Nearly separable behavior of Fermi-Pasta-Ulam chains through the stochasticity threshold
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For the periodic Fermi-Pasta-Ulam chain with quartic potential we prove the relation 〈p k 2 〉 T ≈ (1+α) 〈ω k 2 q k 2 〉τ , i.e., the proportionality, already at early timesT, between averaged kinetic and harmonic energies of each mode. The factor α depends on the parameters of the model, but not on the mode and the number of degrees of freedom. It grows with the anharmonic strength from the value α=0 of the harmonic limit (virial theorem) up to few units for the system much above the threshold. In the stochastic regime the time necessary to reduce the fluctuations ink to a fixed percentage remains at least one order of magnitude smaller than the time necessary to reach a similar level of equipartition. The persistence of such a behavior even above the stochasticity threshold clarifies a number of previous numerical results on the relaxation to equilibrium: e.g., the existence of several time scales and the relevance of the harmonic frequency spectrum. The difficulties in the numerical simulation of the thermodynamic limit are also discussed.
Key WordsVirial theorem approach to equilibrium Fermi-Pasta-Ulam model stochasticity threshold rates of energy exchanges thermodynamic limit
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