Thermalization of isotopically disordered chain

Recently, the equipartition hypothesis in statistical physics has been verified in the case of one-dimensional lattices based on the wave turbulence theory. It has been shown that thermalization should occur for arbitrarily small nonlinearity. Particularly, in the lattices with the polynomial interaction potential $V(x)=x^2/2+\lambda x^n/n$ for $n>4$, the thermalization time $T_{eq}$ is proportional to $\lambda^{-2}$ in the thermodynamic limit. In this paper, we apply wave turbulence theory to study the dynamics of one-dimensional disordered chains with the polynomial potential. We show that the universal scaling law can be extended to $n=3$ since the three-wave resonances are allowable due to the space-translation symmetry broken. Our results indicate that wave turbulence theory is still applicable to disordered systems.