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 $N$ insufficiently large for the thermodynamic limit, the velocity distribution that maximises Havrda-Charv\'at entropy departs markedly from the Maxwell-Boltzmann (Gaussian) form obtained in that limit. We explore how five standard normality tests-Kolmogorov-Smirnov, Anderson-Darling, Cram\'er-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 $N$ and sample size $n$, providing practical reference tables for deciding when finite-size effects remain detectable.