  This article examines large time behaviour of finite state mean-field
interacting particle systems. Our first main result is a sharp estimate (in the
exponential scale) on the time required for convergence of the empirical
measure process of the $N$-particle system to its invariant measure; we show
that when time is of the order of $\exp\{N\Lambda\}$ for a suitable constant
$\Lambda \geq 0$, the process has mixed well and it is close to its invariant
measure. We then obtain large-$N$ asymptotics of the second largest eigenvalue
of the generator associated with the empirical measure process when it is
reversible with respect to its invariant measure. We show that its absolute
value scales as $\exp\{-N\Lambda\}$. The main tools used in establishing our
results are the large deviation properties of the empirical measure process
from its large-$N$ limit. As an application of the study of large time
behaviour, we also show convergence of the empirical measure of the system of
particles to a global minimum of a certain `entropy' function when particles
are added over time in a controlled fashion. The controlled addition of
particles is analogous to the cooling schedule associated with the search for a
global minimum of a function using the simulated annealing algorithm.
