Conclusions

We have studied the statistical behavior of the nonlinear periodic lattice with the nearest neighbor interactions in thermal equilibrium. We have extended the notion of normal modes to the nonlinear system by showing that regardless of the strength of nonlinearity, the system in thermal equilibrium can still be effectively characterized by a complete set of renormalized waves, in the sense that those renormalized waves possess the Rayleigh-Jeans distribution and vanishing correlations between different wave modes. In addition, we have studied the property of dispersion relation of the renormalized waves. The results we obtained in Section II are general and can be applied to the large class of nonlinear systems with the nearest neighbor interactions in thermal equilibrium.

We have further focused our attention on the particular system with the nearest neighbor interactions -- the famous FPU chain. We have confirmed that the general renormalization framework that we discussed above is consistent with the numerical observations. In particular, we have shown that the renormalized dispersion of the thermalized $\beta$-FPU chain is in excellent agreement with the numerical one for a wide range of the nonlinearity strength. We have further demonstrated that the renormalized dispersion is a direct consequence of the trivial resonant interactions of the renormalized waves. Using a self-consistency argument, we have found an approximation of the renormalization factor via a mean-field approximation. In addition, we have used the multiple time-scale, statistical averaging method to obtain the theoretical prediction of the spatiotemporal spectrum and demonstrated that the renormalized waves have long lifetimes. We note that the results obtained here can be extended to general nonlinear potentials with the nearest neighbor interactions.

The scenario of the wave behavior in the thermal equilibrium we obtained here may suggest a theoretical framework for the application of the wave turbulence to $\beta$-FPU in the situation of near-equilibrium.