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The Fermi-Pasta-Ulam system as a model for glasses

We show that the standard Fermi--Pasta--Ulam system, with a suitable choice for the interparticle potential, constitutes a model for glasses, and indeed an extremely simple and manageable one. Indeed, it allows one to describe the landscape of the minima of the potential energy and to deal concretely with any one of them, determining the spectrum of frequencies and the normal modes. A relevant role is played by the harmonic energy $\mathcal E$ relative to a given minimum, i.e., the expansion of the Hamiltonian about the minimum up to second order. Indeed we find that there exists an energy threshold in $\mathcal E$ such that below it the harmonic energy $\mathcal E$ appears to be an approximate integral of motion for the whole observation time. Consequently, the system remains trapped near the minimum, in what may be called a vitreous or glassy state. Instead, for larger values of $\mathcal E$ the system rather quickly relaxes to a final equilibrium state. Moreover we find that the vitreous states present peculiar statistical behaviors, still involving the harmonic energy $\mathcal E$. Indeed, the vitreous states are described by a Gibbs distribution with an effective Hamiltonian close to $\mathcal E$ and with a suitable effective inverse temperature. The final equilibrium state presents instead statistical properties which are in very good agreement with the Gibbs distribution relative to the full Hamiltonian of the system.