Abstract: 
A longstanding issue in the study of outofequilibrium systems in statistical mechanics is the validity of Fourier`s law. In this talk we will present a model introduced for this purpose, i.e. to describe properly heat diffusion. It consists of a $1$dimensional chain of $N$ oscillators coupled at its ends to heat baths at different
temperatures. Here, working with a weakly anharmonic homogeneous chain, we will show how it is possible to prove exponential convergence to the nonequilibrium steady state in Wasserstein$2$ distance and in
Relative Entropy. The method we follow is a generalised version of the theory of $\Gamma$ calculus. It has the advantage to give quantitative results, thus we will discuss how the convergence rates depend on the number of the particles $N$. 
