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

We study a long-range–interaction generalisation of the one-dimensional Fermi-PastaUlam (FPU) β-model, by introducing a quartic interaction coupling constant that decays as 1/r (α ≥ 0) (with strength characterised by b > 0). In the α → ∞ limit we recover the original FPU model. Through molecular dynamics we show that i) for α ≥ 1 the maximal Lyapunov exponent remains finite and positive for an increasing number of oscillators N , whereas, for 0 ≤ α < 1, it asymptotically decreases as N; 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 q(α) 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 1/N ∝ b/tc with γ > 0 and δ > 0, in such a way that the q = 1 (BG) behaviour dominates in the limN→∞ limt→∞ ordering, while in the limt→∞ limN→∞ ordering q > 1 statistics prevails. Copyright c © EPLA, 2014 More than one century ago, in his historical book Elementary Principles in Statistical Mechanics [1], Gibbs remarked that systems involving long-range interactions will be intractable within his and Boltzmann’s theory, due to the divergence of the partition function. This is of course the reason why no standard temperaturedependent thermostatistical quantities (e.g., specific heat) can possibly be calculated for the free hydrogen atom, for instance. Indeed, unless a box surrounds the atom, an infinite number of excited energy levels accumulate at the ionisation value, which yields a divergent canonical partition function at any finite temperature. Related discussions can be seen in [2–5], for instance. To investigate the deep consequences of Gibbs’ remark, we focus on the influence of the linear and nonlinear long-range interactions (LRI) within an isolated system. In particular, we use the classical Fermi-Pasta-Ulam (FPU) β-model [6–10], which combines linear and nonlinear nearest-neighbor interactions and allows to study separately the linear and nonlinear nature of the LRI. In the present paper we focus primarily on the FPU β-model with nonlinear LRI, since the most interesting phenomena appear in this case, although we write the Hamiltonian in a compact form which includes both kinds of long range: