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We study a $1$-dimensional chain of $N$ weakly anharmonic classicaloscillators coupled at its ends to heat baths at different temperatures. Eachoscillator is subject to pinning potential and it also interacts with itsnearest neighbors. In our set up both potentials are homogeneous and bounded(with $N$ dependent bounds) perturbations of the harmonic ones. We show how ageneralized version of Bakry-Emery theory can be adapted to this case of ahypoelliptic generator which is inspired by F. Baudoin (2017).  By that we prove exponential convergence to non-equilibrium steady state(NESS) in Wasserstein-Kantorovich distance and in relative entropy withquantitative rates. We estimate the constants in the rate by solving aLyapunov-type matrix equation and we obtain that the exponential rate, for thehomogeneous chain, has order bigger than $N^{-3}$. For the purely harmonicchain the order of the rate is in $ [N^{-3},N^{-1}]$. This shows that, in thisset up, the spectral gap decays at most polynomially with $N$.

Quantitative Rates of Convergence to Non-Equilibrium Steady State for a Weakly Anharmonic Chain of Oscillators