  We give a lower bound on the spectral gap for a class of stochastic energy
exchange models. In 2011, Grigo et al. introduced the model and showed that,
for a class of stochastic energy exchange models with a uniformly positive rate
function, the spectral gap of an $N$-component system is bounded from below by
a function of order $N^{-2}$. In this paper, we consider the case where the
rate function is not uniformly positive. For this case, the spectral gap
depends not only on $N$ but also on the averaged energy $\mathcal{E}$, which is
the conserved quantity under the dynamics. Under some assumption, we obtain a
lower bound of the spectral gap which is of order $C(\mathcal{E})N^{-2}$ where
$C(\mathcal{E})$ is a positive constant depending on $\mathcal {E}$. As a
corollary of the result, a lower bound of the spectral gap for the mesoscopic
energy exchange process of billiard lattice studied by Gaspard and Gilbert [J.
Stat. Mech. Theory Exp. 2008 (2008) p11021, J. Stat. Mech. Theory Exp. 2009
(2009) p08020] and the stick process studied by Feng et al. [Stochastic
Process. Appl. 66 (1997) 147-182] are obtained.
