Localization of energy in FPU chains

by Luisa Berchialla, Luigi Galgani and Antonio Giorgilli

**Abstract:**
We revisit the celebrated model of Fermi, Pasta and Ulam
with the aim of investigating, by numerical computations, the trend
towards equipartition in the thermodynamic limit. We concentrate our
attention on a particular class of initial conditions, namely, with
all the energy on the first mode or the first few modes. We observe
that the approach to equipartition occurs on two different time
scales: in a short time the energy spreads up by forming a packet
involving all low--frequency modes up to a cutoff frequency
$\omega_c$, while a much longer time is required in order to reach
equipartition, if any. In this sense one has an energy localization
with respect to frequency. The crucial point is that our numerical
computations suggest that this phenomenon of a fast formation of a
natural packet survives in the thermodynamic limit. More precisely we
conjecture that the cutoff frequency $\omega_c$ is a function of the
specific energy $\epsilon = E/N$, where $E$ and $N$ are the total
energy and the number of particles, respectively. Equivalently, there
should exist a function $\epsilon_c(\omega)$, representing the minimal
specific energy at which the natural packet extends up to frequency
$\omega$. The time required for the fast formation of the natural
packet is also investigated.

preprint (2002)

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