# American Institute of Mathematical Sciences

October  2004, 11(4): 855-866. doi: 10.3934/dcds.2004.11.855

## Localization of energy in FPU chains

 1 Università di Milano Bicocca, Piazza dell'Ateneo Nuovo 1, 20126 Milano, Italy 2 Dipartimento di Matematica, Via Saldini 50, 20133 Milano, Italy 3 Dipartimento di Matematica e Applicazioni, Via R. Cozzi 53, 20126 Milano, Italy

Received  February 2003 Revised  March 2004 Published  September 2004

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.
Citation: Luisa Berchialla, Luigi Galgani, Antonio Giorgilli. Localization of energy in FPU chains. Discrete & Continuous Dynamical Systems, 2004, 11 (4) : 855-866. doi: 10.3934/dcds.2004.11.855
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