# Non-Equilibrium Steady States for Networks of Oscillators

Non-equilibrium steady states for chains of oscillators interacting with stochastic heat baths at different temperatures have been the subject of several studies. In this talk I will discuss how to generalize these results to multidimensional networks of oscillators. I will first introduce the model and motivate it from a physical point of view. Then, I will present conditions on the topology of the network and on the interaction potentials which imply the existence and uniqueness of the non-equilibrium steady state, as well as exponential convergence to it. The two main ingredients of the proof are (1) a controllability argument using Hörmander's bracket criterion and (2) a careful study of the high-energy dynamics which leads to a Lyapunov-type condition. I will also mention cases where the non-equilibrium steady state is not unique, and cases where its existence is an open problem. This is joint work with J.-P. Eckmann, M. Hairer and L. Rey-Bellet, Electronic Journal of Probability 23(55): 1-28, 2018 (arXiv:1712.09413).