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 $P(\epsilon) \sim e^{ -\beta \epsilon}$ describes its equilibrium distribution of one-body energies, and its velocity distribution is Maxwellian, i.e., $P(v) \sim e^{ - \beta v^2/2}$. We consider here a generalized system where the quartic coupling constant between sites decays 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, Boltzmann statistics (based on the { \it additive} entropic functional $S_B[P(z)]=-k \int dz P(z) \ln P(z)$) is verified. However, for small values of $\alpha$ (below $\alpha \simeq 1$), i.e., long-range interactions, Boltzmann statistics dramatically fails and is replaced by q-statistics (based on the { \it nonadditive} 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 $P(v) \sim e_{ q_v}^{ - \beta_v v^2/2}$ with $q_v > 1$. Moreover, within small error bars, we verify $q_{ \epsilon} = q_v = q$, which decreases from an extrapolated value q $\simeq$ 5/3 to q=1 when $\alpha$ increases from zero to $\alpha \simeq 1$, and remains q = 1 thereafter.
Categories: cond-mat.stat-mech