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Fermi-Pasta-Ulam model with long-range interactions: Dynamics and thermostatistics

We introduce and numerically study a long-range-interaction generalization of the one-dimensional Fermi-Pasta-Ulam (FPU) $β-$ model. The standard quartic interaction is generalized through a coupling constant that decays as $1/r^α$ ($α\ge 0$)(with strength characterized by $b>0$). In the $α\to\infty$ limit we recover the original FPU model. Through classical molecular dynamics computations we show that (i) For $α\geq 1$ the maximal Lyapunov exponent remains finite and positive for increasing number of oscillators $N$ (thus yielding ergodicity), whereas, for $0 \le α<1$, it asymptotically decreases as $N^{- κ(α)}$ (consistent with violation of ergodicity); (ii) The distribution of time-averaged velocities is Maxwellian for $α$ large enough, whereas it is well approached by a $q$-Gaussian, with the index $q(α)$ monotonically decreasing from about 1.5 to 1 (Gaussian) when $α$ increases from zero to close to one. For $α$ small enough, the whole picture is consistent with a crossover at time $t_c$ from $q$-statistics to Boltzmann-Gibbs (BG) thermostatistics. More precisely, we construct a &#34;phase diagram&#34; for the system in which this crossover occurs through a frontier of the form $1/N \propto b^δ/t_c^γ$ with $γ>0$ and $δ>0$, in such a way that the $q=1$ ($q>1$) behavior dominates in the $\lim_{N \to\infty} \lim_{t \to\infty}$ ordering ($\lim_{t \to\infty} \lim_{N \to\infty}$ ordering).