  We show that the empirical process associated with a system of weakly
interacting diffusion processes exhibits a form of noise-induced metastability.
The result is based on an analysis of the associated McKean--Vlasov free
energy, which, for suitable attractive interaction potentials, has at least two
distinct global minimisers at the critical parameter value $\beta=\beta_c$. On
the torus, one of these states is the spatially homogeneous constant state, and
the other is a clustered state. We show that a third critical point exists at
this value. As a result, we obtain that the probability of transition of the
empirical process from the constant state scales like $\exp(-N \Delta)$, with
$\Delta$ the energy gap at $\beta=\beta_c$. The proof is based on a version of
the mountain pass theorem for lower semicontinuous and $\lambda$-geodesically
convex functionals on the space of probability measures $\mathcal{P}_2(M)$
equipped with the $2$-Wasserstein metric, where $M$ is a complete, connected,
and smooth Riemannian manifold.
