Fermi-Pasta-Ulam model with long-range interactions: Dynamics and thermostatistics

Abstract

We study a long-range–interaction generalisation of the one-dimensional Fermi-Pasta-Ulam (FPU) β-model, by introducing a quartic interaction coupling constant that decays as (with strength characterised by b > 0). In the limit we recover the original FPU model. Through molecular dynamics we show that i) for the maximal Lyapunov exponent remains finite and positive for an increasing number of oscillators N, whereas, for , it asymptotically decreases as ; ii) the distribution of time-averaged velocities is Maxwellian for α large enough, whereas it is well approached by a q-Gaussian, with the index monotonically decreasing from about 1.5 to 1 (Gaussian) when α increases from zero to close to one. For α small enough, a crossover occurs at time tc from q-statistics to Boltzmann-Gibbs (BG) thermostatistics, which defines a “phase diagram” for the system with a linear boundary of the form with and , in such a way that the q = 1 (BG) behaviour dominates in the ordering, while in the ordering q > 1 statistics prevails.