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

Starting from the probability distribution of finite N-body systems, which maximises the Havrda--Charv\'at 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.