Universality of Energy Equipartition in One-dimensional Lattices

Abstract

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.