Quasi-stationary-state duration in the classical d-dimensional long-range inertial XY ferromagnet

A classical α − XY inertial model, consisting of N two-component rotators and characterized by interactions decaying with the distance rij as 1/rαij (α ≥ 0) is studied through first-principle molecular-dynamics simulations on d - dimensional lattices of linear size L (N ≡ Ld and d = 1, 2, 3). The limits α = 0 and α → ∞ correspond to infinite-range and nearest-neighbor interactions, respectively, whereas the ratio α/d > 1 (0 ≤ α/d ≤ 1) is associated with short-range (long-range) interactions. By analyzing the time evolution of the kinetic temperature T (t) in the long-range-interaction regime, one finds a quasi-stationary state (QSS) characterized by a temperature TQSS; for fixed N and after a sufficiently long time, a crossover to a second plateau occurs, corresponding to the Boltzmann-Gibbs temperature TBG (as predicted within the BG theory), with TBG > TQSS. It is shown that the QSS duration (tQSS) depends on N, α, and d, although the dependence on α appears only through the ratio α/d; in fact, tQSS decreases with α/d and increases with both N and d. Considering a fixed energy value, a scaling for tQSS is proposed, namely, tQSS ∝ N A (α/d) e − B(N)(α/d)2, analogous to a recent analysis carried out for the classical α-Heisenberg inertial model. It is shown that the exponent A (α/d) and the coefficient B (N) present universal behavior (within error bars), comparing the XY and Heisenberg cases. The present results should be useful for other long-range systems, very common in nature, like those characterized by gravitational and Coulomb forces.