$N(t) = \lfloor \frac{\log (2+t)}{c^* + \delta}\rfloor$ for a suitable $c^*$ and any $\delta > 0$. The convergence to a global minimum holds for all starting points (see Theorem \[thm:conv-globalmin\]). Such approaches may be of use in situations where a population growth schedule is applied in order to engineer the mean-field system’s move to a desired equilibrium point as time $t \to \infty$. One can also use this approach to study numerically the most likely region in which the process $\mu_N$ spends time for large values of $N$, under stationarity. Again, our proof is inspired by the analysis of the simulated annealing algorithm in [[@hwang-sheu-90 Part III]]{}.

The rest of the paper is organised as follows. In Section \[section:ldp\_finite\], we discuss a known large deviation principle for the empirical measure process $\mu_N$ over a finite time horizon. This result plays an important role in the study of large time behaviour of $\mu_N$ and the large deviation principle for the invariant measure $\{\wp_N\}_{N\geq 1}$. We then study the large time behaviour of the process $\mu_N$ in Section \[section:large\_time\_behaviour\], and prove our first main result on the proximity of the law of $\mu_N$ to its invariant measure. In Section \[section:eval\_problem\], we study 