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

We investigate the information-geometric structure of the classical van der Waals gas 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 is found to be strictly negative throughout the physically accessible parameter space, indicating a smooth, saddle-like deformation of the fluctuation landscape and ruling out the existence of Fisher-geometric crossover behavior. The scalar curvature is negative throughout the stable domain and diverges at the spinodal boundary, revealing a hyperbolic structure of the statistical manifold. In the dilute regime, we derive an explicit analytical expression for the curvature, showing how interactions modify the geometry in a controlled manner. The only singular behavior arises at the spinodal line, where a divergence of the curvature signals the breakdown of mechanical stability. 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.