ABSTRACT
This chapter introduces the basic physical concepts and mathematical quantities needed to describe systems composed of many particles, when only a statistical description remains possible. The knowledge of the statistical distributions for the different ensembles is sufficient to characterize the macroscopic properties of a physical system in thermal equilibrium. The Gaussian distribution is a generic characteristic of equilibrium states: in thermal equilibrium, a system will be in a state minimizing the corresponding thermodynamic potential, and the fluctuations around this stable state will be Gaussian. The analytic solution of the 1d-Ising model is so easy to obtain since the coupling between any two spins is the same. The Ginzburg-Landau model is, from a computational point of view, a very nice topic.
