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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