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When Stein-Type Test Detects Equilibrium Distributions of Finite N-Body Systems

Jae Wan Shim

Abstract

Starting from the probability distribution of finite N-body systems, which maximises the Havrda--Charvát entropy, we build a Stein-type goodness-of-fit test. The Maxwell--Boltzmann distribution is exact only in the thermodynamic limit, where the system is composed of infinitely many particles as N approaches infinity. For an isolated system with a finite number of particles, the equilibrium velocity distribution is compact and markedly non-Gaussian, being restricted by the fixed total energy. Using Stein's method, we first obtain a differential operator that characterises the target density. Its eigenfunctions are symmetric Jacobi polynomials, whose orthogonality yields a simple, parameter-free statistic. Under the null hypothesis that the data follows the finite-N distribution, the statistic converges to a chi-squared law, so critical values are available in closed form. Large-scale Monte Carlo experiments confirm exact size control and give a clear picture of the power. These findings quantify how quickly a finite system approaches the classical limit and provide a practical tool for testing kinetic models in regimes where normality cannot be assumed.

When Stein-Type Test Detects Equilibrium Distributions of Finite N-Body Systems

Abstract

Starting from the probability distribution of finite N-body systems, which maximises the Havrda--Charvát entropy, we build a Stein-type goodness-of-fit test. The Maxwell--Boltzmann distribution is exact only in the thermodynamic limit, where the system is composed of infinitely many particles as N approaches infinity. For an isolated system with a finite number of particles, the equilibrium velocity distribution is compact and markedly non-Gaussian, being restricted by the fixed total energy. Using Stein's method, we first obtain a differential operator that characterises the target density. Its eigenfunctions are symmetric Jacobi polynomials, whose orthogonality yields a simple, parameter-free statistic. Under the null hypothesis that the data follows the finite-N distribution, the statistic converges to a chi-squared law, so critical values are available in closed form. Large-scale Monte Carlo experiments confirm exact size control and give a clear picture of the power. These findings quantify how quickly a finite system approaches the classical limit and provide a practical tool for testing kinetic models in regimes where normality cannot be assumed.
Paper Structure (47 sections, 58 equations, 2 figures, 5 tables)

This paper contains 47 sections, 58 equations, 2 figures, 5 tables.

Table of Contents

  1. Introduction
  2. Finite-$N$ Distribution and Exact Geometry
  3. Location--scale extension.
  4. Theoretical Justification: Consistency and Entropy Equivalence
  5. Specialising to one dimension
  6. Derivation of the Likelihood Ratio for a Typical Configuration
  7. Exact Likelihoods
  8. Null hypothesis $H_0$ (finite $N$).
  9. Alternative hypothesis $H_1$ (Gaussian).
  10. Likelihood ratio (in favour of the Gaussian).
  11. Typical-State Approximation
  12. Closed-Form Result
  13. Interpretation as a Probability
  14. Consistency with Sanov's Theorem
  15. Stein Characterisation for Finite-N Laws
  16. ...and 32 more sections

Figures (2)

  • Figure 1: Empirical detection power (colour gradient) plotted together with the Sanov rate prediction (dashed curve). Empirical results are obtained using the Stein-type test with truncation level $m=4$ and the theoretical $\chi^2$ critical values at $\alpha=0.05$; on our simulation grid, the corresponding calibrated cutoff yields nearly indistinguishable power. The blue dashed curve indicates the theoretical $80\%$-power boundary derived from the Sanov exponent $\exp\{-n D_{\mathrm{KL}}(p_N\|p_\infty)\}$. The empirical contours follow the same overall trend as the bound; the remaining gap for larger systems reflects the inherent trade-off between extracting higher-order information and maintaining robust Type I error control.
  • Figure 2: Power comparison between the proposed Stein-type test ($m=4$) and standard omnibus goodness-of-fit tests (Anderson--Darling, Kolmogorov--Smirnov, Cramér--von Mises) stephens1974edf for a system with $N=20$. To ensure a fair comparison at the nominal level $\alpha=0.05$, all curves report calibrated power, i.e., rejection rates computed using Monte Carlo calibrated critical values under $H_0$. The horizontal axis shows the sample size $n$ on a logarithmic scale. Over the displayed range ($n\le 5000$), the Stein-type test exhibits consistently higher power.