#### Bagchi, D.,Tsallis, C.

The Fermi-Pasta-Ulam (FPU) one-dimensional Hamiltonian includes a quartic term which guarantees ergodicity of the system in the thermodynamic limit. Consistently, the Boltzmann factor P(is an element of) similar to e-(beta is an element of) describes its equilibrium distribution of one-body energies, and its velocity distribution is Maxwellian, i.e., P(v) similar to e-(beta v2/2). We consider here a generalized system where the quartic coupling constant between sites decays as 1/d(ij)(alpha) (alpha >= 0; d(ij) = 1, 2,...). Through first-principle molecular dynamics we demonstrate that, for large a (above alpha 1), i.e., short-range interactions, Boltzmann statistics (based on the additive entropic functional S-B[P(z)] = k integral dzP(z) In P(z)) is verified. However, for small values of a (below alpha 1), i.e., long-range interactions, Boltzmann statistics dramatically fails and is replaced by q-statistics (based on the nonadditive entropic functional S-q[P(z)] = k(1 integral dz[P(z)](q))/(q - 1), with S-1 = S-B). Indeed, the one-body energy distribution is q-exponential, P(is an element of) similar to e(q is an element of)-[1 + (q(is an element of) - 1)beta(is an element of)is an element of]-(1/(q is an element of)-(1)) with q(is an element of) > 1, and its velocity distribution is given by P(v) similar to e(qv)-(beta vv2/2) with q(v) > 1. Moreover, within small error bars, we verify q(is an element of) = q(v) = q, which decreases from an extrapolated value q5/3 to q = 1 when a increases from zero to alpha 1, and remains q = 1 thereafter.