  We consider a diatomic chain characterized by a cubic anharmonic potential.
After diagonalizing the harmonic case, we study in the new canonical variables,
the nonlinear interactions between the acoustical and optical branches of the
dispersion relation. Using the {\it wave turbulence} approach, we formally
derive two coupled wave kinetic equations, each describing the evolution of the
wave action spectral density associated to each branch. An $H$-theorem shows
that there exist an irreversible transfer of energy that leads to an
equilibrium solution characterized by the equipartition of energy in the new
variables. While in the monoatomic cubic chain, in the large box limit, the
main nonlinear transfer mechanism is based on four-wave resonant interactions,
the diatomic one is ruled by a three wave resonant process (two acoustical and
one optical wave): thermalization happens on shorter time scale for the
diatomic chain with respect to the standard chain. Resonances are possible only
if the ratio between the heavy and light masses is less than 3. Numerical
simulations of the deterministic equations support our theoretical findings.
