Universal scaling of the thermalization time in one-dimensional lattices

We show that, in the thermodynamic limit, a one-dimensional (1D) nonlinear lattice can always be thermalized for arbitrarily small nonlinearity, thus proving the equipartition theorem for a class of systems. Particularly, we find that in the lattices with nearest-neighbor interaction potential V(x)=x^{2}/2+λx^{n}/n with n≥4, the thermalization time, T_{eq}, follows a universal scaling law; i.e., T_{eq}∝λ^{-2}ε^{-(n-2)}, where ε is the energy per particle. Numerical simulations confirm that it is accurate for an even n, while a certain degree of deviation occurs for an odd n, which is attributed to the extra vibration modes excited by the asymmetric interaction potential. This finding suggests that although the symmetry of interactions will not affect the system reaching equipartition eventually, it affects the process toward equipartition. Based on the scaling law found here, a unified formula for the thermalization time of a 1D general nonlinear lattice is obtained.