down phenomenon associated with positive $\Lambda$. In fact, Panageas and Vishnoi [@panageas-vishnoi-16] and Panageas et al. [@panageas-etal-16] show that the mixing time is $O(\log N)$ in the discrete-time setting. Kifer [@kifer-90] considers a more restrictive discrete-time model, which does not cover the mean-field model, and identifies the constant analogous to $\Lambda$ [@kifer-90 Theorem 4.3]. The restriction is that the state space of $\mu_N$ is the same for each $N$ and that a certain uniform finite duration large deviation principle should hold with the rate function satisfying a continuity property. One can view our result as an extension of Kifer’s [@kifer-90 Theorem 4.3] to the continuous time *mean-field* setting, where the state space of the Markov process $\mu_N$ changes with $N$. Hwang and Sheu [@hwang-sheu-90] establish a result similar to ours on the scaling of the second eigenvalue of a reversible small noise diffusion process, and our method of proof is inspired by their approach.

As an application of the large time behaviour of $\mu_N$, we also study convergence of the empirical measure process to a global minimum of a natural ‘entropy’ function when particles are injected over time at a specific rate reminiscent of the simulated annealing algorithm’s cooling schedule, 