Dynamics and Statistics of the Fermi--Pasta--Ulam $\beta $--model with different ranges of particle interactions

In the present work we study the Fermi--Pasta--Ulam (FPU) $\beta$--model involving long--range interactions (LRI) in both the quadratic and quartic potentials, by introducing two independent exponents $\alpha_1$ and $\alpha_2$ respectively, which make the {forces decay} with distance $r$. Our results demonstrate that weak chaos, in the sense of decreasing Lyapunov exponents, and $q$--Gaussian probability density functions (pdfs) of sums of the momenta, occurs only when long--range interactions are included in the quartic part. More importantly, for $0\leq \alpha_2<1$, we obtain extrapolated values for $q \equiv q_\infty >1$, as $N\rightarrow \infty$, suggesting that these pdfs persist in that limit. On the other hand, when long--range interactions are imposed only on the quadratic part, strong chaos and purely Gaussian pdfs are always obtained for the momenta. We have also focused on similar pdfs for the particle energies and have obtained $q_E$-exponentials (with $q_E>1$) when the quartic-term interactions are long--ranged, otherwise we get the standard Boltzmann-Gibbs weight, with $q=1$. The values of $q_E$ coincide, within small discrepancies, with the values of $q$ obtained by the momentum distributions.