We present a new straightforward algorithm that calculates the energy minimization of a melt-present system and incorporates newly calibrated silicate melt thermodynamic parameters. This algorithm searches for equilibrium phase assemblages, fractions, and compositions that lead to the system having a global minimum of total Gibbs free energy (G). It calculates changes in G with respect to minimal amounts of dissolution or solidification of melt and solid end-member components using a constant bulk composition constraint. In addition, we have formulated a set of solid-melt end-member components and dissolution-precipitation stoichiometry that enables the modeling of a melt-present system. Melt thermodynamic properties are calibrated based on an ideal mixing model using ΔCp and ΔV (the differences in molar specific heat and volume between the corresponding melt and solid end-member components, respectively), based on solid properties established during previous studies. We also describe the application of the energy minimization algorithm and thermodynamic melt parameters to melting of spinel lherzolite at 1 GPa, in a SiO2–Al2O3–FeO–Fe3O4–MgO–CaO system, including olivine, clinopyroxene, orthopyroxene, and spinel. Our calculations agree well with experimentally determined melting phase relations, and temperature and phase fraction relationships, including solidus temperatures, indicating that direct calibration of thermodynamic melt parameters at pressures and temperatures corresponding to melting conditions is a useful approach. The energy minimization algorithm and thermodynamic configuration presented here will allow the modeling of mantle melting in a variety of geodynamic settings.
 Partial melting is an essential process involved in both material differentiation and heat transportation by the production and migration of melts, and causes present day mid-ocean ridge, hotspot, and subduction zone magmatism. Partial melting in the early Earth may have been even more important than in the present day [e.g., Labrosse et al., 2007; Lee et al., 2010], meaning that detailed knowledge of the processes involved in melting is an important step in improving our understanding of the material differentiation and the thermal evolution of the Earth.
 In general, melting is associated with dynamic processes, such as relative motion and chemical reactions between melts and solids, both of which redistribute mass and energy, with these processes being coupled and evolving with time. This means that modeling of these processes needs an approach that simultaneously solves both mass and energy balance. One consistent approach to modeling of these systems is thermodynamic calculations that minimize the free energy of the system.
 Several thermodynamic models have been presented to calculate melting phase relations, including MELTS and pMELTS [Ghiorso, 1994; Ghiorso and Sack, 1995; Ghiorso et al., 2002], and Perplex [Connolly, 2005]. MELTS and pMELTS have been widely used to calculate equilibrium phase relations for melt-present systems. Interaction parameters for silicate melt end-member components are calibrated assuming a regular solution model, with reference state properties derived by extrapolation from pure end-member component systems at 1 bar pressure [Ghiorso and Sack, 1995; Ghiorso et al., 2002; Hirschmann et al., 1998b]. Phase relations and compositions at fixed bulk compositions are then calculated to satisfy equilibrium melt and mineral compositions by evaluating “affinity”, which represents the degree of saturation of each phase in a melt [Ghiorso, 1994]. Put simply, the energy minimization algorithm used in MELTS/pMELTS modeling calculates the saturation state and equilibrium composition of solid phases in a melt. In this sense, MELTS and pMELTS could be referred to as “composition-based” models constructed specifically for melt-present systems.
 In comparison, Perplex calculates the phase equilibria of an arbitrary chosen system based on direct minimization of total Gibbs free energy [Connolly, 2005]. The nonlinear relationship between chemical potential and composition of the phase is described at finite discrete compositions for each phase, instead of as a continuous function. The thermodynamic models and properties can be chosen from a previously compiled database, all of which are readily incorporated into Perplex. As defaults, the THERMOCALC data set [e.g., Holland and Powell, 1998] for rock-forming minerals, and pMELTS or MELTS for mafic silicate melts is recommended [Connolly, 2005]. Although the flexibility of the system is a useful advantage, internally consistent thermodynamic models and properties need to be used to ensure accurate modeling, especially when modeling silicate melts.
 These thermodynamic models use different energy minimization algorithms, thermodynamic configurations, and parameters. These models should ideally reproduce experimental results within experimental uncertainties. However, as will be shown in this paper, these models exhibit significant variability in reproducibility. In order to improve these modeling approaches, we propose a new algorithm for system energy minimization and a relatively simple thermodynamic model that is appropriate for modeling of melt-present systems and that incorporates newly calibrated parameter values. For bulk-composition-constrained minimization of system energy, we have developed a straightforward formulation in continuous pressure, temperature, and composition space, and as an example, we model a peridotite system at a fixed pressure using the formulation and algorithm presented here. Our model accurately reproduces the relationship between temperature and phase fractions, including solidus temperature and degree of melting, variables that if constrained can be useful for dynamical modeling of melt generation, including melting and melt migration. The widespread nature of polybaric melting, for example at mid-ocean ridges and hotspots, means that we are currently expanding the calibration database and the range of pressures that our modeling encompasses, using the energy minimization algorithm and thermodynamic configuration outlined in this study; this expansion will be presented separately.
 Here we describe a general formulation for energy minimization in a system that consists of multiple solution phases and components. We then describe more specific configurations and equations, such as the chemical components within melt and solid phases, melt-solid reactions, and corresponding thermodynamic expressions, including chemical potentials for individual components. The formulation for energy minimization is valid irrespective of the specific configuration used; here, we use a dry peridotitic system as a representative example of the modeling possible using this approach. This system, melting under conditions of the uppermost upper mantle, is experimentally well constrained, and we consider this system to be an appropriate context for the formulation and parameters employed in calculating the phase equilibria.
 The energy minimization algorithm used in this model is modified for systems including solid solutions and silicate melts, extending the method and formulation of Albaréde , who modeled a multicomponent system that only contained phases with fixed compositions.
2.1 General Formulation for Energy Minimization
 Here we consider a multicomponent system with a fixed bulk chemical composition, using the variables and constants listed in Table 1. The total Gibbs free energy of the system, including m end-member components, is defined as G, and, at a given P, T, and bulk composition, is expressed as
where ni represents the molar content of an ith end-member component in the system (i = 1, 2, ⋯, m), n is a column vector consisting of n1,2,⋯,m, μi represents the molar Gibbs free energy of the ith end-member component, and μ is an m column vector consisting of n1,2,⋯,m. Stable phase assemblages and compositions at given P, T, and bulk compositions are calculated by minimizing the Gibbs free energy of the system in equation (1), with mass conservation with stoichiometric constraints in a closed system defined as
where B is the stoichiometric coefficient of a m × c dimensional matrix, q denotes the bulk composition of the system (molar contents of oxides) of a c-dimensional vector, c is the number of oxides included in the given system, and m is the number of end-member components within the system. Each n component must have a non-negative value to satisfy the mass conservation constraint involved in this modeling.
Table 1. Definition of Notation Used in the Text
Activity coefficient of an end-member component i, which is written as ai = γiXi where γi is a nonideality coefficient of an end-member component i and Xi is a molar fraction of an end-member component i in a phase. γi = 1 corresponds to ideal solution.
m × c dimensional matrix that constraints the stoichiometry of end-member components. m denotes a number of chemical species and c denotes a number of end-member components included in the system.
Molar specific heat of an end-member component i.
Difference between molar specific heats of solid end-member component j and melt end-member component i, written as .
Total Gibbs free energy of the system, written as .
Change in G against the of mass perturbation δn.
Molar enthalpy of an end-member component i.
Reference state molar enthalpy of an end-member component i.
Enthalpy of fusion of a solid end-member component i at and 1 bar.
Molar content of an end-member component i in the system.
One-dimensional compositional vector consisting of c rows.
One-dimensional compositional vector consisting of m−c rows.
One-dimensional mass perturbation vector.
Chemical potential of an end-member component i in a phase, which is written as μi = μi0 + RTlnai, where μi0 is a chemical potential of a pure end-member component i at the reference state.
Difference between chemical potentials of melt end-member component i and solid end-member component j, written as .
Change in μi with respect to the mass perturbation vector δn.
m dimensional vector describing a bulk composition of the system (molar contents of oxide end-member components).
Molar entropy of an end-member component i.
Reference state molar entropy of an end-member component i.
Entropy of fusion of a solid end-member component i at and 1 bar.
Melting temperature of a solid end-member component i at 1 bar.
Molar volume of an end-member component i.
Difference between molar volumes of solid end-member component j and melt end-member component i, written as .
 Assuming a c-component system involving m end-member components (m ≥ c), mass conservation therefore provides c equations, with the number of independent end-member components given by m − c. In order to find the value of n that minimizes the Gibbs free energy of the system, we first split n into two blocks, nB and nF, as
where nB is a one-dimensional matrix consisting of c rows, and nBTBB yields a minimum set of equations that reproduce the given bulk composition q, where BB represents a c × c dimensional matrix. The c end-member components in nB are chosen to enable BB to be a regular matrix, with vector nF for the remaining m − c end-member components regarded as free parameters, with a relationship to nB that is expressed as
 Equation (4) expresses the mass conservation involved in stoichiometric reactions. The relationship of a tiny perturbation of mass, δnF to δnB, is therefore written as
where δnB and δnF are vectors with c and m − c rows, respectively. When mass perturbation is applied to all end-member components in nF in equation (5), the components in nB will automatically be perturbed. This means that it is possible to test all possible combinations of n under constant bulk composition conditions.
δG, a change in G with respect to a tiny perturbation in mass δn, is written as
By substituting equation (5) into equation (6), δG with respect to a given perturbation of mass δnF is written as
 The molar content of each end-member component in the system (i.e., n) is modified according to the steepest gradient, quantified as the direction that reduces G most effectively during each calculation step. This minimization procedure reaches its goal if δG becomes positive with respect to any mass perturbation, with a solution vector n that contains the molar content of each end-member component that is stable at a given temperature, pressure, and bulk composition of the system. It should be noted that end-member components that are expected to appear in the system must be imported in vector n, and if all end-member components of a single phase are zero after minimization, this specific phase is unstable under these conditions.
 The chemical potential of an ith end-member component (μi) is expressed as , where μi0 is the chemical potential of a pure end-member component i at an arbitrary chosen reference T and P, ai is an activity coefficient, and R is the gas constant. ai is further written as ai = γiXi, where γi is a nonideality coefficient, and Xi is the molar fraction of an ith end-member component in a phase. This means that, δμi is defined as
with equation (8) substituted into equation (7) to calculate δG.
2.2 Specific Configurations: Melt-Solid Reaction
 In order to practically implement this algorithm, we need to explore a specific model that describes the dissolution of solid end-member components into a melt, where solid end-member components are assigned to a matrix nF, and melt end-member components are assigned to a matrix nB. Assuming an m end-member component system where melt end-member components have identical compositions to their corresponding solid end-member components, the number of solid end-member components is equal to the number of melt end-member components in a system with m end-member components. Hence, nB and nF are vectors with m/2 rows, and we can arrange nB and nF to satisfy
In this specific case, equation (7) can be simplified as
The chemical potential of an ith end-member component (μi) at a given temperature T and pressure P is written as
where is the molar enthalpy and is the molar entropy of a pure end-member component i at an arbitrary chosen reference T and P.
 and values for a pure solid end-member component j are calculated using
where and are the molar enthalpy and entropy of an end-member component j at 298 K and 1 bar, is the specific heat of an end-member component j, which is a function of T, and is the molar volume of an end-member component j, which is a function of P and T.
 By substituting equations (12) and (13) into equation (11), we can determine the chemical potential of a solid end-member component j as
In addition, and of a pure melt end-member component i can be written using the melting properties of a corresponding solid end-member component j at 1 bar:
where a melt end-member component i has a composition identical to that of a solid end-member component j, is the melting temperature, and and are the enthalpy and entropy values during melting of a solid component j at 1 bar. By substituting equations (15) and (16) into equation (11), we can obtain the chemical potential of a melt end-member component i as
 We also define a vector Δμ as the differences in chemical potentials between the melt and the corresponding solid end-member components. Δμ consists of Δμ1,2,⋯,m/2, and is expressed as
By substituting equations (13) and (14) into equation (18), we have derived the following equation to calculate the molar Gibbs free energy of the silicate melt end-member component:
Furthermore, we can rearrange equation (19) by substituting in ΔCpi and ΔVi
where and . For simplicity, we assume that the difference between the specific heats of melt and solid end-member components, ΔCpi, is independent of temperature or pressure in the relatively limited range of P and T considered in this study (i.e., 1 GPa and 1230°C–1500°C). We similarly assume that the difference between the molar melt and solid end-member component volumes, ΔVi, is constant. It should be noted that proper formulations for ΔCpi and ΔVi should be chosen for the specific system being modeled, with the values chosen here being representative of the peridotite system that is the focus of this study.
 By introducing Δμ into equation (10), the change in G with respect to δnsolid, representing the solidification or dissolution of solid end-member components, can be expressed as
The G of melt-present systems is minimized using equations (8), (20), and (21).
 Here we consider a simple melting of a Mg2SiO4 forsterite end-member component system as an example. Below the solidus temperature at a fixed pressure, is greater than zero, meaning that δG with respect to a given positive δnsolid value is negative, resulting in a tiny amount of solidification and an increase in nsolid. This process then repeats until complete solidification is reached. Above the solidus temperature, the reaction proceeds in the opposite direction, resulting in total melting.
2.3 Specific Configurations: Solid and Melt Thermodynamic Modeling
 An anhydrous spinel lherzolite system at 1 GPa was selected as a test of the energy minimization algorithm outlined in sections 2.1 and 2.2. The system is experimentally well constrained and is also suitable for parametric calibration. A sufficient number of experimental results allows us to test the accuracy of various aspects of our model, including phase composition, phase stability (including solidus temperature), and phase content, including melt fraction. We will evaluate the pressure dependence of the thermodynamic melt parameters elsewhere as described in the introduction.
 A total of six oxide components, SiO2–MgO–FeO–Fe3O4–CaO–Al2O3, allows the description of solid solution relationships between mineral end-member components in a spinel lherzolite assemblage, and are used to describe a system with a maximum of five solid solution phases: olivine, clinopyroxene (cpx), orthopyroxene (opx), spinel, and silicate melt.
2.3.1 Solid Phase End-Member Components and Activity Modeling
 In order to cover the whole compositional range of experimentally determined mineral compositions involved in the melting of spinel lherzolite, the following end-member components are defined: forsterite and fayalite in olivine; enstatite, ferrosilite, diopside, and Ca-tschermak in cpx; enstatite, ferrosilite and Mg-tschermak in opx; and spinel and magnetite in spinel (Table 2), with minor elements in each mineral (e.g., Ca in olivine and opx, and Fe3+ in pyroxene) disregarded in this model. The following activity models are employed with these end-member components: a two-site regular solution mixing model for olivine [Sack and Ghiorso, 1989], a nonideal activity model for spinel [Sack, 1982], and a three-site ideal mixing model for opx and cpx [e.g., Newton, 1983; Nielsen and Drake, 1979]. Additional explanations that justify our selection of mixing models are given in Appendix A.
Table 2. Compositions and Parametric Values of Solid Phases Used in This Study, Including References for Thermodynamic Parameters for Pure End-Member Components (μ0) and for Activity Models
2.3.2 End-Member Components and Melt Activity Model
 As shown in section 2.2, melt end-member components are defined to have identical compositions to their corresponding solid end-member components. We compiled and calibrated thermodynamic parameters, and calculated the molar Gibbs free energy for each melt end-member component using equation (20).
 Previous attempts to model silicate melts indicated that mixing between oxide components can describe their thermodynamic behaviors, including phase relations, immiscibility, and melt-solid compositions [e.g., Barron, 1981; Flood and Knapp, 1968; Ghiorso et al., 2002]. In order to cover the entire compositional space of experimentally determined spinel lherzolite partial melts, the following melt oxide end-member components are used here: SiO2, Al2O3, Mg2SiO4, Fe2SiO4, Fe3O4, CaSiO3. These melt oxide end-member components allow consistent descriptions of melting reactions, as outlined in section 2.2, including congruent and incongruent melting (see Appendix B), and are suitable for calibration of thermodynamic melt parameters. The relationship between melt end-member components with identical composition solid end-member components, and melt oxide end-member components can be expressed using the following matrix D (hereafter referred to as the dissolution matrix, Table 3):
where superscript “oc” indicates the melt oxide end-member component, and noc describes the melt composition.
Table 3. Dissolution Matrix D (Equation (22)) Used to Define Melt End-Member Component Compositions
 Ideal and nonideal mixing behaviors of end-member components in silicate melts have both been reported in previous studies, with ideal mixing models suggested for orthoclase-quartz, albite-quartz, and anorthite-quartz systems based on phase relation reproducibilities and melt calorimetric data [Ryerson, 1985; Navrotsky et al., 1980; Flood and Knapp, 1968]. In contrast, SiO2–MgO–metal oxide [Ryerson, 1985], Mg0.5Ca0.5SiO3 − CaAl2Si2O8 [Navrotsky et al., 1980; Navrotsky et al., 1989; Osborn, 1942; Stebbins et al., 1983; Sugawara, 2005; Sugawara and Akaogi, 2003], and CaSiO3 − CaAl2Si2O8 [Kosa et al., 1992; Tarina et al., 1994] systems potentially include significant deviations from ideal mixing. Melt immiscibility is also suggestive of the nonideal behavior of constituent components in basaltic melts [e.g., Philpotts, 1982]. This has led to regular solution models [Hildebrand and Salstrom, 1932] being widely used to describe the thermodynamic relations of silicate melts [e.g., Bottinga and Richet, 1978; Ghiorso et al., 1983; Médard and Grove, 2007], where the molar Gibbs free energies of pure melt oxide end-member components are calculated by extrapolating enthalpy, entropy, and specific heat in these simple systems at 1 bar. These models assume that the calibration of regular solution parameters may compensate for uncertainties in the speciation and structure of the melt end-member components and their interactions [e.g., Hirschmann et al., 1998b].
 Here we take a different approach that assumes ideal mixing of melt oxide end-member components (SiO2, Al2O3, Mg2SiO4, Fe2SiO4, Fe3O4, CaSiO3) during calculation of the melt composition-chemical potential relationship. We calibrate the specific heat (Cpi) and molar volume (Vi) of the melt end-member components, which are assumed to have constant offsets compared with their corresponding solid end-member components, as justified later. This indicates that ΔCpi and ΔVi are independent of temperature and pressure. No immiscibility gap has been observed in anhydrous spinel lherzolite-derived partial melts at 1 GPa, which is the focus of this study. In addition, peridotitic partial melts and residual minerals have a relatively restricted compositional range that involves a limited magnitude of excess Gibbs free energy. In the anhydrous spinel lherzolite system, calibration of the Cpi and Vi of silicate melt means that melting phase relations that might involve nonideality can be reproduced with reasonable accuracy, as shown below. In order to expand this thermodynamic calculation to hydrous or alkali-bearing systems, melt nonideality and parametric calibration is essential, primarily because these systems have significantly nonlinear relationships between melt compositions and phase relations [e.g., Médard and Grove, 2007; Hirschmann et al., 1998a].
 The large uncertainty in the relationship between melt composition and actual chemical species prevents the construction of a unique universal thermodynamic model for multicomponent silicate melts. Here we assume that parametric calibration can compensate for silicate melt structure uncertainties, and therefore calibrate thermodynamic parameters exclusive for the P-T-compositional range being considered in this study. This means that the set of thermodynamic melt parameters calibrated in this study cannot be applied to systems outside this range.
3 Reference State Properties and Parametric Calibration
3.1 Molar Gibbs Free Energy of Pure Solid and Melt End-Member Components
 The enthalpy, entropy, and volume of solid end-member components have been calculated using reported reference state properties, including molar specific heat, thermal expansion coefficients, and compressibility. The thermodynamic properties for forsterite, ferrosilite, orthoenstatite, orthoferrosilite, diopside, Ca-tschermak, spinel, and magnetite end-member components are taken from Berman , with those for clinoenstatite and clinoferrosilite taken from Sack and Ghiorso [1994b]; these properties are internally consistent with the thermodynamic properties reported by Berman . For the Mg-tschermak component, internally consistent thermodynamic properties have not been provided, and we use those of Holland and Powell . The thermodynamic properties reported by Berman  and Holland and Powell  are not mutually consistent, meaning that the molar Gibbs free energy of the Mg-tschermak component may to some extent be inconsistent with other values.
 As formulated in section 2.2, ΔCpi and ΔVi are assumed to be constant. As the target pressure is fixed at 1 GPa, the constant ΔVi may represent an average value over the relatively narrow temperature range discussed here (1230°C–1500°C). For ΔCpi, a weak temperature dependence is suggested based on the empirical relation for temperature-dependent Cp of crystallized basalt and basaltic melt [Bouhifd et al., 2007], where the standard deviation of ΔCpi values between basaltic melts and solids is only ± 4.32% in the 1000°C–1500°C temperature range.
 We optimize ΔCpi and ΔVi values for all melt end-member components, as well as and for some pyroxene end-member components (see below for details) to satisfy equilibrium constraints (Δμi = 0). A least squares method was used during parametric optimization. Fixed values were used for phase compositions from high-pressure melting experiments, previously reported activity coefficients for solid end-member components ( and ), and thermodynamic properties related to melting of solid end-member components at 1 bar (, , and ) during least squares regression optimization. The resulting parameter values, shown in bold in Table 4, and the standard deviations of calibration misfits for all melt end-member components are summarized in Table 4. These parameters were optimized to achieve a zero average with the lowest Δμi variance (Figure 1). The parametric calibrations for olivine, spinel, clinoenstatite and clinoferrosilite, diopside and Ca-tschermak, and opx are shown in Figures 1a–1e, respectively, where horizontal axes represent the Mg# (molar MgO/(FeO*+MgO) ratio) values of each mineral produced, and vertical axes represent optimal calculated Δμi values.
Table 4. Thermodynamic Parameters for Melt End-Member Components Used in This Study. Parameters Shown in Bold Were Derived by Calibration in This Study (Figure 1), With Data Sources for Other Thermodynamic Parameters Detailed in the Footnotes. These Values are Regarded as Input Parameters During Calibration. σ Denotes the Standard Deviation of the Misfits of Δμ During Calibration (see Figure 1), and the Results of the Following Melting Experiments Were Used During Parametric Calibration: Falloon et al. [1997, 1999, 2001], Kinzler and Grove , Laporte et al. , Pickering-Witter and Johnston , Schwab and Johnston , Wagner and Grove  and Wasylenki et al. 
Δμi is calibrated for ranges in composition, pressure, and temperature appropriate for melting of peridotite in the uppermost mantle. This calibration method is particularly useful with the thermodynamic configuration for silicate melts outlined here, more so than the alternative method of determining melt properties by extrapolation from pure melt end-member components, or from glasses, at 1 bar [e.g., Ghiorso and Sack, 1995], as this study incorporates several fictive melt end-member components; clinoferrosilite, orthoferrosilite, Ca-tschermak, and Mg-tschermak components are either unstable at 1 bar or do not undergo simple congruent melting. This indicates that and values at 1 bar for these mineral components cannot be experimentally determined. In addition to ΔCpi and ΔVi, and were calibrated for the mineral components, and are shown in bold in Table 4. An iterative process was used to optimize these parameters.
 The standard deviations of residuals during optimization are relatively small, comparable to or less than the uncertainties of experimentally determined thermodynamic parameters. This is exemplified by the experimental uncertainties on enthalpy of fusion values of ± 2.6 kJ for forsterite [Sugawara, 2005], ± 1.1 kJ for fayalite [Stebbins and Carmichael, 1984], ± 5.8 kJ for orthoenstatite [Sugawara, 2005], ± 1.0 kJ for diopside [Lange et al., 1991], ± 8.4 kJ for magnetite [Darken and Gurry, 1946], and ± 11.0 kJ for spinel [Richet et al., 1993], the majority of which are larger than the standard deviations of the current calibration range (from ± 0.6 to ± 4.4 kJ). The standard deviation of residuals during regression is thought to represent an estimate of the uncertainty on the calibrated thermodynamic parameters. For forsterite, the standard deviation of Δμ (= ± 1.64 kJ) corresponds to ± ∼ 5 [J/K-mol] of the maximum error on ΔCp (when assuming no error on ΔV) or ± 1.64 [kJ/GPa] of the maximum error on ΔV.
 Throughout this study, melt ferric-ferrous ratios are calculated from oxygen fugacity, using a set of revised equations involving oxygen (see Appendix C). The oxygen fugacity of each experimental run was calculated using the methods of Chou  and Myers and Eugster , although oxygen fugacity was not precisely controlled in some experiments, and this may affect the calibration for Fe-bearing end-member components such as ferrosilite, fayalite, and magnetite.
 The data sources used during calibration are shown in Table 4, with experimental data selected from the Library of Experimental Phase Relations open database of melting experiments [Hirschmann et al., 2008] using the following three criteria:
Nominally anhydrous run at 1 GPa.
An experimental system consisting of at least the following six oxide components: SiO2, Al2O3, MgO, FeO*, CaO, and Na2O.
The inclusion of olivine, orthopyroxene, clinopyroxene, and/or spinel without plagioclase and garnet.
 As a result, 138 individual runs from a total of 9 papers were used during calibration. The SiO2 and MgO contents of partial melts range from 47.9 to 54.7 wt %, and 7.7 to 21.4 wt %, respectively, with temperatures from 1230°C to 1500°C. These criteria, especially 1 and 3, limit the application of this model significantly, as polybaric melting occurs extensively in natural tectonic and magmatic settings (e.g., mid-ocean ridges and hotspots). The primary scope of this study is, however, to propose a new algorithm and to test it with well-constrained experimental data. The next section demonstrates that the algorithm and parameters presented here reproduce experimental results well within the calibrated P-T-compositional range.
4 Results and Discussion
 The calculated relationships between temperature and phase fractions, including melt, at 1 GPa are shown with experimentally determined phase relations and melt compositions (Figures 2 to 9). Three anhydrous bulk compositions, ranging from relatively depleted to fertile peridotitic compositions, were selected to test the reproducibility of the model: depleted MORB mantle DMM1 [Wasylenki et al., 2003], and relatively fertile mantle mm3 [Baker and Stolper, 1994; Baker et al., 1995; Falloon et al., 1999] and KLB-1 compositions [Takahashi, 1986; Hirose and Kushiro, 1993; Falloon et al., 1999]. The melting relationships of DMM1 and mm3 at 1 GPa are especially well constrained by a large number of experiments conducted over a wide range of temperatures. The experimental results of Wasylenki et al.  for DMM1 and Falloon et al.  for mm3 and KLB-1 were used to calibrate the parameters described in section 3. Baker and Stolper  and Baker et al.  reported the relationship between temperature, phase fractions, and phase compositions for mm3, with Takahashi  and Hirose and Kushiro  reporting the relationship between temperature, degree of melting, and melt composition for KLB-1, but not the compositions and proportions of residual solid phases. It should be noted that the experimental results for mm3 by Baker and Stolper  and Baker et al.  and KLB-1 by Takahashi  and Hirose and Kushiro  have not been used for parameter calibration, but were only used for testing the reproducibility of the model.
 In the following arguments, the uncertainties of the experimental results are important, and are noted below for the modal compositions of experimental run products: ± 0.1–2.0 wt % for silicate melt, ± 0.1–1.0 wt % for olivine, ± 0.1–2.0 wt % for opx, ± 0.12–1.1 wt % for cpx, and ± 0.1–1.2 wt % for spinel, based on mass balance using electron microprobe analyses and least squares calculations, with phase identification mainly by backscattered electron imaging. Thermocouple temperature measurements incorporated an uncertainty of ±5°–15°C [Baker and Stolper, 1994; Falloon et al., 1999; Wasylenki et al., 2003].
4.1 Fertile mm3 and KLB-1 Compositions
 In terms of the mm3 fertile spinel lherzolite bulk composition, (Figure 2a), the melt fractions calculated by our model compared with experimental results [Baker and Stolper, 1994; Falloon et al., 1999] are 12.0 versus 9.2 wt % at 1280°C, 27.4 versus 22.4 wt % at 1350°C, and 31.4 versus 31.2 wt % at 1400°C, respectively (Figures 2a and 3a). As temperature increases from 1280°C to 1330°C, our model results indicate a rapid decrease in the modal content of cpx, coinciding well with experimental results. However, our model results indicate a trace amount of cpx remaining at temperatures higher than 1330°C, whereas no cpx has been reported in experimental run products above 1330°C. The mineral proportions reproduced in the model compared with experimental results are 54 versus 59 wt % for olivine, 10 versus 9.4 wt % for opx, 1.0 versus 0.0 wt % for cpx, and 0.0004 versus 0.0 wt % for spinel at 1400°C. It is unclear whether the discrepancy in cpx and spinel abundances is due to uncertainty in our modeling or is related to the difficulty in detecting such a small amount of cpx in experimental products.
 Below the solidus temperature, the mineral proportions calculated by our modeling compared with experimental results are 52.2 versus 55.0 wt % for olivine, 22.5 versus 26.0 wt % for opx, 23.7 versus 13.4 wt % for cpx, and 0.0005 versus 1.0 wt % for spinel, respectively. A plagioclase abundance of 4.2 wt % was documented in a subsolidus experimental run [Falloon et al., 1999], although this mineral was not considered in our calculations.
 The results for modeling of KLB-1, which has a bulk composition similar to mm3, show a similar relationship to experimental results as was observed for mm3 (Figures 2b and 3b). The melting phase relationship of KLB-1 is also well reproduced by our model (Figure 2b). The modal content of cpx rapidly decreases between 1300°C–1350°C, coinciding well with the experimentally determined cpx-out temperature of 1350°C–1380°C. However, a trace amount of cpx remains at higher temperatures. Spinel behaves slightly differently in KLB-1 and mm3, as follows. For the KLB-1 bulk composition, spinel disappears between 1250°C and 1275°C, whereas spinel coexists with melt over the entire investigated temperature range for mm3. The experimentally determined spinel out temperature is 1300°C–1325°C for KLB-1, which is 50°C higher than our model would suggest.
4.2 DMM1 Depleted Peridotite Compositions
 Calculated melting degrees and experimentally determined melt fractions [Wasylenki et al., 2003] coincide well within uncertainties for the DMM1 bulk composition (Figures 2c and 3c); 3.7 versus 1.6–4.0 wt % at 1275°C, 8.5 versus 6.6 wt % at 1300°C, 18.5 versus 12.5 wt % at 1350°C, and 21.9 versus 16.6 wt % at 1390°C, respectively. Calculated subsolidus modal mineral abundances also broadly agree with experimental results: 64.6 versus 62.8 wt % for olivine, 17.4 versus 28.4 wt % for cpx, 17.4 versus 8.1 wt % for opx and 0.0 versus 0.6 wt % for spinel. These results reflect the depleted composition of the starting material, leading to a lower modal abundance of pyroxene and a higher modal abundance of olivine in both our modeling and experimental results. As melting proceeds, the modal content of cpx constantly decreases from 17 wt % before melting to 1.9 wt % at 1325°C. Above 1325°C, approximately 1 wt % of cpx remains in the system up to 1400°C, which is the maximum temperature tested in our calculations. Our model results of DMM1 compositions do not produce spinel, whereas melting experiments indicate that spinel is stable up to 1250–1300°C.
4.3 Phase Compositions
 Experimentally observed trends for major element concentrations in partial melts with varying temperatures are successfully reproduced by our model results for all tested bulk compositions (Figure 4), with increasing MgO and FeO*, decreasing Al2O3, and convex-up variations in CaO concentrations with increasing temperature.
 Our modeling also reproduces systematic differences in partial melt compositions between depleted and fertile sources [Wasylenki et al., 2003], with DMM1 calculated partial melt compositions (Figure 4c) having higher FeO*, CaO, and MgO, and lower Al2O3 contents than those produced by melting of mm3 (Figure 4a) and KLB-1 (Figure 4b) at identical degrees of partial melting.
 At degrees of melting higher than 5 wt %, major element concentrations in partial melt are reproduced well for SiO2, Al2O3, and CaO, whereas calculated FeO* and MgO concentrations are systematically offset from experimental results. To examine the reproducibility of melt compositions with changing temperature, calculated melt compositions are plotted against experimental results at given temperatures in Figure 5. A linear correlation between experimental and calculated melt compositions is present for all bulk compositions tested during this study, although systematic shifts are also present, especially for MgO and FeO*. In terms of mm3 bulk composition, the calculated FeO* and MgO concentrations at given temperatures are 2.8–4.6 wt % lower, and 4–7 wt % higher than the experimental results, respectively (Figures 4a and 5a). Accordingly, calculated partial melts have higher Mg# (molar MgO/(FeO*+MgO) ratio) values than the experimental melts.
 The degree of overestimation of MgO and underestimation of FeO* by our model is equivalent to an overestimation of 100°C~150°C, based on calculations using PRIMELT2 [Herzberg and Asimow, 2008]. It should be noted that the total molar content of FeO*+MgO is reproduced well by our modeling (Figure 6), suggesting that the misfit may be related to Fe–Mg exchange partitioning, as is discussed later.
 Near-solidus melt compositions are not well reproduced by our model results, with Al2O3 contents at degrees of partial melting less than 10 wt % being overestimated by up to ~15 wt %, and SiO2 contents underestimated by up to ~10 wt % (Figure 4), when compared with experiments with a low degree of partial melting [Baker et al., 1995], although solidus temperatures are well reproduced by our model results, as discussed in the following section.
 Residual minerals compositions are shown in Figure 7, with Mg# values for olivine and Al2O3 contents of opx plotted against temperature for both modeled and experimental results. The relationships between temperature and residual mineral compositions are broadly reproduced in our model results of both mm3 and DMM1 systems, although both olivine Mg# and opx Al2O3 contents are systematically lower than those of experimental results, indicating that Fe–Mg exchange partitioning between residual minerals and melt is not well reproduced by our modeling, as also suggested by melt compositions.
4.4 Solidus Temperatures and Near-Solidus Melting Behavior
 Solidus temperatures (hereafter referred to as Ts) and near-solidus behaviors of a system are critical in determining the melting processes and dynamics of the mantle, and are investigated here in detail. We have plotted melt fractions against temperature using a logarithmic scale in Figure 8 to compare near-solidus behaviors in our model with experimental results for both DMM1 and mm3 bulk compositions. Our modeling reproduces both temperature-melt fraction relationships and the steep drop of the melt fraction at near-solidus temperatures better than pMELTS calculations. However, in both our model and pMELTS, a small amount of melt is stable at temperatures lower than the experimentally determined Ts; this small amount of melt is hereafter termed the “low-melt-fraction tail”. Within the P-T range of our calibration database (1 GPa and 1230°C–1500°C), silicate melts are always stable in our model results. In the DMM1 melting experiments, 3.3 wt % of silicate melt was observed at 1270°C, whereas no silicate melt was observed in an experimental run at 1250°C, suggesting that the experimental Ts was 1270±10°C [Wasylenki et al., 2003]. In our calculations, the calculated degrees of melting were 2.5 wt % at 1270°C, with G = − 16, 521 kJ/kg, and 0.0022 wt % at 1250°C, with G = − 16, 475 kJ/kg. It is important to know whether the experiment has sufficient resolution to allow the determination of the presence of this low melt fraction, or whether there are errors inherent in our modeling calculations that artificially produce this tail. In our model calculations, G is approximately 10 kJ/kg less than in the melt-absent assemblage at the same P-T conditions, both at 1250°C and 1270°C. Although our modeling incorporates an uncertainty of ~1.5 kJ/kg, based on fitting errors during parameter calibration, the difference in G between melt-absent and melt-present calculations is significantly larger than this.
 At low degrees of partial melting, aluminous accessory phases may contribute to the energy budget in the system more than at higher degrees of melting. In addition to spinel, plagioclase (which is not considered in this study) is a common phase within peridotitic systems in sub- and near-solidus experiments at 1 GPa [Hirose and Kushiro, 1993; Falloon et al., 1999; Borghini et al., 2010]. However, a low-melt-fraction tail is also observed in pMELTS result (Figure 8), which included plagioclase and garnet in addition to spinel. This suggests that the low-melt-fraction tail is not a feature specific to this study, in which spinel is the only aluminous accessory phase considered.
 In experimental partial melts of fertile peridotite [Baker et al., 1995; Kushiro, 1996; Pickering-Witter and Johnston, 2000; Schwab and Johnston, 2001], an increase in SiO2 content with decreasing degree of melting at temperatures just above the solidus is observed, with this increase not being reproduced in our model results. The increase in SiO2 at lower degrees of melting of fertile peridotite has been attributed to the influence of alkali elements on SiO2 activity in mafic melts [Baker et al., 1995; Falloon et al., 2008;Hirschmann et al., 1998a; Kushiro, 1975; Ryerson, 1985; Walter et al., 1995]. Hirschmann et al. [1998a] suggested that a negative enthalpy of mixing between Na2SiO3 and SiO2 in silicate melts is responsible for the increase in SiO2 in MELTS results. The negative enthalpies of mixing between alkali end-member components and SiO2 in silicate melts are also incorporated in pMELTS, again leading to increases in SiO2 contents in calculated near-solidus melts (Figure 12). This indicates that introducing an activity model that accounts for these compositional effects might be required to accurately predict compositions, at least for near-solidus conditions in alkali-bearing systems.
 The experimental detection limit for melt fractions in peridotitic systems is ~2 wt %, below which the temperature-melt fraction curve is simply extrapolated without any direct experimental constraints [Laporte et al., 2004]. In addition, the identification of minor phases in experimental run products is difficult, primarily because phase identification is based on back-scattered electron imaging. One exception to this is provided by Laporte et al. , who resolved melt fractions as low as 0.2 wt % by trapping melts in microdikes formed in the run charge. For the sake of practical comparisons between modeling and experimental results, the temperature at which the model-calculated melt fraction is ~0.2 wt % is hereafter referred to as a practical solidus temperature, or simply Ts. Figure 9 compares calculated and experimentally determined Ts values, with the experimental uncertainties shown in Wasylenki et al. . Our modeled Ts for DMM1 is 1260°C, coinciding well with (and within the experimental uncertainty of) the experimentally determined Ts (1270±10°C). In addition, we modeled the Ts for mm3 to be 1225°C, whereas it was experimentally determined to be 1240°C by extrapolation of the curve representing temperature versus melt fraction [Baker and Stolper, 1994]. Falloon et al.  estimated Ts for a synthetic mm3 composition (without K2O and P2O5) to be 1260°C at 1 GPa, and observed no melt at 1250°C. The discrepancy between the two experiments is attributed to the absence of K2O and P2O5 in one of the experiments [Falloon et al., 1999], as well as uncertainties in thermocouple temperature measurements [Schwab and Johnston, 2001; Ghiorso et al., 2002]. Our modeled Ts for mm3 is 1225°C, slightly lower than the experimentally estimated range of 1240°C to 1260°C. In addition, our model result with KLB-1 composition yielded a practical Ts of 1220°C, coinciding well with experimental results (1220±30°C), again based on extrapolation of the melt fraction-temperature curve [Hirose and Kushiro, 1993; Wasylenki et al., 2003].
 This indicates that our model reproduces the relationship between phase relations and temperatures at 1 GPa well, including Ts and degree of melting, although it should be noted that melt thermodynamic parameters were not directly obtained from phase fractions, and instead have been determined from phase compositions, as shown in Figure 1. Although our model results indicate melt fractions slightly higher than experimental values (Figures 2 and 3), this offset is notably smaller than those calculated with previous thermodynamic models, as discussed in section 4.6. The maximum offset is 5 to 6 wt % at temperatures of 1325°C to 1350°C, or approximately 50°C for a fixed melt fraction. The differences associated with variations in bulk compositions are also well reproduced by our modeling, with fertile mm3 and KLB-1 composition exhibiting higher degrees of melting, whereas relatively refractory DMM1 composition, with a higher Ts by 40°C (Figure 9), yielded lower degrees of partial melting for a given temperature.
4.5 Minimization and Total Energy of the System
 Figure 10 shows the total Gibbs free energy of the melt-absent system as a function of the molar contents of forsterite and orthoenstatite with the DMM1 bulk composition at 1200°C and 1 GPa, consisting of olivine, opx, and cpx [Wasylenki et al., 2003]. This system has six degrees of freedom, and allows us to map the Gibbs free energy within the entire compositional space. To illustrate an energy-composition relationship, four parameters are fixed and the molar contents of orthoenstatite and forsterite are varied in Figure 10. This energy map indicates that the system has a single global minimum, rather than multiple local minima possibly associated with nonideality of solid end-member components. This global minimum is elliptical, indicating that the total Gibbs free energy of the system is less sensitive to the orthoenstatite content than the forsterite content.
 The poorer sensitivity for orthoenstatite content in Figure 10 could be attributed to the source data of Sack and Ghiorso [1994b], where μ0 values for clinoenstatite and orthoenstatite end-member components, as well as clinoferrosilite and orthoferrosilite end-member components, are assumed to be identical. This means that our calculations, involving a three-site ideal solution activity model for pyroxene, have only a weak energetic constraint on Fe–Mg partitioning between opx and cpx. Significant misfits are observed in clinoferrosilite and Mg-tschermak end-member component calibrations, in addition to a clear correlation between Δμ and the Mg# of experimental cpx (Figure 1). The presence of immiscibility in the pyroxene system is also indicative of nonideal mixing of components [e.g., Lindsley, 1983], indicating that the future introduction of a thermodynamic model for pyroxene, including nonideal solid solutions, could result in better reproducibility of Fe–Mg partitioning and pyroxene compositions. The higher Mg# values of the modeled melt, compared with the experimental melt, could also be related to inaccuracies associated with pyroxene end-member components.
 Figure 11 shows the melt fraction [wt %] versus total Gibbs free energy of the system G [kJ/kg] calculation path during minimization at 1400°C and at a pressure of 1 GPa for the DMM1 bulk composition. Point A on the vertical axis minimizes the total energy of the system at a given P-T, when melt is assumed to be absent. Minimal amount of melt phase is added in the system to ensure effective minimization (Point B), and then reduce G of the system along the steepest gradient (B–C), refining the molar end-member component contents of the coexisting phases. A global minimum is located where changes in the molar fraction of phases (δn) results in an increase of G (Point C). In order to display the structure in a melt-present system, we conducted further calculation beyond the global minimum (C–D) by forcing mass perturbation. The resultant structure indicates that the system has a single trough-shaped minimum, rather than a number of local minima. Although Figure 11 is simply an example within a vast parameter space, based on a number of modeling runs with variable starting condition, we suggest that the global peridotite melting minimum can be properly identified for given P-T bulk compositions using the algorithm presented in this study.
4.6 Comparison With Previous Models
 The combination of the energy minimization algorithm developed in this study with newly calibrated thermodynamic melt parameters can enable accurate descriptions of mantle melting, especially in terms of phase stability and proportions. The energy minimization algorithm and thermodynamic configuration are equally important in thermodynamic calculations, and comparisons between the results of thermodynamic modeling can demonstrate how model results are affected by different sets of thermodynamic parameters and minimization algorithms. For this purpose, we compare our model results with modeling undertaken using pMELTS [Ghiorso et al., 2002] and Perplex [Connolly, 2005], both of which are widely used for petrological and geodynamic modeling. This comparison highlights the characteristics of each model, and will be useful in understanding the advantages and disadvantages of using a specific model.
 Modeled phase relations and partial melt compositions determined using pMELTS with mm3 and DMM1 bulk compositions are plotted against temperature in Figure 12, with calculated melt compositions plotted against experimental results at given temperatures in Figures 5d–5f for mm3, KLB-1, and DMM1 bulk compositions, respectively, for comparison. Although minor systematic shifts in SiO2 concentrations are observed [Ghiorso et al., 2002], the composition of partial melts at given temperatures is relatively well reproduced by pMELTS for the mm3 and KLB-1 bulk compositions (Figures 5d and 5e). For the DMM1 bulk composition, the reproducibility of the melt composition is relatively poor, especially at near-solidus conditions (Figure 12f). In addition, the reproducibility of the relationship between melt fraction and temperature T is poor in pMELTS modeling. As indicated by Ghiorso et al.  for mm3, melt fraction is underestimated by 5–10 wt % at a given T when melt fraction is lower than 20 wt %, whereas this variable is estimated more accurately when melt fraction is higher than 20 wt % (Figures 3a and 12a). The reproducibility of pMELTS models is worse for DMM1, with melt fraction values underestimated by 2–18 wt % at a given T (Figures 3c and 12e). In addition, although calculated Ts (practical solidus) values estimated using pMELTS fall within an experimentally determined range for mm3 and KLB-1 bulk compositions, Ts is overestimated by 50°C compared with experimentally determined solidus values for the DMM1 bulk composition (Figure 9). Similar overestimation, by 20°C to 50°C, has also been documented for several other bulk compositions [Wasylenki et al., 2003]. These deviations are greater than those expected from the residual of the parameter calibration (estimated to be ~5°C) or from experimental errors, and could be related to an overestimation of cpx modal content by 5~10 wt % [Ghiorso et al., 2002]. This indicates that pMELTS should be used with caution during modeling where the T-melt fraction relationship is critical.
 Calculated phase relations and melt compositions determined using Perplex with mm3, and DMM1 bulk compositions are shown in Figure 12. These calculations used the THERMOCALC [e.g., Holland and Powell, 1998] and pMELTS [Ghiorso et al., 2002] thermodynamic databases for minerals and silicate melt, respectively. These data are recommended for use when modeling peridotitic systems [Connolly, 2005], although they are unable to satisfactorily reproduce peridotite melting phase relations. At 1 GPa, opx disappears prior to cpx with increasing degrees of partial melting in bulk compositions ranging from fertile to depleted, as modeled during this study (Figures 12c and 12g). As observed in pMELTS results, calculated melt fractions at given temperatures are systematically lower than experimental results (Figures 3 and 12), with calculated solidus temperatures having the largest deviations from experimental results of all the models discussed here (Figure 9). Calculated partial melt compositions also significantly deviate from those determined experimentally (Figures 5 and 12), with SiO2 contents being systematically higher, and FeO* and CaO contents systematically lower than experimental results. Calculated SiO2 contents range from 64 to 60 wt % between 1350°C and 1400°C in the DMM1 system, some ~10 wt % higher than experimental values (Figure 12h). This indicates that Perplex-based model results deviate significantly from the corresponding pMELTS results, demonstrating that the mineral thermodynamic models and parameter values must be internally consistent with these of silicate melt.
4.6.3 Model Characteristics
 The minimization algorithm integral to Perplex may work effectively during phase diagram construction in P-T composition. However, these phase compositions may not be accurately reproduced in some cases, such as melt-present systems, if phase compositions are defined at discrete grid-points as in the Perplex program. Section 4.5 shows that the magnitude of a change in G associated with phase composition modification is relatively small (Figures 10 and 11), requiring a relatively high accuracy during modeling.
 MELTS/pMELTS can effectively calculate equilibrium mineral compositions in a melt-present system, possibly because melt interaction parameters were calibrated and its algorithm searches for a mineral composition at which the plane tangent to the composition-chemical potential surface is equal to that of the melt with a given composition. However, if MELTS and pMELTS are used this way, checking if the total system energy is efficiently reduced to reach the global minimum is not trivial, with difficulty in accurately determining the phase fractions.
 Our algorithm performs direct minimization of the total Gibbs free energy of the system, including phases with compositionally continuous solutions, and describes mass conservation, stoichiometric constraint, and energy minimization in a straightforward manner. This approach allows the calculation of energy-mass balance relationships during open system melting associated with melt migration.
 On the other hand, the modeling approach outlined in this study requires a relatively long computation time, primarily as this modeling searches for the steepest energy gradient in all possible components. To calculate an equilibrium phase assemblage from a melt-absent initial state at a given P-T typically takes 30~60 s of calculation time on a desktop computer. In addition, the currently available thermodynamic parameters are probably not accurate enough to reproduce Fe–Mg partitioning, as discussed in section 4.5, meaning that our modeling approach could be improved by the use of better thermodynamic models of relevant phases, especially pyroxene.
 In summary, we have shown that various minimization algorithms using different thermodynamic parameter sets can lead to significant variations in calculated phase equilibria. In principle, any thermodynamic modeling should yield similar, if not identical, results; in reality, however, the approximations and errors involved in modeling can have a significant impact on the results. This indicates that thermodynamic models, phase parameters, and energy minimization algorithms need to be carefully chosen to reflect the purpose of the study.
 We presented a general energy minimization algorithm for use with multicomponent systems. This algorithm describes the total Gibbs free energy of the system gradient in a straightforward manner with respect to any tiny dissolution or solidification of melt and solid end-member components at constant P, T, and bulk composition. This study has also provided a specific set of thermodynamic formulations for molar Gibbs free energy and newly calibrated thermodynamic parameters for silicate melts. For a silicate melt activity model, we have employed an ideal solution model to test the algorithm and configuration. The energy minimization algorithm and melt thermodynamic model were applied to the SiO2–Al2O3–FeO–Fe3O4–MgO–CaO system, consisting of olivine, cpx, opx, spinel, and silicate melt at 1 GPa. Calculated phase relations agree well with experimentally determined melting phase relations for mantle peridotites of fertile to depleted compositions. The calculated variations in melt compositions with temperature also broadly agree with experimental results. However, the Fe/Mg ratio of partial melt is not well reproduced, suggesting that the Fe–Mg exchange reaction between solid and melt is poorly reproduced in our model. To establish a more accurate Fe–Mg exchange reaction, nonideality for phases appropriate for the system to be modeled is required, especially for pyroxene. Modeling results are also strongly dependent on the thermodynamic models used (pMELTS, Perplex, and our model), highlighting the importance of establishing a coherent set of minimization algorithms, thermodynamic phase configurations, and parametric values in order to produce more accurate predictions.
Appendix A: Activity Model for Solid Phases
 Here we briefly describe the activity models and solid phase thermodynamic parameters employed in this study.
 Olivine: The two-site regular solution mixing model of Sack and Ghiorso  for Mg2SiO4 (forsterite)– Fe2SiO4 (fayalite) olivine was employed to model composition-activity relationships, with nonideal activity parameters calibrated using experimentally determined olivine-opx equilibria.
 Spinel: The two-site mixing model of Sack  was employed to calculate the MgAl2O4 (spinel) and Fe3O4 (magnetite) composition-activity relationship, where nonideal activity parameters were calibrated using spinel-olivine equilibria and the experimentally determined spinel immiscibility gap. The end-member components considered in Sack  are identical to those used here.
 Pyroxene: Here the site occupancy model of clinopyroxene is assumed, after for example Deer et al. , to be
with that of orthopyroxene assumed as
 The wide pyroxene immiscibility gap [Lindsley, 1983] is indicative of significant nonideality. The activity-composition relationship of pyroxenes also contains a significant number of uncertainties [e.g., Berman et al., 1995; Sack and Ghiorso, 1994a], partly due to the large compositional space and insufficient experimental data for pyroxenes, in particular for Al-bearing pyroxenes at high pressures. There is also considerable disagreement regarding the site-occupancy of cations between experimentally determined data and theoretical predictions [Berman et al., 1995]. However, this study use a relatively small compositional range of pyroxenes compared with the entire possible range in the SiO2–Al2O3–FeO*–MgO–CaO system. Therefore, we assume that the effects of nonideality are small enough to justify use of an ideal site-mixing model in the range of pyroxene compositions used in this study. Considering this, a simple three-site ideal mixing model [e.g., Newton, 1983; Nielsen and Drake, 1979], for pyroxene is employed here.
Appendix B: Incongruent Melting
 The following mineral components have two or more nonzero coefficients in the melting reactions outlined in Table 3: enstatite, ferrosilite, Mg-tschermak, Ca-tschermak, diopside, and spinel will decompose into several melt oxide end-member components upon melting. This dissociation and total energy minimization of the system means that incongruent melting may be naturally reproduced in this study.
 The energy balance associated with incongruent melting is explained here using enstatite dissolution as an example. Dissolution matrix D (Table 3) indicates that the enstatite melting reaction is assumed to be
 If we set the reference state to be the melting temperature of enstatite () at 1 bar, ΔμEn at 1 bar is written as
where “px” indicates pyroxene, “En” denotes the MgSiO3 end-member component, “Qz” denotes SiO2, and “Fo” denotes Mg2SiO4 oxide end-member components in a melt.
 Here “fictive” enthalpy, entropy, ΔCp, and ΔV for a melt end-member component are defined as follows:
 This means that at and 1 bar,
 The entropy of a reaction can be obtained by differentiating Gibbs free energy in terms of temperature, leading to the derivation of
 Equations (B6) and (B7) indicate that and can be derived from experimentally determined enthalpy and entropy of fusion results for enstatite at 1 bar.
 Equation (B1) can be rearranged and expanded to the pressure P using the fictive parameters in equations (B2) to (B5) as
 By calibrating and in the same manner as other congruently melting end-member components (see section 3.1), we can calculate the chemical potential of enstatite melt end-member components using equation (B8).
 The Δμ values for ferrosilite, Mg-tschermak, Ca-tschermak, diopside, and spinel melt end-member components were obtained using the procedure outlined above for enstatite. The instability of ferrosilite, Mg-tschermak, and Ca-tschermak melt end-member components at 1 bar meant that and were also calibrated in addition to the fictive parameters in equations (B2) to (B5).
Appendix C: Oxygen Fugacity and Ferric-Ferrous Ratio
 Both Fe2+ and Fe3+ are considered here, meaning that before parameter calibration, the ferric-ferrous ratio of experimental silicate melts must be estimated, primarily because ferric-ferrous ratios for experimental melts are generally not determined.
 Empirical relationships between oxygen fugacity and ferric-ferrous ratio in silicate melts have been obtained by several studies and are commonly used. The parameterization of Kress and Carmichael  has been widely used to obtain the ferric-ferrous ratio of a melt at a given oxygen fugacity [e.g., Ghiorso and Sack, 1995]. Specific oxygen-gas buffering reactions [e.g., Chou, 1978; Myers and Eugster, 1983] have been used to obtain exact oxygen-gas fugacity values, because fixing of oxygen gas contents during experimental runs is difficult.
Kress and Carmichael  assumes that the reaction between gaseous oxygen and the Fe-species dissolved in a silicate melt are described as
 This indicates that the equilibrium constant of the reaction is a function of . The reaction outlined in equation (C1) does not give a reasonable fit with experimental results, meaning that the empirical parameterization of Kress and Carmichael  is based on the following equation:
where fO2 is oxygen fugacity, α, β, ε, and ζ are constants, and ΣX is the sum of the molar concentrations of oxide components in a melt. Consequently, although apparent fitting is achieved with equation (C2), their parameterization does not satisfy the energy balance or the charge balance of the reaction. Kress and Carmichael  indicate that an oxidizing reaction of Fe-species in silicate melt can be accurately described using
 This reaction is equivalent to the “QFM reaction” in terms of the oxidization state of Fe; as such, we propose the following reaction:
for the dissolution of gaseous oxygen and the oxidization of iron in a silicate melt. The energy balance of the reaction (39) is written as
 We have optimized Δμ0 in equation (C5) using a least squares method to reproduce the experimentally determined relationship between the oxygen fugacity and ferric-ferrous ratios [Kress and Carmichael, 1988]. Then we obtain
as an equilibrium constant of the reaction (C4). The ferric-ferrous ratio of silicate melts at a given oxygen fugacity is then calculated using equation (C7) during parametric calibration.
Appendix D: Minimization Procedure
 To derive the initial state of a melt-present minimization, we first calculate a stable phase assemblage by minimizing G without considering a melt phase. A given bulk composition vector (in oxide wt %) is converted to n using equation (2) to derive an initial state of melt-absent minimization. We then calculate the change in the Gibbs free energy of the system (δG) with respect to all possible tiny mass perturbations of solid end-member components (δn) using equation (7). The molar concentration of each end-member component is then recalculated along the steepest gradient at each calculation step, with the equilibrium state of the system being determined where the gradient is zero with respect to any δn, minimizing G.
 In melt-present systems, calculations start with n derived by melt-absent minimization (A in Figure 11), meaning that zero-melt end-member components are present in n at this stage. Positive molar concentrations assigned to melt end-member components are added to n as per equation (9), ensuring effective minimization (B in Figure 11). This procedure uses an arbitrarily chosen melt composition, with the dissolution vector δn magnitudes fixed as equal for all solid end-member components. This leads to the G value of this melt-present initial state being higher than the melt-absent G, as shown in B in Figure 11.
 Next, δG values with respect to all possible δn of melt and solid end-member components, as per equation (25), are calculated at each step, with the magnitude of δn refined according to the magnitude of δG during each step. The equilibrium state of the melt-present system is defined when δG equals zero with respect to any dissolution or solidification of solid end-member components (C in Figure 11). The vector n derived by this minimization describes the equilibrium molar melt and solid end-member component contents at a given P, T, and bulk composition.
 We thank Keith Putirka and an anonymous reviewer for constructive reviews and Joel Baker for handling the manuscript. Toshitsugu Fujii and Kazuhito Ozawa are thanked for discussions and encouragement during this research, and Guillaume Richard is thanked for assistance during this study. This work has been partly supported by JSPS International Bilateral Programs (Joint Research Projects and Seminars) to H.I.