When Stein-Type Test Detects Equilibrium Distributions of Finite N-Body Systems
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arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2026
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| _version_ | 1866914327519297536 |
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| author | Shim, Jae Wan |
| author_facet | Shim, Jae Wan |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2602_12297 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | When Stein-Type Test Detects Equilibrium Distributions of Finite N-Body Systems Shim, Jae Wan Mathematical Physics Statistical Mechanics Statistics Theory Data Analysis, Statistics and Probability 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. |
| title | When Stein-Type Test Detects Equilibrium Distributions of Finite N-Body Systems |
| topic | Mathematical Physics Statistical Mechanics Statistics Theory Data Analysis, Statistics and Probability |
| url | https://arxiv.org/abs/2602.12297 |