  We show that a general one-dimensional (1D) lattice with nonlinear
inter-particle interactions can always be thermalized for arbitrarily small
nonlinearity in the thermodynamic limit, thus proving equipartition hypothesis
in statistical physics for an important class of systems. Particularly, we find
that in the lattices of interaction potential $V(x)=x^2/2+\lambda x^n/n$ with
$n\geq4$, there is \textit{a universal scaling law} for the thermalization time
$T^{eq}$, i.e., $T^{eq}\propto\lambda^{-2}\epsilon^{-(n-2)}$, where $\epsilon$
is the energy density. Numerical simulations confirm that it is accurate for an
even $n$. A slight correction is needed for an odd $n$, which is due to the
Chirikov overlap occurring in the weakly nonlinear regime between extra
vibration modes excited by the asymmetry of potential. Based on this scaling
law, as well as previous prediction for the case of $n=3$, a universal formula
for the thermalization time for a 1D lattice with a general interaction
potential is obtained.
