  This paper is devoted to some qualitative descriptions and some numerical
results for ergodic Mean Field Games systems which arise, e.g., in the
homogenization with a small noise limit. We shall consider either power type
potentials or logarithmic type ones. In both cases, we shall establish some
qualitative properties of the effective Hamiltonian $\bar H$ and of the
effective drift $\bar b$. In particular we shall provide two cases where the
effective system keeps/looses the Mean Field Games structure, namely where
$\nabla_P \bar H(P,\alpha)$ coincides or not with $\bar b(P, \alpha)$.
  On the other hand, we shall provide some numerical tests validating the
aforementioned qualitative properties of $\bar H$ and $\bar b$. In particular,
we provide a numerical estimate of the discrepancy $\nabla_P \bar
H(P,\alpha)-\bar b(P, \alpha)$.
