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

Abstract

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 $S_B[P(z)]=-k\int dzP(z)\ln P(z)$) 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.

7 pages including 4 figures

References in corpus (2)

• Thermostatistics of overdamped motion of interacting particles
• Fermi-Pasta-Ulam model with long-range interactions: Dynamics and thermostatistics