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

TL;DR

This paper investigates how standard normality tests detect deviations from Gaussian distributions in finite N-body systems, providing a detailed analysis of test power and practical guidelines for identifying finite-size effects.

Contribution

It introduces a comprehensive Monte Carlo analysis of normality tests applied to finite-N equilibrium distributions, offering new insights into their effectiveness and practical detection thresholds.

Findings

Normality tests vary in sensitivity to finite-N deviations

A heuristic scaling law predicts detectability of finite-size effects

Practical reference tables assist in experimental detection

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 and a heuristic scaling law, visualised as a contour plot, that together indicate when finite-size effects remain detectable.