Hilbert's 6th problem consists in justifying rigourously the mechanisms leading from classical, molecular dynamics to fluid motion. Using Boltzmann's equation as an intermediate description, we shall explain in this mini-course how this is possible in the simplified, linear settings of the heat and Stokes equations. The main difficulty is the justification of Boltzmann's equation starting from Newton's laws, on diffractive times. We shall therefore concentrate in the presentation of Lanford's proof of convergence to the Boltzmann equation, insisting on the particularities of the linear and linearized case (leading respectively to the heat and Stokes equations). This corresponds to joint works with Thierry Bodineau and Laure Saint-Raymond.