Recent results on the Fermi–Pasta–Ulam problem

The celebrated model of Fermi, Pasta, and Ulam with the aim of investigating the thresholds to equipartition in the thermodynamic limit is revisited. Starting with a particular class of initial conditions, i.e., with all the energy on the first mode, we observe that in a short time the system splits in two separate subsystems. We conjecture the existence of a function $\epsilon_c(\omega)$, independent on the number $N$ of particles in the chain, such that if the initial energy $E$ satisfies $E/N<\epsilon_c(\omega)$ then only the packet of modes with frequency not exceeding $\omega$ shares most of the energy.