and is derived from a set of internally consistent thermodynamic models for solid solutions of the igneous rock forming minerals, including: (Mg,Fe2+,Ca)-olivines, (Na,Mg,Fe2+,Ca)M2 (Mg,Fe2+, Ti, Fe3+, Al)M1 (Fe3+, Al,Si)2TETO6-pyroxenes, (Na,Ca,K)-feldspars, (Mg,Fe2+) (Fe3+, Al, Cr)2O4-(Mg,Fe2+)2 TiO4 spinels and (Fe2+, Mg, Mn2+)TiO3-Fe2O3 rhombohedral oxides. The calibration utilizes over 2,500 experimentally determined compositions of silicate liquids coexisting at known temperatures, pressures and oxygen fugacities with apatite ±feldspar ±leucite ±olivine ±pyroxene ±quartz ±rhombohedral oxides ±spinel ±whitlockite ±water. The model is applicable to natural magmatic compositions (both hydrous and anhydrous), ranging from potash ankaratrites to rhyolites, over the temperature (T) range 900°–1700°C and pressures (P) up to 4 GPa. The model is implemented as a software package (MELTS) which may be used to simulate igneous processes such as (1) equilibrium or fractional crystallization, (2) isothermal, isenthalpic or isochoric assimilation, and (3) degassing of volatiles. Phase equilibria are predicted using the MELTS package by specifying bulk composition of the system and either (1) T and P, (2) enthalpy (H) and P, (3) entropy (S) and P, or (4) T and volume (V). Phase relations in systems open to oxygen are determined by directly specifying the fo2 or the T-P-fo2 (or equivalently H-P-fo2, S-P-fo2, T-V-fo2) evolution path. Calculations are performed by constrained minimization of the appropriate thermodynamic potential. Compositions and proportions of solids and liquids in the equilibrium assemblage are computed.

Chemical mass transfer in magmatic processes IV. A revised and internally consistent thermodynamic model for the interpolation and extrapolation of liquid-solid equilibria in magmatic systems at elevated temperatures and pressures

Major and trace element composition of the depleted MORB mantle (DMM)

Knowledge of temperature and pressure, however qualitative, has been central to our views of geology since at least the early 19th century. In 1822, for example, Charles Daubeny presented what may be the very first “Geological Thermometer,” comparing temperatures of various geologic processes (Torrens 2006). Daubeny (1835) may even have been the first to measure the temperature of a lava flow, by laying a thermometer on the top of a flow at Vesuvius—albeit several months following the eruption, after intervening rain (his estimate was 390°F). In any case, pressure ( P ) and temperature ( T ) estimation lie at the heart of fundamental questions: How hot is Earth, and at what rate has the planet cooled. Are volcanoes the products of thermally driven mantle plumes? Where are magmas stored, and how are they transported to the surface—and how do storage and transport relate to plate tectonics? Well-calibrated thermometers and barometers are essential tools if we are to fully appreciate the driving forces and inner workings of volcanic systems.

This chapter presents methods to estimate the P-T conditions of volcanic and other igneous processes. The coverage includes a review of existing geothermometers and geobarometers, and a presentation of approximately 30 new models, including a new plagioclase-liquid hygrometer. Our emphasis is on experimentally calibrated “thermobarometers,” based on analytic expressions using P or T as dependent variables. For numerical reasons (touched on below) such expressions will always provide the most accurate means of P-T estimation, and are also most easily employed. Analytical expressions also allow error to be ascertained; in the absence of estimates of error, P-T estimates are nearly meaningless. This chapter is intended to complement the chapters by Anderson et al. (2008), who cover granitic systems, and by Blundy and Cashman (2008) and Hansteen and Klugel (2008), who consider additional methods …

Thermometers and Barometers for Volcanic Systems

[1] We describe a newly calibrated model for the thermodynamic properties of magmatic silicate liquid. The new model, pMELTS, is based on MELTS [Ghiorso and Sack, 1995] but has a number of improvements aimed at increasing the accuracy of calculations of partial melting of spinel peridotite. The pMELTS algorithm uses models of the thermodynamic properties of minerals and the phase equilibrium algorithms of MELTS, but the model for silicate liquid differs from MELTS in the following ways: (1) The new algorithm is calibrated from an expanded set of mineral-liquid equilibrium constraints from 2439 experiments, 54% more than MELTS. (2) The new calibration includes mineral components not considered during calibration of MELTS and results in 11,394 individual mineral-liquid calibration constraints (110% more than MELTS). Of these, 4924 statements of equilibrium are from experiments conducted at elevated pressure (200% more than MELTS). (3) The pMELTS model employs an improved liquid equation of state based on a third-order Birch-Murnaghan equation, calibrated from high-pressure sink-float and shockwave experiments to 10 GPa. (4) The new model employs a revised set of end-member liquid components. The revised components were chosen to better span liquid composition-space. Thermodynamic properties of these components are optimized as part of the mineral-liquid calibration. Comparison of pMELTS to partial melting relations of spinel peridotite from experiments near 1 GPa indicates significant improvements relative to MELTS, but important outstanding problems remain. The pMELTS model accurately predicts oxide concentrations, including SiO2, for liquids from partial melting of MM3 peridotite at 1 GPa from near the solidus up to � 25% melting. Compared to experiments, the greatest discrepancy is for MgO, for which the calculations are between 1 and 4% high. Temperatures required to achieve a given melt fraction match those of the experiments near the solidus but are � 60� C high over much of the spinel lherzolite melting interval at this pressure. Much of this discrepancy can probably be attributed to overstabilization of clinopyroxene in pMELTS under these conditions. Comparison of pMELTS calculations to the crystallization and partial melting experiments of Falloon et al. [1999] shows excellent agreement but also suffers from exaggerated calculated stability of clinopyroxene. Finally, comparison of pMELTS

https://agupubs.onlinelibrary.wiley.com/doi/epdf/10.1029/2001GC000217

The pMELTS: A revision of MELTS for improved calculation of phase relations and major element partitioning related to partial melting of the mantle to 3 GPa

Silicic magma systems are of great scientific interest and societal importance owing to their role in the evolution of the crust and the hazards posed by volcanic eruptions. MELTS is a powerful and widely used tool to study the evolution of magmatic systems over a wide spectrum of compositions and conditions. However, the current calibration of MELTS fails to correctly predict the position of the quartz þ feldspar saturation surface in temperature, pressure and composition space, making it unsuitable to study silicic systems. We create a modified calibration of MELTS optimized for silicic systems, dubbed rhyolite-MELTS, using early erupted Bishop pumice as a reference. Small adjustments to the calorimetrically determined enthalpy of formation of quartz and of the potassium end-member of alkali feldspar in the MELTS calibration lead to much improved predictions of the quartz þ feldspar saturation surface as a function of pressure. Application of rhyolite-MELTS to the Highland Range Volcanic Sequence (Nevada), the Peach Spring Tuff (Arizona^Nevada^California), and the late-erupted Bishop Tuff (California), using compositions that vary from trachydacite to high-silica rhyolite, shows that the calibration is appropriate for a variety of fluid-bearing silicic systems. Some key observations include the following. (1) The simulated evolutionary paths are consistent with petrographic observations and glass compositions; further work is needed to compare predicted and observed mineral compositions. (2) The nearly invariant nature of silicic magmas is well captured by rhyolite-MELTS; unusual behavior is observed after extensive pseudo-invariant crystallization, suggesting that the new calibration works best for relatively small (i.e.550 wt %) crystallization intervals, comparable with what is observed in volcanic rocks. (3) Our success with rhyolite-MELTS shows that water-bearing systems in which hydrous phases do not play a critical role can be appropriately handled; simulations are sensitive to initial water concentration, and although only a pure-H2O fluid is modeled, suitable amounts of water can be added or subtracted to mimic the effect of CO2 in fluid solubility. Our continuing work on natural systems shows that rhyolite-MELTS is very useful in constraining crystallization conditions, and is particularly well suited to explore the eruptive potential of silicic magmas. We show that constraints placed by rhyoliteMELTS simulations using late-erupted Bishop Tuff whole-rock and melt inclusion compositions are inconsistent with a vertically stratified magma body.

/pdf/rhyolite-melts-a-modified-calibration-of-melts-optimized-for-3k4qdt5pkg.pdf

Rhyolite-MELTS: a Modified Calibration of MELTS Optimized for Silica-rich, Fluid-bearing Magmatic Systems

Algorithmic modifications to the MELTS software package are presented in order that calculations of heterogeneous phase equilibria can be performed in the subsolidus. Methods
are presented for: (1) selecting an "initial guess assemblage" that satisfies the bulk composition constraints; (2) detecting saturation of new phases (including liquid) in an assemblage; (3) adding and removing phases from the assemblage without adjusting the system bulk composition; and (4) constraining the assemblage to a fixed f_(O2). These methods have O2 been added to MELTS, allowing it to calculate heterogeneous phase equilibria with or without liquid, closed or open to O, and with fixed intensive variables (P,T), (P,S), (P,H), or (V,T). Applications include fractional melting calculations, metamorphic phase equilibria, and geophysical models of subsolidus regions of the Earth.

/pdf/algorithmic-modifications-extending-melts-to-calculate-1kuwncn6va.pdf

Algorithmic modifications extending MELTS to calculate subsolidus phase relations

Thermodynamic and mathematical relations are presented to facilitate the description of an algorithm for the calculation of chemical mass transfer in magmatic systems. This algorithm extends the silicate liquid solution model of Ghiorso et al. (1983) to allow for the quantitative modelling of natural magmatic processes such as crystal fractionation, equilibrium crystallization, magma mixing and solid-phase assimilation. The algorithm incorporates a new method for determining the saturation surface of a non-ideal multicomponent solid-solution crystallizing from a melt. It utilizes a mathematical programming (optimization) approach to determine the stable heterogeneous (solids+liquid) equilibrium phase assemblage at a particular temperature and pressure in magmatic systems both closed and open to oxygen. Closed system equilibria are computed by direct minimization of the Gibbs free energy of the system. Open system equilibria are determined by minimization of the Korzhinskii potential (Thompson 1970), where oxygen is treated as a perfectly mobile component. Magmatic systems undergoing chemical mass transfer processes are modelled in a series of discrete steps in temperature, pressure or bulk composition, with each step characterized by heterogeneous solid-liquid equilibrium. A numerical implementation of the algorithm has been developed (in the form of a FORTRAN 77 computer program) and calculations demonstrating its utility are provided in an accompanying paper (Ghiorso and Carmichael 1985).

Mark S. Ghiorso

Papers

Chemical mass transfer in magmatic processes IV. A revised and internally consistent thermodynamic model for the interpolation and extrapolation of liquid-solid equilibria in magmatic systems at elevated temperatures and pressures

The pMELTS: A revision of MELTS for improved calculation of phase relations and major element partitioning related to partial melting of the mantle to 3 GPa

Rhyolite-MELTS: a Modified Calibration of MELTS Optimized for Silica-rich, Fluid-bearing Magmatic Systems

Algorithmic modifications extending MELTS to calculate subsolidus phase relations

Chemical mass transfer in magmatic processes