Markov Chains

Abstract

In this chapter we will start dealing with stochastic processes, which are the mathematical models for phenomena whose temporal evolution contains some randomness. Starting from the celebrated example of the random walk, we will introduce the central definition of Markov chains, which, although simple, provide extremely important models for physical systems. The description of Markov chains will allow us to introduce the central topic of thermalization and approach to equilibrium of random motions, that is the existence of asymptotic laws to which the Markov chains converge, in a sense that will be made rigorous. Finally, we will introduce Metropolis theorem, a cornerstone of numerical simulations, as will be discussed in the following chapter.