Nature has a hierarchical structure with macroscopic behavior arising from the dynamics of atoms and molecules. The connection between different levels of the hierarchy is however not always straightforward, as seen in the emergent phenomena, such as phase transition and heat convection. Establishing in a mathematical precise way the connection between the different levels is the central problem of rigorous statistical mechanics. One of the methods leading to such results is to introduce some stochasticity inside the system.

We summarise some of the results obtained recently concerning the derivation of the macroscopic heat equation from the microscopic behaviour of a harmonic chain with a stochastic perturbation. We focus our attention on the emergence of macroscopic boundary conditions. The results have been obtained in collaboration with Joel Lebowitz, Stefano Olla, Marielle Simon.