Central Limit Theorem for continuous $N$-particle systems in fixed- and high-temperature: single level

We characterize central limit theorems for polynomial linear statistics of continuous $N$-particle systems on $\mathbb R$, in both the fixed-temperature regime $θ=β/2>0$ and the high-temperature regime $Nθ\toγ>0$. In each regime, we give necessary and sufficient conditions in terms of the coefficients of the logarithmic Bessel generating function, together with an explicit combinatorial formula for the limiting covariance. At fixed temperature, this formula recovers the classical second-order free moment–cumulant relation, up to the factor $1/θ$; at high temperature, it yields new second-order free $γ$-cumulants and their covariance relations. The proof combines moment extraction through Dunkl operators with a topological expansion of the logarithmic multivariate Bessel function indexed by ribbon graphs on surfaces. This expansion gives a topological interpretation of the free-cumulants: only genus zero contributes at fixed temperature, whereas arbitrarily high genera contribute at high temperature. As applications, we prove that $β$-addition preserves the CLT in both regimes and determine its limiting covariance from the input free cumulants, and establish a suitably rescaled transition from the high-temperature covariance to its fixed-temperature counterpart as $γ\to\infty$.