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) $\beta-$ model. The standard quartic interaction is generalized through a coupling constant that decays as $1/r^\alpha$ ($\alpha \ge 0$)(with strength characterized by $b>0$). In the $\alpha \to\infty$ limit we recover the original FPU model. Through classical molecular dynamics computations we show that (i) For $\alpha \geq 1$ the maximal Lyapunov exponent remains finite and positive for increasing number of oscillators $N$ (thus yielding ergodicity), whereas, for $0 \le \alpha <1$, it asymptotically decreases as $N^{- \kappa(\alpha)}$ (consistent with violation of ergodicity); (ii) The distribution of time-averaged velocities is Maxwellian for $\alpha$ large enough, whereas it is well approached by a $q$-Gaussian, with the index $q(\alpha)$ monotonically decreasing from about 1.5 to 1 (Gaussian) when $\alpha$ increases from zero to close to one. For $\alpha$ 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 "phase diagram" for the system in which this crossover occurs through a frontier of the form $1/N \propto b^\delta /t_c^\gamma$ with $\gamma >0$ and $\delta >0$, 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).