Information geometry for Fermi–Dirac and Bose–Einstein quantum statistics

Abstract

Information geometry is an emergent branch of probability theory that consists of assigning a Riemannian differential geometry structure to the space of probability distributions. We present an information geometric investigation of gases following the Fermi–Dirac and the Bose–Einstein quantum statistics. For each quantum gas, we study the information geometry of the curved statistical manifolds associated with the grand canonical ensemble. The Fisher–Rao information metric and the scalar curvature are computed for both fermionic and bosonic models of non-interacting particles. In particular, by taking into account the ground state of the ideal bosonic gas in our information geometric analysis, we find that the singular behavior of the scalar curvature in the condensation region disappears. This is a counterexample to a long held conjecture that curvature always diverges in phase transitions.