Phase transitions and finite-size effects in integrable virial statistical models

We analyze thermodynamic models for fluid systems in equilibrium based on a virial expansion of the internal energy in terms of the volume density. We prove that the models, formulated for finite-size systems with N particles, are exactly solvable to any expansion order, as expectation values of physical observables (e.g., volume density) are determined from solutions to nonlinear C-integrable partial differential equations (PDEs) of hydrodynamic type. In the limit where the particle number goes to infinity, phase transitions emerge as classical shock waves in the space of thermodynamic variables. Near critical points, we argue that the volume density exhibits a scaling behavior consistent with the Universality Conjecture in viscous transport PDEs. As an application, we employ our framework to nuclear and quark matter, constructing a global quantum chromodynamics (QCD) phase diagram that reveals critical points for the nuclear liquid-gas transition and the hadron gas–quark-gluon plasma transition. We demonstrate how finite-size effects smear critical signatures, implying their potential impact on the search for the QCD critical point.