We study the statistical behavior of the 1D nonlinear \beta-FPU chain.
When the nonlinearity is weak the thermal equilibrium state of the
 system can be described as weakly interacting linear waves, and
the wave action n_k is proportional to 1/w_k with w_k - linear
dispersion (Rayleigh-Jeans distribution).We demonstrate numerically that
surprisingly even in the strongly nonlinear limit the system still
can be effectively described like weakly interacting waves and hence
the same distribution holds. This arises because strong nonlinearity
effectively renormalizes the linear dispersion frequency. We find
the theoretical prediction for the frequency renormalizing factor and
show that it is in good agreement with numerical experiments. Moreover
the presence of nonlinearity widens the resonance peaks which results
in the occurrence of near-resonance wave-wave interactions due to
discreteness.

    We also employ Random Phase Approximation
and use the Wiener-Khinchin formula and the perturbation methods to
derive the form of the resonance width as a function of the wave
number and the nonlinearity strength. The resulting analytical
prediction for the resonance width is in good agreement with
numerical experiments.Finally we numerically observed that in the
thermodynamical equilibrium \beta-FPU exhibits spatially localized
oscillations - Discrete Breathers.