When Normality Tests Detect Equilibrium Distributions of Finite N-Body Systems

Abstract

The particle number N can be used as a quantitative gauge of non-Gaussianity. This idea extends to systems that are not literally finite by assigning them a notional N that captures the same deviation. For an ideal gas with insufficiently large for the thermodynamic limit, the velocity distribution that maximises Havrda–Charvát entropy departs markedly from the Maxwell–Boltzmann (Gaussian) form obtained in that limit. We explore how five standard normality tests—Kolmogorov–Smirnov, Anderson–Darling, Cramér–von Mises, Jarque–Bera and Shapiro–Wilk—respond to samples drawn from this finite-N equilibrium distribution. A large-scale Monte Carlo study maps the tests’ statistical power across system size and sample size , providing practical reference tables and a heuristic scaling law, visualised as a contour plot, that together indicate when finite-size effects remain detectable.