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

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(ϵ)∼e−βϵ describes its equilibrium distribution of one-body energies, and its velocity distribution is Maxwellian, i.e., P(v)∼e−βv2∕2. We consider here a generalized system where the quartic coupling constant between sites decays as 1∕dijα(α≥0;dij=1,2,…). Through first-principle molecular dynamics we demonstrate that, for large α (above α≃1), i.e., short-range interactions, Boltzmann statistics (based on the additive entropic functional SB[P(z)]=−k∫dzP(z)lnP(z)) is verified. However, for small values of α (below α≃1), i.e., long-range interactions, Boltzmann statistics dramatically fails and is replaced by q-statistics (based on the nonadditive entropic functional Sq[P(z)]=k(1−∫dz[P(z)]q)∕(q−1), with S1=SB). Indeed, the one-body energy distribution is q-exponential, P(ϵ)∼eqϵ−βϵϵ≡[1+(qϵ−1)βϵϵ]−1∕(qϵ−1) with qϵ>1, and its velocity distribution is given by P(v)∼eqv−βvv2∕2 with qv>1. Moreover, within small error bars, we verify qϵ=qv=q, which decreases from an extrapolated value q≃5∕3 to q=1 when α increases from zero to α≃1, and remains q=1 thereafter.