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The Fermi-Pasta-Ulam (FPU) one-dimensional Hamiltonian includes a quarticterm which guarantees ergodicity of the system in the thermodynamic limit.Consistently, the Boltzmann factor $P(\epsilon) \sim e^{-\beta \epsilon}$describes its equilibrium distribution of one-body energies, and its velocitydistribution is Maxwellian, i.e., $P(v) \sim e^{- \beta v^2/2}$. We considerhere a generalized system where the quartic coupling constant between sitesdecays as $1/d_{ij}^{\alpha}$ $(\alpha \ge 0; d_{ij} = 1,2,\dots)$. Through{\it first-principle} molecular dynamics we demonstrate that, for large$\alpha$ (above $\alpha \simeq 1$), i.e., short-range interactions, Boltzmannstatistics (based on the {\it additive} entropic functional $S_B[P(z)]=-k \intdz P(z) \ln P(z)$) is verified. However, for small values of $\alpha$ (below$\alpha \simeq 1$), i.e., long-range interactions, Boltzmann statisticsdramatically fails and is replaced by q-statistics (based on the {\itnonadditive} entropic functional $S_q[P(z)]=k (1-\int dz [P(z)]^q)/(q-1)$, with$S_1 = S_B$). Indeed, the one-body energy distribution is q-exponential,$P(\epsilon) \sim e_{q_{\epsilon}}^{-\beta_{\epsilon} \epsilon} \equiv[1+(q_{\epsilon} - 1) \beta_{\epsilon}{\epsilon}]^{-1/(q_{\epsilon}-1)}$ with$q_{\epsilon} > 1$, and its velocity distribution is given by $P(v) \sime_{q_v}^{ - \beta_v v^2/2}$ with $q_v > 1$. Moreover, within small error bars,we verify $q_{\epsilon} = q_v = q$, which decreases from an extrapolated valueq $\simeq$ 5/3 to q=1 when $\alpha$ increases from zero to $\alpha \simeq 1$,and remains q = 1 thereafter.

Fermi--Pasta--Ulam--Tsingou problems: Passage from Boltzmann to $q$-statistics