  The Peierls equation is considered for the Fermi-Pasta-Ulam $\beta$ lattice.
Explicit form of the linearized collision operator is obtained. Using this form
the decay rate of the normal mode energy as a function of wave vector $k$ is
estimated to be proportional to $k^{5/3}$. This leads to the $t^{-3/5}$ long
time behavior of the current correlation function, and, therefore, to the
divergent coefficient of heat conductivity. These results are in good agreement
with the results of recent computer simulations. Compared to the results
obtained though the mode coupling theory our estimations give the same $k$
dependence of the decay rate but a different temperature dependence. Using our
estimations we argue that adding a harmonic on-site potential to the
Fermi-Pasta-Ulam $\beta$ lattice may lead to finite heat conductivity in this
model.
