Nell'ambito dei "Tè di Matematica", per celebrare il pensionamento di Fabio Martinelli, mercoledì *28 ottobre* alle *ore 16:00*, presso il Dipartimento di Matematica e Fisica di Roma Tre, Pietro Caputo terrà un seminario dal titolo *"Entropy and Gibbs samplers: from functional inequalities to mixing times".* *Abstract:* Functional inequalities provide a powerful framework for studying convergence to equilibrium in stochastic dynamics and Markov chains, at the intersection of probability theory and statistical mechanics. For interacting spin systems such as the Ising and hard-core models, spectral gaps and logarithmic Sobolev inequalities relate structural properties of their Gibbs measures to the rate at which the dynamics used to sample from them approach equilibrium. These themes have also become central to the analysis of randomized algorithms for sampling and counting in theoretical computer science. Fabio Martinelli has made highly influential contributions to this field since the early 1990s. On the occasion of his retirement, I will introduce some of its central ideas and discuss more recent developments, with particular emphasis on entropy factorization and Gibbs samplers. Il seminario si svolgerà in presenza presso il Dipartimento di Matematica e Fisica, Lungotevere Dante 376, aula *M1*.