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

We study a long-range–interaction generalisation of the one-dimensional Fermi-Pasta-Ulam (FPU) ?-model, by introducing a quartic interaction coupling constant that decays as $1/r^\alpha\ (\alpha \ge 0)$ (with strength characterised by b > 0). In the $\alpha \to\infty$ limit we recover the original FPU model. Through molecular dynamics we show that i) for $\alpha \geq 1$ the maximal Lyapunov exponent remains finite and positive for an increasing number of oscillators N, whereas, for $0 \le \alpha <1$ , it asymptotically decreases as $N^{-\kappa(\alpha)}$ ; ii) the distribution of time-averaged velocities is Maxwellian for ? 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 ? increases from zero to close to one. For ? small enough, a crossover occurs at time tc from q-statistics to Boltzmann-Gibbs (BG) thermostatistics, which defines a phase diagram for the system with a linear boundary 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 (BG) behaviour dominates in the $\lim_{N \to\infty} \lim_{t \to\infty}$ ordering, while in the $\lim_{t \to\infty} \lim_{N \to\infty}$ ordering q > 1 statistics prevails.