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We consider a diffusion on a potential landscape which is given by a smoothHamiltonian $H:\mathbb {R}^n\to \mathbb {R}$ in the regime of low temperature$\varepsilon$. We proof the Eyring-Kramers formula for the optimal constant inthe Poincar\'{e} (PI) and logarithmic Sobolev inequality (LSI) for theassociated generator $L=\varepsilon \Delta -\nabla H\cdot\nabla$ of thediffusion. The proof is based on a refinement of the two-scale approachintroduced by Grunewald et al. [Ann. Inst. Henri Poincar\'{e} Probab. Stat. 45(2009) 302-351] and of the mean-difference estimate introduced by Chafa\"{i}and Malrieu [Ann. Inst. Henri Poincar\'{e} Probab. Stat. 46 (2010) 72-96]. TheEyring-Kramers formula follows as a simple corollary from two main ingredients:The first one shows that the PI and LSI constant of the diffusion restricted tometastable regions corresponding to the local minima scales well in$\varepsilon$. This mimics the fast convergence of the diffusion to metastablestates. The second ingredient is the estimation of a mean-difference by aweighted transport distance. It contains the main contribution to the PI andLSI constant, resulting from exponentially long waiting times of jumps betweenmetastable states of the diffusion.

Poincar\'{e} and logarithmic Sobolev inequalities by decomposition of the energy landscape