2017/09/25 by Debarshee Bagchi, Constantino Tsallis · 1 citation
Physics and Astronomy · #cond-mat.stat-mech
paper · pdf · doi:10.1016/j.physa.2017.09.098
7 pages including 4 figures
arxiv created 2017/09/25 · arxiv updated 2017/11/22
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 \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 ∫ dz P(z) ln P(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) ∼ eqε-βε ε ≡ [1+(qε - 1) βεε]-1/(qε-1) with qε > 1, and its velocity distribution is given by P(v) ∼ eqv - βv v2/2 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.