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We consider a drift-diffusion process on a smooth potential landscape
with small noise. We give a new proof of the Eyring-Kramers formula
which asymptotically characterizes the spectral gap of the generator of
the diffusion. The proof is based on a refinement of the two-scale
approach introduced by Grunewald, Otto, Villani, and Westdickenberg and
of the mean-difference estimate introduced by Chafai and Malrieu. The
new proof exploits the idea that the process has two natural
time-scales: a fast time-scale resulting from the fast convergence to a
metastable state, and a slow time-scale resulting from exponentially
long waiting times of jumps between metastable states. A nice feature
of the argument is that it can be used to deduce an asymptotic formula
for the log-Sobolev constant, which was previously unknown.