[en] Density fluctuations in small phase space bins for centrally produced particles are conveniently parameterized by a Gaussian free energy functional of the density fluctuation field. This observation leads to the investigation of the O(N) linear σ-model; the O(4) version serves as a prototype theory of the QCD chiral order parameter. We derive an exact, non-perturbative and finite equation for its finite temperature effective action. In the large-N limit, well-known results on the nature of the phase transition are recovered. Thus, we obtain the Ginzburg-Landau theory for a QCD system in thermodynamic equilibrium. (orig.)