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 r_{ij} as 1/r_{ij}^{α} (α≥0) is studied through first-principle molecular-dynamics simulations on d-dimensional lattices of linear size L (N≡L^{d} 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 T_{QSS}; for fixed N and after a sufficiently long time, a crossover to a second plateau occurs, corresponding to the Boltzmann-Gibbs temperature T_{BG} (as predicted within the BG theory), with T_{BG}>T_{QSS}. It is shown that the QSS duration (t_{QSS}) depends on N, α, and d, although the dependence on α appears only through the ratio α/d; in fact, t_{QSS} decreases with α/d and increases with both N and d. Considering a fixed energy value, a scaling for t_{QSS} is proposed, namely, t_{QSS}∝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.