  Consider a lattice gas evolving according to the conservative Kawasaki
dynamics at inverse temperature $\beta$ on a two dimensional torus
$\Lambda_L=\{0,..., L-1\}^2$ . We prove the tunneling behavior of the process
among the states of minimal energy. More precisely, assume that there are
$n^2\ll L$ particles and that the initial state is the configuration in which
all sites of the square $\mb x + \{0,..., n-1\}^2$ are occupied. We show that
in the time scale $e^{2\beta}$ the process is close to a Markov process on
$\Lambda_L$ which jumps from any site $\mb x$ to any other site $\mb y\not =\mb
x$ at a strictly positive rate which can be expressed in terms of the jump
rates of simple random walks.
