Predicting measurable thermodynamic and kinetic properties of solids from first-principles requires the use of statistical mechanics. A major challenge for materials of technological importance arises from the fact that first-principles electronic structure calculations of elementary excited states are computationally very demanding. Hence statistical mechanical averaging over the spectrum of excited states must rely on the use of effective Hamiltonians that are parameterized by a limited number of first-principles electronic structure calculations, but nevertheless predict energies of excited states with a high level accuracy. Here we review important effective Hamiltonians that account for vibrational and configurational degrees of freedom in multi-component crystalline solids and show how they can be used to predict phase stability as a function of composition and temperature as well as kinetic transport constants such as diffusion coefficients in non-dilute crystalline solids.