Wilson equation is applied to the ternary ilmenite solid solution, (Fe, Mn, Mg) TiO₃, for describing the excess molar Gibbs free energy of mixing (G^ex). Non-linear programming technique is used to derive the parameters of the equation from the ilmenite-fluid equilibrium experiments at 600°C and 1000 kgf/cm² by Kubo et al. (1992). The excess molar Gibbs free energy of mixing for ilmenite is obtained as G^ex=-873.15R{X_Fe in (X_Fe+1.056X_Mn+ 0.878X_Mg)+X_Mn ln (0.704X_Fe+X_Mn+0.424X_MS)+X_Mg ln (0.450X_Fe+0.326X_Mn+X_Mg)}, where X stands for the mole fraction of the subscript component (Fe as FeTiO₃, Mn as MnTiO₃, Mg as MgTiO₃) and R denotes the gas constant. Fluid compo-sitions in equilirium with the ternary ilmenite solid solution are also obtained. The calculated results of the GeX values and the fluid compositions agree well with the experimental results, suggesting the usefulness of the Wilson equation.

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With the increasing interest on nonlinear complex phenomena which are lack of explicit symmetry and are not treatable with established methods of physics, new methodology is now required.Here, using a recent computational method with a new point of view, the global pattern dynamics of dunes is discussed. The present procedure of study is expected to give a new prototype for searching complex systems and to give a way to make collaboration between researchers on different fields.

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Various patterns are produced when precipitates emerge from the supersaturated solution. A morphology of a single precipitate can change from a spherical droplet to a Fractal-like structure according to certain environmental condi-tions. As a group, precipitates also form macroscopic patterns. For example, precipitates obey the competitive growth at the late stage of the coarsening process due to the mass conservation in the whole system, and the dynamical scaling laws hold for physical quantities, such as the size distribution of precipitates. These dynamical scaling behaviors hold even when the morphology of each precipitate is fractal. The formation of the macroscopic spatial pattern of precipitates is considered to be related with the coarsening process in non-uniform systems.

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Diffusion in silicate melts often shows uphill diffusion, the diffusion of a component up its own concentration gradient. As discussed by Zhang (1993: J. Geophys. Res., 98, 11901-11920), there has been no practical model until now that can predict whether uphill diffusion will occur in a given system. This paper presents a new nonlinear diffusion equation which can describe uphill diffusion in a ternary system and provide a criterion in what conditions uphill diffusion will take place. Cross coefficients D_ij(i≡j) in a multicomponent diffusion equation describe the strength in coupling between diffusional flows, which may cause uphill diffusion. Irreversible thermodynamics shows that cross coefficients are dependent on con-centration, although their functional form is unknown. To estimate it, in this paper an irreversible thermodynamic approach by Miller (1959: J. Phys. Chem., 63, 570-578) and an eigenvector analysis by Gupta and Cooper (1971: Physica, 54, 39-59) is combined. The functional form satisfying the two conditions above is:D₁₂=Φ₂(D₂₂-D₁₁) and D₂₁=Φ₁(D₁₁-D²²). These cross coefficients give a nonlinear diffusion equation which satisfactorily describes uphill diffusion.

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Theoretical studies of zonal structure are reviewed. It is stressed that the origin of zonal structure formed during crystal growth can be understood fully only if crystal growth theories including variables such as chemical composition and order parameters were provided.

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The size of minerals in volcanic rocks is typically shown to be exponentially distributed. Rnadolph and Larson first proposed the CSD theory to explain the crystal size distribution. Marsh applied the CSD theory to explain the size distributions in volcanic rocks. Some problems on master equation that Marsh used, however, found, and related with the initial and boundary conditions. I show the CSD theory to use Rnadolph and Larson's method, and point out difference of Marsh's method. According to the CSD theory, an exponential size distribution may be obtained at steady state condition in open system, if growth rate and nucleation rate are kept constant. Such conditions may occur in magma chambers. Other models that explain size distribution of minerals in rocks are also reviewed. For instance, growth probability model that was proposed by Maaloe also explains the exponential size distribution in a close system.

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This paper presents textural evolution of metamorphic rocks due to Ostwald ripening. First, Crystal size distributions (CSDs) of garnets in various metamorphic rocks are compared with the LSW (Lifshitz-Slyozov-Wagner) distribution for Ostwald ripening. There are two types of CSD. One is consistent with LSW distribution, the other is wider than LSW distribution. The former one has a homogeneous spatial disposition of garnets, and represents the progress of Ostwald ripen-ing. That CSD of the latter type is wider than LSW, is either; 1) because duration of Ostwald ripening is not sufficient, or 2) because spatial disposition of garnets is not homogeneous. LSW theory needs homogeneous spatial disposition of precipi-tated phase. Ostwald ripening in spatially heterogeneous system is researched by numerical simulation. The results suggest that heterogeneous structure, such as layered structure, are developed by Ostwald ripening. Shape of size distribution is important in two reasons; 1) material transport between layers by Ostwald ripening can be evaluated by comparisons of size distributions between layers, and 2) the width of initial size distribution strongly influences an evolution of layering structure.

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Rim replacement texture is commonly observed in hydrothermal ore deposits. Examples are replacements of native bismuth by bismuthinite, ferberite by scheelite, galena by sphalerite and cassiterite by stannoidite and stannite. There are three main rate-determining mechanisms of replacement. They are surface reaction, diffusion in boundary layer and diffusion in altered layer. Theoretical considerations on these mechanisms indicate that the variations in the amount of altered product with time for each mechanism are different. Experimental study on the reaction of electrum and NaHS and Na₂S solutions at 150°C by Nakata et al. (1994)(Resource Geol., 44, 395-407) demonstrated that surface reaction is rate determining mechanism for the formation of Ag₂S which is the product of the reaction between electrum and NaHS solution. Such replacement experiments are necessary to clarify the mechanism of replacement and metasomatism.

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Fractal geometrical analyses of fractures, grain boundaries and pore size ditributions in a typical fresh granite from Inada, Japan have been studied in order to quantitatively characterize transport properties of rock textures. Box counting method reveals that these textures have fractal nature for scale ranges from 25 nm to 10 mm and their box fractal dimensions are estimated to be around 1.5. Microcrack ditributions in feldspars have higher fractal dimensions than those in quartz showing more tortuous nature. This observation is in agreement with two diffusion pathways having different tortuosities proposed on the same granite. Fractal analyses of fracture networks in a granite bore hole have indicated that the fractal dimension can also be an indicator of groundwater permeability by representing regularity and connectivity of the networks. The quantitative treatment of rock textures are thus important in predicting material transport properties such as diffusivity and permeability. They are also fundamental in evaluating the rock deformation rates.

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The methods based on thermodynamics of irreversible processes are discussed in order to analyze non-equilibrium textures formed via diffusion-controlled solid-state reactions in meteorites. Simple expressions for rate constants of reaction between enstatite and metallic iron and internal reduction of olivine are presented.

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