Monte Carlo Simulations in Statistical Physics

In these notes I discuss Monte Carlo simulations for the study of classical models in statistical mechanics. I include a simple and direct proof that the method converges to the Boltzmann distribution. Usually, physics articles discuss this important point by just giving a reference to the mathematical literature on " Markov chains " , where the proof is rather abstract. In these notes I give a proof of convergence which is self contained and uses only elementary algebra. In statistical mechanics one computes averages of a quantity A from the Boltzmann distribution, i.e. A = l P eq l A l , (1) where l denotes a state, A l is the value of A in that state, and P eq l is the equilibrium (Boltzmann) probability distribution for the system to be in state l, i.e.