Recent results on the Fermi-Pasta--Ulam problem
by L. Galgani and A. Giorgilli

Abstract: We revisit the celebrated model of Fermi, Pasta and Ulam with the aim of investigating the thresholds to equipartition in the thermodynamic limit. 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.

presented at the "Workshop on differential equations dedicated to the memory of Vladimir Lazutkin", St.Petersburg, August 18-20, 2002.
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