We investigate thermalization of one-dimensional lattices with asymmetric interactions. For the FPUT-$\alpha \beta$ lattices with different boundary conditions, we observe distinct thermalization behaviors. Specifically, we find that the Toda lattice with free boundary conditions exhibits asymptotic integrability, which accounts for these differences. The FPUT-$\alpha \beta$ lattice and the Toda lattice with free boundary conditions should be classified as a perturbed Harmonic lattice. The scaling relationship between the thermalization time and the specified perturbation strength remains consistent with the predictions of the wave turbulence theory. The impact of internal pressure on the scaling relationship is also briefly discussed.

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• Chaos
• Classical statistical mechanics
• Nonequilibrium statistical mechanics