this paper and highlight our main results.

Suppose that the limiting McKean-Vlasov equation (\[eqn:MVE\]) has multiple $\omega$-limit sets (multiple stable equilibria and/or limit cycles). If we focus on a fixed time interval $[0,T]$, let the number of particles $N \to \infty$ and let the initial conditions $\mu_N(0)$ converge weakly to a deterministic limit $\nu$, then the mean-field convergence suggests that the empirical measure process tracks the solution to the McKean-Vlasov equation (\[eqn:MVE\]) over $[0,T]$ starting at $\nu$. If we then let $t \to \infty$, the solution to the McKean-Vlasov equation goes to an $\omega$-limit set of (\[eqn:MVE\]) depending on the initial condition $\nu$. On the other hand, for a large but fixed $N$, the process would track the McKean-Vlasov equation with very high probability, and as time becomes large, would thus enter a neighbourhood of the $\omega$-limit set corresponding to the initial condition $\nu$; however, since $N$ is finite, the process can exit the basin of attraction of this $\omega$-limit set. It is then likely to remain in a neighbourhood of another $\omega$-limit set for a large amount of time before transiting to the next one, and so on. These are examples of *metastable phenomena,* and it turns out that 