Non-Extensive Statistical Mechanics in Long-Range Interacting Systems

Abstract

Systems with long-range interactions, where the pairwise coupling decays as a power law with an exponent smaller
than the system's spatial dimension, violate the additivity assumption underlying Boltzmann-Gibbs statistical
mechanics and have been proposed as natural candidates for Tsallis's non-extensive q-entropy framework. We
tested this proposal numerically in the Hamiltonian mean-field (HMF) model, simulating N-body dynamics for
system sizes N from 500 to 8,000 particles with pairwise coupling decaying as r^-alpha for six values of the range
parameter alpha between 0 (fully long-range) and 1.5 (short-range-like), following each system's relaxation from a
fixed out-of-equilibrium initial condition (water-bag distribution) toward its final equilibrium state. For alpha <= 0.6, all
system sizes tested showed an extended quasi-stationary state (QSS) before final relaxation to Boltzmann-Gibbs
equilibrium, with QSS lifetime scaling with system size as N^1.68 (95% CI: 1.59-1.77) at alpha = 0, consistent with
the divergent-lifetime signature expected for genuinely long-range coupling, and momentum distributions during the
QSS phase were fit substantially better by a q-Gaussian (Tsallis) distribution than by a Maxwell-Boltzmann
distribution, with mean fitted q = 1.42 (SD 0.08) across the six N values tested at alpha = 0. The quality advantage
of the q-Gaussian fit, measured as the reduction in reduced chi-squared relative to the Maxwell-Boltzmann fit, was
68% at alpha = 0, fell to 34% at alpha = 0.6, and became negligible (4%, not significant at p = 0.31) for alpha >= 1.0,
tracing a clear crossover from non-extensive to extensive statistical behavior as the interaction range parameter
increased. QSS lifetime scaling exponents decreased systematically with alpha, from 1.68 at alpha = 0 to 0.71 at
alpha = 0.6, and QSS states were no longer statistically distinguishable from rapid Boltzmann-Gibbs relaxation for
alpha >= 1.0 across all tested N. A finite-size scaling collapse of the QSS lifetime data using the fitted exponents
achieved a mean residual sum of squares 76% lower than a fixed-exponent (N^1) collapse, confirming that the
N-dependence of QSS lifetime is genuinely alpha-dependent rather than a uniform finite-size artifact. These results
provide quantitative numerical support for the applicability of q-statistics to the quasi-stationary states of long-range
interacting systems specifically in the regime alpha <= 0.6, while showing that this non-extensive signature vanishes
smoothly as the interaction range shortens toward the additive, short-range regime.