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

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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/dαij (α≥0;dij=1,2,…). Through {\it first-principle} molecular dynamics we demonstrate that, for large α (above α≃1), i.e., short-range interactions, Boltzmann statistics (based on the {\it 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 {\it 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)∼e−βϵϵqϵ≡[1+(qϵ−1)βϵϵ]−1/(qϵ−1) with qϵ>1, and its velocity distribution is given by P(v)∼e−βvv2/2qv 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.

D. Bagchi, C. Tsallis, Fermi–Pasta–Ulam–Tsingou problems: Passage from Boltzmann to q-statistics, Physica A, Vol. 491 (2018) 869–873