Freezing of the optical--branch energy in a diatomic FPU chain

We study the exchange of energy between the modes of the optical branch and those of the acoustic one in a diatomic chain of particles, with masses $m_1$ and $m_2$. We prove that, at small temperature and provided $m_1\gg m_2$, for the majority of the initial data the energy of each branch is approximately constant for times of order ${\beta }^{S\mathrm{/}2}$, where $S=\lfloor \sqrt{m_1/m_2}/2\rfloor$and$\beta$ is the inverse temperature. The result is uniform in the thermodynamic limit.