How Interactions Affect Many-Body States Distinguishability: A Geometric Viewpoint

We investigate the information-geometric structure of a self-consistent second-virial approximation with the van der Waals form of the second virial coefficient, using the Fisher information as a generating potential. By constructing the associated Hessian metric and scalar curvature, we obtain a geometric description of thermodynamic fluctuations that complements the standard equation-of-state approach. We show that interaction effects enter the Fisher information selectively through their temperature dependence, with attractive interactions playing the dominant role while excluded-volume contributions remain subleading. The Hessian determinant of the Fisher information changes sign along well-defined crossover loci in the reduced parameter space, separating regions of saddle-like and locally convex curvature of the fluctuation landscape, and diverges as D˜−12 at the closure boundary. The scalar curvature diverges at the closure boundary as a simple pole, with a leading term that is universal and independent of the interaction parameters. We derive its exact closed-form expression, which shows that the curvature is negative for weak-to-moderate effective coupling but turns positive in the strong-coupling regime, where the underlying Fisher–Rao metric loses positive-definiteness. The curvature diverges on approach to this metric boundary, which lies strictly inside the mechanically stable region, short of the closure boundary itself. These results demonstrate that Fisher geometry provides a selective probe of interaction-induced fluctuations, capturing how thermal states reorganize under temperature variations rather than encoding phase transitions directly. This approach offers a complementary perspective on interacting systems and suggests new avenues for extending information-geometric methods to more strongly correlated and quantum regimes.