  In this article, we study the mixing properties of metastable diffusion
processes which possess a Gibbs invariant distribution. For systems with
multiple stable equilibria, so-called metastable transitions between these
equilibria are required for mixing since the unique invariant distribution is
concentrated on these equilibria. Consequently, these systems exhibit slower
mixing compared to those with a unique stable equilibrium, as analyzed in
Barrera and Jara (Ann. Appl. Probab. 30:1164-1208, 2020). . Our proof is based
on the theory of metastability, which is a primary tool for studying systems
with multiple stable equilibria. Within this framework, we compute the total
variation distance between the distribution of the diffusion process and its
invariant distribution for any time scale larger than $\epsilon^{-1}$. Finally,
we derive precise asymptotics for the mixing time.
