The Eyring-Kramers formula for Poincaré and logarithmic Sobolev inequalities

Let us consider a diffusion process on a potential landscape which is given by a smooth Hamiltonian function in the regime of small noise. We give a new proof of the Eyring-Kramers formula for the spectral gap of the associated generator of the diffusion. The proof consists of a divide-and-conquer strategy, which mimics the two-scale approach introduced by Grunewald, Otto, Villani, and Westdickenberg. The Eyring-Kramers formula follows as a simple corollary from two main ingredients: The first one shows that the Gibbs measure restricted to a basin of attraction has a ”good” Poincaré constant providing the fast convergence of the diffusion to metastable states. The second ingredient is a refinement of the mean-difference estimate introduced by Chafai and Malrieu. There, we propose a new weighted transportation distance, which contains the main contribution of the spectral gap, resulting from exponential long waiting times of jumps between metastable states of the diffusion. This new approach also results in a sharp estimate of the constant of the logarithmic Sobolev inequality.

(joint work with Georg Menz)