  The dispersive interacting waves in Fermi-Pasta-Ulam (FPU) chains of
particles in \textit{thermal equilibrium} are studied from both statistical and
wave resonance perspectives. It is shown that, even in a strongly nonlinear
regime, the chain in thermal equilibrium can be effectively described by a
system of weakly interacting \textit{renormalized} nonlinear waves that possess
(i) the Rayleigh-Jeans distribution and (ii) zero correlations between waves,
just as noninteracting free waves would. This renormalization is achieved
through a set of canonical transformations. The renormalized linear dispersion
of these renormalized waves is obtained and shown to be in excellent agreement
with numerical experiments. Moreover, a dynamical interpretation of the
renormalization of the dispersion relation is provided via a self-consistency,
mean-field argument. It turns out that this renormalization arises mainly from
the trivial resonant wave interactions, i.e., interactions with no momentum
exchange. Furthermore, using a multiple time-scale, statistical averaging
method, we show that the interactions of near-resonant waves give rise to the
broadening of the resonance peaks in the frequency spectrum of renormalized
modes. The theoretical prediction for the resonance width for the thermalized
$\beta$-FPU chain is found to be in very good agreement with its numerically
measured value.
