Abstract Most of the Earth’s material exists at high pressures and temperatures inside the planet. Since experiments in this p-T regime often turn out to be difficult or plain impossible, it is often necessary to do simulations, which avoid some of the important problems encountered in experiments. For a more specific look on the topics of this chapter we refer the reader to Kohn (1999), Martin (2004) , Oganov et al. (2002) , Payne et al. (1992) and Stixrude et al. (1998) . In this chapter we describe how to calculate the energy of a crystal with ab initio methods. It will shortly touch the historical origins of today’s methods and will end with the state-of-the-art quantum-mechanical calculations. When we want to investigate a mineral system we start with the Gibbs free energy. Every system in equilibrium likes to be in the state with the lowest Gibbs free energy G at given pressure and temperature condition. The Gibbs free energy is then given by minimising the following equation: where E is the energy, p the pressure, V the volume, T the temperature and S the entropy of the system. From statistical mechanics, knowing the energy of different states of the system (i.e. the energies of different vibrational and electronic quantum levels or of different atomic configurations) one can calculate the entropy and the free energy 1 , the link being provided by the partition function Z : The total energy of a non-relativistic electron-nuclear system and all its energy levels can be calculated by solving the Schrodinger equation, where H is the Hamilton operator and ψ is the wave function for the N electrons and the M nuclei.

Abstract Thermodynamic modelling of a solid solution requires a description of the mixing functions, of the Gibbs free energy, enthalpy, volume, vibrational and configurational entropy. The enthalpy of mixing, volume and the vibrational entropy of silicate solid solutions can be measured directly. For example, the vibrational entropy can be determined from calorimetric experiments through integration of the heat capacity. The same method does not work for the configurational entropy, however. Due to the slow kinetics of cationic order/disorder in silicates, the configurational degrees of freedom of atoms remain unchanged in the calorimetric experiments and do not contribute to the heat capacity. Therefore, the configurational entropy can only be determined from indirect experimental data. There are two approaches to this problem. The first one is to use the classical thermodynamic relationship Δ G mix = Δ H mix − T Δ S mix + P Δ V mix , where Δ S mix = Δ S mix,conf + Δ S mix, vib . when the Δ G mix term is constrained by phase equilibrium experiments and the Δ H mix , Δ S mix,vib and Δ V mix terms are known from calorimetric and X-ray diffraction measurements, Δ S mix,conf can be calculated. The second approach is to calculate the configurational entropy using its statistical-thermodynamic definition, by counting the number of available configurational states. To do this one needs to know which configurational states are available. Such information can be obtained from diffraction and spectroscopic data. The configurational entropy can be described with a variety of models with different levels of accuracy and simplicity. The choice of a model depends largely on the availability of experimental data.

Summary Bayesian statistics provide a formalism for inversion of magnetotelluric (MT) data in 3-D structures composed of elementary homogeneous domains. Available information (including assumptions about the model) is put in a probability density function (PDF) for prior values of the conductivities in the region of the search; the parameters to be found are the posterior values of the conductivity. A stochastic algorithm called a Gibbs sampler estimates the posterior PDF. The outer cycle of the iterative inversion consists of scanning the homogeneous domains in the region of the search; the inner cycle involves solution of the forward problem for a set of models. This process represents a Markov chain, whose transition law converges to the marginal PDF of the parameters. The inner cycle uses the finite-difference program FDM3D-MT, which computes electromagnetic responses in the frequency domain for 1-D, 2-D, or 3-D models. Iterative solution of the finite-difference equations is very fast and reduces greately the total CPU time since the results of the previous cycle are used as starting points for the next forward model. In most of our tests, the outer iteration converges in 15–20 iterations, which allows an attack on 3-D problems even on microcomputers. Examples show how the quantity and quality of data and the prior information affect the results of inversion.

An amount of disorder among tetrahedral silicon and aluminum in sillimanite too small to measure readily by means of x-rays may have significant effects on phase relations of the Al 2 SiO 5 minerals. The effect is nearly as great as that due to solid solution of minor constituents and cannot be ignored. A statistical thermodynamic treatment permits evaluation of the Gibbs' free energy of sillimanite as a function of disorder at any temperature on the assumption of a most stable degree of order at a temperature. Reasonable permutations of the assumed order and its associated temperature indicate that the enthalpy change associated with complete disorder is approximately 4 kilocalories and that sillimanite may become completely disordered at temperatures of 1,700°C or greater. The processes of ordering and disordering are likely to occur by homogeneous diffusion rather than by nucleation of domains.