In this talk, we study mixing times and the so-called cutoff phenomenon, which was introduced by Diaconis and Aldous in the study of a quantitative convergence to equilibrium for card-shuffling Markov models. We will estimate the mixing times and show cutoff phenomenon for the ergodic Langevin dynamics with a strongly coercive potential and driven by an additive noise with small amplitude.More precisely, when the driven noise is the Brownian motion, the total variation distance between the current state and its equilibrium decays around the mixing time from one to zero abruptly. When the noise is the alpha-stable with index alpha>3/2, cutoff phenomenon still holds while for alpha\leq 3/2 our coupling techniques do not apply, and therefore we cannot conclude if the cutoff phenomenon still holds. In the case of degenerate potential, cutoff phenomenon does not hold, however precise estimates for the mixing times can be obtained. The talk is based on series of papers with Milton Jara (IMPA, Brazil), Michael Högele (Universidad de los Andes, Colombia), Juan Carlos Pardo (CIMAT, Mexico) and Conrado da Costa (Durham University, UK).