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

Jae Wan Shim

TL;DR

The paper addresses finite-$N$ deviations from Gaussianity in a one-dimensional, ideal-gas–like system by deriving a compact-support $q$-Gaussian via Havrda–Charvát entropy and evaluating five standard normality tests under a custom null across a wide $(N,n)$ grid. Using a large-scale Monte Carlo study, it shows that KS and AD maintain correct size under the custom null but have limited power to reject normality at larger $N$, while moment-based tests JB and SW exhibit much higher power, especially for smaller sample sizes, effectively detecting finite-size effects through skewness and kurtosis. As $N$ increases toward the Gaussian limit, all tests lose power, with ECDF-based tests requiring tens of thousands of observations to reach high power, whereas JB/SW reach high power with far fewer samples. The results provide practical guidance on test choice and required sample sizes for detecting finite-$N$ deviations in equilibrium distributions, with implications for physics and related fields where non-Gaussian finite-size effects are relevant.

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-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 $N$ and sample size $n$, providing practical reference tables for deciding when finite-size effects remain detectable.

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

TL;DR

The paper addresses finite- deviations from Gaussianity in a one-dimensional, ideal-gas–like system by deriving a compact-support -Gaussian via Havrda–Charvát entropy and evaluating five standard normality tests under a custom null across a wide grid. Using a large-scale Monte Carlo study, it shows that KS and AD maintain correct size under the custom null but have limited power to reject normality at larger , while moment-based tests JB and SW exhibit much higher power, especially for smaller sample sizes, effectively detecting finite-size effects through skewness and kurtosis. As increases toward the Gaussian limit, all tests lose power, with ECDF-based tests requiring tens of thousands of observations to reach high power, whereas JB/SW reach high power with far fewer samples. The results provide practical guidance on test choice and required sample sizes for detecting finite- deviations in equilibrium distributions, with implications for physics and related fields where non-Gaussian finite-size effects are relevant.

Abstract

The particle number 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 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- equilibrium distribution. A large-scale Monte-Carlo study maps the tests' statistical power across system size and sample size , providing practical reference tables for deciding when finite-size effects remain detectable.

Paper Structure

This paper contains 14 sections, 4 equations, 2 figures, 1 table.

Table of Contents

  1. Introduction
  2. Simulation Procedure
  3. Target distribution
  4. Loop structure
  5. Simulation results
  6. Size control under the custom null
  7. Power as a function of $N$
  8. Sample-size effects at $N=20$
  9. Summary
  10. Simulation Results: Statistical Power Surface Plots
  11. Conclusion
  12. Practical takeaway.
  13. Outlook.
  14. Tables

Figures (2)

  • Figure 1: Comparison of statistical power surfaces for five goodness-of-fit tests. The moment-based tests (JB and SW) demonstrate significantly higher power across a wider range of parameters compared to the ECDF-based tests (KS, AD, and CvM).
  • Figure 2: Statistical power of five goodness-of-fit tests as a function of sample size ($n$) for the custom distribution with parameter $N=20$. While ECDF-based tests (KS, AD, C-vM) require large sample sizes to achieve high power, moment-based tests (JB, SW) demonstrate high power even with smaller datasets.