  Whether and how a system reaches thermalization is a fundamental issue of
statistical physics. While for one-dimensional lattices this issue has been
intensively studied in terms of energy equipartition for more than half a
century, few work has been performed in the case of two- and three-dimensional
lattices, and thus the thermalization dynamics remains unclear for more
realistic lattices. In this Letter we investigate analytically and numerically
the time-scaling of energy relaxation in these lattices. We show that the
equipartition of energy is generally reached following a universal scheme for
large enough lattices, regardless of its dimensionality, its specific lattice
structure, and whether the system is translation invariant or not. Our results
have practical significance in exploring the effect of high-order
nonlinearities, i.e., the combining effect of multi-phonon process, in solid
materials.
